From bd78325e74648c49f837fe697e70eb63dc1ecfcc Mon Sep 17 00:00:00 2001 From: Wenqi Li Date: Sun, 12 Jul 2026 13:08:00 +0100 Subject: [PATCH 1/4] Fix optimizer convergence and regenerate figures --- .gitignore | 15 +- README.md | 27 +++- cg.png | Bin 134399 -> 46596 bytes drawFunc.py | 27 ++-- func.py | 69 ++++----- main.py | 169 +++++++++------------ optMethods.py | 307 +++++++++++++++++---------------------- pyproject.toml | 3 + quadratic.png | Bin 113462 -> 51332 bytes quasi_newton.png | Bin 145791 -> 52822 bytes requirements-dev.txt | 2 + requirements.txt | 2 + tests/test_optimizers.py | 59 ++++++++ 13 files changed, 336 insertions(+), 344 deletions(-) create mode 100644 pyproject.toml create mode 100644 requirements-dev.txt create mode 100644 requirements.txt create mode 100644 tests/test_optimizers.py diff --git a/.gitignore b/.gitignore index d2d6f36..dc67086 100644 --- a/.gitignore +++ b/.gitignore @@ -1,9 +1,5 @@ *.py[cod] - -# C extensions *.so - -# Packages *.egg *.egg-info dist @@ -17,19 +13,14 @@ develop-eggs .installed.cfg lib lib64 - -# Installer logs pip-log.txt - -# Unit test / coverage reports .coverage .tox nosetests.xml - -# Translations *.mo - -# Mr Developer .mr.developer.cfg .project .pydevproject +.venv/ +__pycache__/ +.pytest_cache/ diff --git a/README.md b/README.md index 8f9ffd2..4908aff 100644 --- a/README.md +++ b/README.md @@ -1,8 +1,19 @@ -gradient -======== - -optimisation practices -- Newton, modified Newton, Quasi Newton (rank one with backtracking) - - - +# gradient + +Small educational implementations of Newton, SR1 quasi-Newton, BFGS, and +Fletcher–Reeves nonlinear conjugate-gradient optimisation. + +## Run + +```sh +python3 -m venv .venv +.venv/bin/pip install -r requirements-dev.txt +.venv/bin/pytest +MPLBACKEND=Agg .venv/bin/python main.py +``` + +Running `main.py` regenerates all three figures below. + +![Quadratic objective](quadratic.png) +![Quasi-Newton methods](quasi_newton.png) +![Conjugate-gradient method](cg.png) diff 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np.meshgrid(dx, dy) -# eval function f_x - Z = [f_x(np.matrix([[x1], [x2]])) \ - for (x1, x2) in zip(np.hstack(X), np.hstack(Y))] - Z = np.array(Z) - Z = Z.reshape(np.shape(X)) -# take this pencil - levels = [0.0, 1, 2, 5, 20, 30, 50, 80, 100, 200, 400] - #cs = plt.contour(X, Y, Z, levels, colors='k') - cs = plt.contourf(X, Y, Z, 20, cmap=plt.cm.YlGnBu) - #plt.clabel(cs, fmt='%.1f', inline=1) +def draw(f_x, ax=None, x_bounds=(-3.0, 2.5), y_bounds=(-4.5, 4.5), resolution=220): + ax = ax or plt.gca() + x_values = np.linspace(*x_bounds, resolution) + y_values = np.linspace(*y_bounds, resolution) + X, Y = np.meshgrid(x_values, y_values) + points = np.column_stack((X.ravel(), Y.ravel())) + Z = np.fromiter((f_x(point) for point in points), dtype=float).reshape(X.shape) + return ax.contourf(X, Y, Z, 20, cmap="YlGnBu") diff --git a/func.py b/func.py index 6a32d9e..2793efc 100644 --- a/func.py +++ b/func.py @@ -1,59 +1,60 @@ +"""Objective functions used by the optimisation examples.""" + import numpy as np -class quadratic: -# trivial (solving linear system) - A = np.zeros([2, 2]) - b = np.zeros([2, 1]) +def _vector(x): + """Return *x* as a one-dimensional floating-point array.""" + return np.asarray(x, dtype=float).reshape(-1) - def __init__(self): - pass +class quadratic: def __init__(self, A, b): - self.A = A - self.b = b + self.A = np.asarray(A, dtype=float) + self.b = _vector(b) + if self.A.shape != (self.b.size, self.b.size): + raise ValueError("A must be square and match the size of b") def f_x(self, x): - f = 0.5 * np.dot(np.dot(x.T, self.A), x) - np.dot(self.b.T, x) - return f[0, 0] + x = _vector(x) + return float(0.5 * x @ self.A @ x - self.b @ x) def g_x(self, x): - return np.dot(self.A, x) - self.b + return self.A @ _vector(x) - self.b def G_x(self, x): - return np.asmatrix(self.A) + return self.A.copy() + + def about_alpha(self, x, s): + return lambda alpha: self.f_x(_vector(x) + alpha * _vector(s)) + + def about_alpha_prime(self, x, s): + return lambda alpha: float(self.g_x(_vector(x) + alpha * _vector(s)) @ _vector(s)) class rosenbrock: - -# f(x_0, x_1) = 10 * (x_1 - x_0^2)^2 + (x_0 - 1)^2 - def __init__(self): - pass + """The two-dimensional Rosenbrock function with coefficient 10.""" def f_x(self, x): - f = 10.0 * (x[1] - (x[0])**2)**2 + (1-x[0])**2 - return f[0, 0] + x0, x1 = _vector(x) + return float(10.0 * (x1 - x0**2) ** 2 + (1.0 - x0) ** 2) def g_x(self, x): - g0 = -40.0 * (x[0]*x[1] - x[0]**3) - 2.0 + 2.0*x[0] - g1 = 20.0 * (x[1] - x[0]**2) - return np.matrix([[g0[0,0]], [g1[0,0]]]) + x0, x1 = _vector(x) + return np.array([ + -40.0 * x0 * (x1 - x0**2) + 2.0 * (x0 - 1.0), + 20.0 * (x1 - x0**2), + ]) def G_x(self, x): - G00 = 120.0 * x[0]**2 - 40.0 * x[1] + 2.0 - G01 = -40.0*x[0] - G10 = -40.0*x[0] - G11 = 20.0 - return np.matrix([[G00[0,0], G01[0,0]], [G10[0,0], G11]]) + x0, x1 = _vector(x) + return np.array([ + [120.0 * x0**2 - 40.0 * x1 + 2.0, -40.0 * x0], + [-40.0 * x0, 20.0], + ]) def about_alpha(self, x, s): - # f(a) = f(x + a*s) at point x, direction s - def along_s(alpha): - return self.f_x(x+s*alpha) - return along_s + return lambda alpha: self.f_x(_vector(x) + alpha * _vector(s)) def about_alpha_prime(self, x, s): - def along_s(alpha): - p = np.dot(self.g_x(x+alpha*s).T, s) - return p[0, 0] - return along_s + return lambda alpha: float(self.g_x(_vector(x) + alpha * _vector(s)) @ _vector(s)) diff --git a/main.py b/main.py index 1685be6..50a394e 100644 --- a/main.py +++ b/main.py @@ -1,100 +1,73 @@ -import optMethods as opt +"""Generate the example figures used by the README.""" + +from pathlib import Path + +import matplotlib.pyplot as plt +import numpy as np + import drawFunc import func -import numpy as np -import matplotlib.pyplot as plt -import pdb - -def quadratic(): - - A = np.matrix([ - [3., 2.], - [2., 6.]]) - b = np.matrix([[1.], [-5.]]) - obj = func.quadratic(A,b) - - plt.figure() - drawFunc.draw(obj.f_x) - - x = np.matrix([[0], [2]]) - track = opt.newton(x, obj, 0, 1) # just one iteration - p1, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0]) - - track = opt.quasi_newton(x, obj, 100) - p2, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - plt.legend([p1, p2], ['Newton', 'Quasi_Newton_Rank_One']) - - plt.show() - -def rosen(): - obj = func.rosenbrock() - - plt.figure() - - drawFunc.draw(obj.f_x) - start_point = np.matrix([[-1.5], [-4.0]]) - plt.annotate('Start', xy=(-1.5, -4.0), xytext=(-1.8, -3.5), - arrowprops=dict(facecolor='black', shrink=0.02, frac=0.5)) - plt.annotate('Optimal', xy=(1, 1), xytext=(1.1, 1.4), - arrowprops=dict(facecolor='black', shrink=0.02, frac=0.5)) - - #v = 0.0 - #str1 = "Newton" - #track = opt.newton(start_point, obj, v) - #p1, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - # 'o-', linewidth=2.0) - - v = 0.1 - str2 = "Modified_Newton %.2f"%v - track = opt.newton(start_point, obj, v) - p2, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - v = 1.5 - str3 = "Modified_Newton %.2f"%v - track = opt.newton(start_point, obj, v, 50) - p3, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - str4 = "Quasi_Newton_trivial" - track = opt.quasi_newton(start_point, obj, 150, 1) - p4, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - str5 = "Quasi_Newton_Armijo" - track = opt.quasi_newton(start_point, obj, 450, 0) - p5, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - str6 = "BFGS_trivial" - track = opt.BFGS(start_point, obj, 350, 1) - p6, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - str7 = "BFGS_Armijo" - track = opt.BFGS(start_point, obj, 350, 0) - p7, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - v = 0.001 - str8 = "CG_FR_%.5f"%v - track = opt.fletcher_reeves(start_point, obj, iteration=30, alpha=v) - p8, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - - v = 0.01 - str9 = "CG_FR_%.5f"%v - track = opt.fletcher_reeves(start_point, obj, iteration=30, alpha=v) - p9, = plt.plot(track[0,:].tolist()[0], track[1,:].tolist()[0], - 'o-', linewidth=2.0) - #plt.legend([p1, p2, p3, p4, p5, p6, p7],\ - # [str1, str2, str3, str4, str5, str6, str7]) - - plt.legend([p2, p3, p4, p5, p6, p7, p8, p9],\ - [str2, str3, str4, str5, str6, str7, str8, str9]) - plt.show() - -#quadratic() -rosen() +import optMethods as opt + + +ROOT = Path(__file__).resolve().parent + + +def _plot_track(ax, track, label, style="o-", **kwargs): + ax.plot(track[0], track[1], style, label=label, markersize=3, **kwargs) + + +def quadratic(output=ROOT / "quadratic.png"): + objective = func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) + start = np.array([0.0, 2.0]) + fig, ax = plt.subplots(figsize=(7, 5)) + drawFunc.draw(objective.f_x, ax) + _plot_track(ax, opt.newton(start, objective, iterations=1), "Newton") + _plot_track(ax, opt.quasi_newton(start, objective, 12), "SR1 quasi-Newton") + ax.set(title="Quadratic objective", xlabel="$x_0$", ylabel="$x_1$") + ax.legend() + fig.tight_layout() + fig.savefig(output, dpi=160) + plt.close(fig) + + +def quasi_newton(output=ROOT / "quasi_newton.png"): + objective = func.rosenbrock() + start = np.array([-1.5, -4.0]) + fig, ax = plt.subplots(figsize=(7, 5)) + drawFunc.draw(objective.f_x, ax) + _plot_track(ax, opt.quasi_newton(start, objective, 250), "SR1") + _plot_track(ax, opt.BFGS(start, objective, 250), "BFGS") + ax.plot(1, 1, "r*", markersize=10, label="Optimum") + ax.set(title="Quasi-Newton methods on Rosenbrock", xlabel="$x_0$", ylabel="$x_1$") + ax.legend() + fig.tight_layout() + fig.savefig(output, dpi=160) + plt.close(fig) + + +def conjugate_gradient(output=ROOT / "cg.png"): + objective = func.rosenbrock() + # The conventional Rosenbrock starting point keeps the full trajectory in + # the plotted domain while still showing the narrow curved valley. + start = np.array([-1.2, 1.0]) + fig, ax = plt.subplots(figsize=(7, 5)) + drawFunc.draw(objective.f_x, ax) + track = opt.fletcher_reeves(start, objective, iteration=500) + _plot_track(ax, track, "Fletcher-Reeves", linewidth=1) + ax.plot(1, 1, "r*", markersize=10, label="Optimum") + ax.set(title="Nonlinear conjugate gradient on Rosenbrock", xlabel="$x_0$", ylabel="$x_1$") + ax.legend() + fig.tight_layout() + fig.savefig(output, dpi=160) + plt.close(fig) + + +def generate_figures(): + quadratic() + quasi_newton() + conjugate_gradient() + + +if __name__ == "__main__": + generate_figures() diff --git a/optMethods.py b/optMethods.py index e5c8380..cd07e7e 100644 --- a/optMethods.py +++ b/optMethods.py @@ -1,179 +1,136 @@ -from numpy import * - -# TODO: convergence check - -def newton(start_point, obj_fun, modified=0, iterations=5): -# with second order information - x = start_point - track = x - - k = 0 - while k < iterations: - - if modified > 0: - vI = modified * matrix([[1, 0], [0, 1]]) - G_ = linalg.inv(obj_fun.G_x(x) + vI) # inverse? - else: - G_ = linalg.inv(obj_fun.G_x(x)) # inverse? - g_ = obj_fun.g_x(x) - delta = -1.0 * dot(G_, g_) - x = x + delta - - track = concatenate((track, x), axis=1) - k += 1 - - return track - -def BFGS(start_point, obj_fun, iteration=10, interpolation=0): -# with first order information - x = start_point - track = x - - k = 0 - #H = dot(obj_fun.g_x(x), obj_fun.g_x(x).T) - H = 1.0*eye(2) - gamma = 1.0 - - while k < iteration and linalg.norm(gamma) > 1e-10: - - p = -dot(H, obj_fun.g_x(x)) - - if interpolation > 0: - alpha_k = _backtracking_line_search(obj_fun, x, p) - else: - alpha_k = _armijo_line_search(obj_fun, x, p) - - s = alpha_k * p - g_k = obj_fun.g_x(x) - x = x + alpha_k * p - g_k_1 = obj_fun.g_x(x) - - y = g_k_1 - g_k - - z = dot(H, y) - sTy = dot(s.T, y) - if sTy > 0: - H += outer(s, s) * (sTy + dot(y.T, z))[0,0]/(sTy**2) \ - - (outer(z, s) + outer(s, z))/sTy - - track = concatenate((track, x), axis=1) - k += 1 - - return track - -def quasi_newton(start_point, obj_fun, iteration=10, interpolation=0): -# with first order information - x = start_point - track = x - - k = 0 - #H = matrix([[.01, 0], [0, .01]]) - H = 1.0*eye(2) - - while k < iteration: - - s = -1.0 * dot(H, obj_fun.g_x(x)) - - if interpolation > 0: - alpha_k = _backtracking_line_search(obj_fun, x, s) - else: - alpha_k = _armijo_line_search(obj_fun, x, s) - - delta = alpha_k * s - x_k_1 = x + delta - - gamma = obj_fun.g_x(x_k_1) - obj_fun.g_x(x) - u = delta - dot(H, gamma) - - scale_a = dot(u.T, gamma) - if scale_a == 0: # :( - scale_a = 0.000001 - H = H + outer(u, u) / scale_a - - track = concatenate((track, x), axis=1) - x = x_k_1 - k += 1 - - return track - -def fletcher_reeves(start_point, obj_fun, iteration=10, alpha=0.1): - x = start_point - track = x - - k = 0 - while k < iteration: - - if k < 1: - g = obj_fun.g_x(x) - beta = 0.0 - s = -g - else: - g = obj_fun.g_x(x) - - beta = dot(g.T, g)/dot(g_old.T, g_old) - beta = beta[0,0] - - s = -g + beta * s - alpha_k = _armijo_line_search(obj_fun, x, s, alpha0=alpha) - g_old = obj_fun.g_x(x) - x = x + alpha_k * s - - track = concatenate((track, x), axis=1) - k += 1 - - return track - -def _backtracking_line_search(obj_fun, x, s, c=1e-4): -# dummy - alpha = 1.0 - p = 0.5 # magic number - c = 0.0001 - diff = 1 - - f_alpha = obj_fun.about_alpha(x, s) - g_alpha = obj_fun.about_alpha_prime(x, s) - i = 0 - while (diff > 0 and i < 50): - f_x = obj_fun.f_x(x) - diff = f_alpha(alpha) - f_alpha(0) - c * alpha * g_alpha(0) - alpha *= p - i += 1 +"""Small, dependency-free implementations of common optimisation methods.""" + +import numpy as np + + +def _initialise(start_point): + x = np.asarray(start_point, dtype=float).reshape(-1).copy() + return x, [x.copy()] + + +def _finish(track): + return np.column_stack(track) + + +def newton(start_point, obj_fun, modified=0, iterations=5, tolerance=1e-8): + x, track = _initialise(start_point) + identity = np.eye(x.size) + for _ in range(iterations): + gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + if np.linalg.norm(gradient) <= tolerance: + break + hessian = np.asarray(obj_fun.G_x(x), dtype=float) + modified * identity + # Solving is both faster and more numerically stable than forming inv(H). + step = np.linalg.solve(hessian, -gradient) + x = x + step + track.append(x.copy()) + return _finish(track) + + +def BFGS(start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-8): + x, track = _initialise(start_point) + inverse_hessian = np.eye(x.size) + gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + + for _ in range(iteration): + if np.linalg.norm(gradient) <= tolerance: + break + direction = -inverse_hessian @ gradient + if direction @ gradient >= 0: + inverse_hessian = np.eye(x.size) + direction = -gradient + alpha = _select_line_search(obj_fun, x, direction, interpolation) + step = alpha * direction + next_x = x + step + next_gradient = np.asarray(obj_fun.g_x(next_x), dtype=float).reshape(-1) + y = next_gradient - gradient + curvature = float(step @ y) + if curvature > 1e-12 * np.linalg.norm(step) * np.linalg.norm(y): + rho = 1.0 / curvature + identity = np.eye(x.size) + left = identity - rho * np.outer(step, y) + inverse_hessian = ( + left @ inverse_hessian @ left.T + rho * np.outer(step, step) + ) + x, gradient = next_x, next_gradient + track.append(x.copy()) + return _finish(track) + + +def quasi_newton(start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-8): + """Symmetric-rank-one (SR1) quasi-Newton method.""" + x, track = _initialise(start_point) + inverse_hessian = np.eye(x.size) + gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + + for _ in range(iteration): + if np.linalg.norm(gradient) <= tolerance: + break + direction = -inverse_hessian @ gradient + # SR1 updates need not stay positive definite. Restart when that makes + # the proposed direction point uphill. + if direction @ gradient >= 0: + inverse_hessian = np.eye(x.size) + direction = -gradient + alpha = _select_line_search(obj_fun, x, direction, interpolation) + step = alpha * direction + next_x = x + step + next_gradient = np.asarray(obj_fun.g_x(next_x), dtype=float).reshape(-1) + y = next_gradient - gradient + residual = step - inverse_hessian @ y + denominator = float(residual @ y) + # A tiny denominator makes SR1 unstable; the correct action is to skip it. + if abs(denominator) > 1e-8 * np.linalg.norm(residual) * np.linalg.norm(y): + inverse_hessian += np.outer(residual, residual) / denominator + x, gradient = next_x, next_gradient + track.append(x.copy()) + return _finish(track) + + +def fletcher_reeves(start_point, obj_fun, iteration=10, alpha=1.0, tolerance=1e-8): + x, track = _initialise(start_point) + gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + direction = -gradient + + for k in range(iteration): + if np.linalg.norm(gradient) <= tolerance: + break + # Restart if round-off has destroyed the descent direction. + if direction @ gradient >= 0: + direction = -gradient + step_size = _armijo_line_search(obj_fun, x, direction, alpha0=alpha) + next_x = x + step_size * direction + next_gradient = np.asarray(obj_fun.g_x(next_x), dtype=float).reshape(-1) + denominator = float(gradient @ gradient) + beta = float(next_gradient @ next_gradient) / denominator if denominator else 0.0 + # Periodic restarts are required for robust nonlinear CG in finite precision. + direction = -next_gradient if (k + 1) % x.size == 0 else -next_gradient + beta * direction + x, gradient = next_x, next_gradient + track.append(x.copy()) + return _finish(track) + + +def _select_line_search(obj_fun, x, direction, interpolation): + if interpolation > 0: + return _backtracking_line_search(obj_fun, x, direction) + return _armijo_line_search(obj_fun, x, direction) + + +def _backtracking_line_search(obj_fun, x, s, c=1e-4, alpha0=1.0, contraction=0.5): + """Return the first step satisfying the Armijo sufficient-decrease rule.""" + alpha = float(alpha0) + f0 = obj_fun.f_x(x) + slope = float(np.asarray(obj_fun.g_x(x)).reshape(-1) @ np.asarray(s).reshape(-1)) + if slope >= 0: + raise ValueError("line search direction must be a descent direction") + for _ in range(60): + if obj_fun.f_x(x + alpha * s) <= f0 + c * alpha * slope: + return alpha + alpha *= contraction return alpha def _armijo_line_search(obj_fun, x, s, c=1e-4, alpha0=1.0): -# adapted from scipy.optimisation - amin = 0.0 - f_alpha = obj_fun.about_alpha(x, s) - g_alpha = obj_fun.about_alpha_prime(x, s) - - if(f_alpha(alpha0) <= f_alpha(0) + c * alpha0 * g_alpha(0)): - return alpha0 - - alpha1 = -(g_alpha(0)) * alpha0**2 / \ - 2.0 / (f_alpha(alpha0) - f_alpha(0) - g_alpha(0) * alpha0) - if(f_alpha(alpha1) <= f_alpha(0) + c * alpha1 * g_alpha(0)): - return alpha1 - - while alpha1 > amin: - - factor = alpha0**2 * alpha1**2 * (alpha1 - alpha0) - a = alpha0**2 * (f_alpha(alpha1) - f_alpha(0) - g_alpha(0) * alpha1) - \ - alpha1**2 * (f_alpha(alpha0) - f_alpha(0) - g_alpha(0) * alpha0) - a = a / factor - - b = -alpha0**3 * (f_alpha(alpha1) - f_alpha(0) - g_alpha(0) * alpha1)+ \ - alpha1**3 * (f_alpha(alpha0) - f_alpha(0) - g_alpha(0) * alpha0) - b = b / factor - - alpha2 = (-b + sqrt(abs(b**2 - 3 * a * g_alpha(0)))) / (3.0 * a) - if(f_alpha(alpha2) <= f_alpha(0) + c * alpha2 * g_alpha(0)): - return alpha2 - - if(alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96: - alpha2 = alpha1 / 2.0 - - alpha0 = alpha1 - alpha1 = alpha2 - - return 0.0001 + # The previous cubic interpolation could divide by zero and take sqrt(abs(...)), + # which masks invalid interpolants. 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a/requirements.txt b/requirements.txt new file mode 100644 index 0000000..057269d --- /dev/null +++ b/requirements.txt @@ -0,0 +1,2 @@ +matplotlib>=3.7 +numpy>=1.24 diff --git a/tests/test_optimizers.py b/tests/test_optimizers.py new file mode 100644 index 0000000..5a14b16 --- /dev/null +++ b/tests/test_optimizers.py @@ -0,0 +1,59 @@ +import numpy as np +import pytest + +import func +import optMethods as opt + + +@pytest.fixture +def quadratic_objective(): + return func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) + + +def test_quadratic_derivatives_and_optimum(quadratic_objective): + optimum = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) + assert np.linalg.norm(quadratic_objective.g_x(optimum)) < 1e-12 + assert quadratic_objective.f_x(optimum) < quadratic_objective.f_x([0, 2]) + + +def test_newton_reaches_quadratic_optimum_in_one_step(quadratic_objective): + track = opt.newton([0, 2], quadratic_objective, iterations=1) + expected = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) + np.testing.assert_allclose(track[:, -1], expected, atol=1e-12) + + +@pytest.mark.parametrize("method", [opt.BFGS, opt.quasi_newton]) +def test_quasi_newton_methods_converge_on_quadratic(method, quadratic_objective): + track = method([0, 2], quadratic_objective, iteration=30) + expected = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) + np.testing.assert_allclose(track[:, -1], expected, atol=1e-6) + assert not np.array_equal(track[:, 0], track[:, 1]) + + +def test_backtracking_returns_an_accepted_step(quadratic_objective): + x = np.array([0.0, 2.0]) + direction = -quadratic_objective.g_x(x) + alpha = opt._backtracking_line_search(quadratic_objective, x, direction) + slope = quadratic_objective.g_x(x) @ direction + assert quadratic_objective.f_x(x + alpha * direction) <= ( + quadratic_objective.f_x(x) + 1e-4 * alpha * slope + ) + + +@pytest.mark.parametrize("method, iterations", [ + (opt.BFGS, 300), + (opt.quasi_newton, 300), + (opt.fletcher_reeves, 600), +]) +def test_first_order_methods_reduce_rosenbrock(method, iterations): + objective = func.rosenbrock() + start = np.array([-1.5, -4.0]) + track = method(start, objective, iteration=iterations) + assert objective.f_x(track[:, -1]) < 1e-5 + assert np.linalg.norm(objective.g_x(track[:, -1])) < 1e-2 + + +def test_line_search_rejects_ascent_direction(quadratic_objective): + x = np.array([0.0, 2.0]) + with pytest.raises(ValueError, match="descent"): + opt._backtracking_line_search(quadratic_objective, x, quadratic_objective.g_x(x)) From 49c9a6a3b9a2431527508037dae6d4f62f6397a4 Mon Sep 17 00:00:00 2001 From: Wenqi Li Date: Sun, 12 Jul 2026 13:10:26 +0100 Subject: [PATCH 2/4] Improve optimization figure design --- cg.png | Bin 46596 -> 139353 bytes drawFunc.py | 25 +++++++++++++++-- main.py | 71 ++++++++++++++++++++++++++++++++++------------- quadratic.png | Bin 51332 -> 101329 bytes quasi_newton.png | Bin 52822 -> 145036 bytes 5 files 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y_values) points = np.column_stack((X.ravel(), Y.ravel())) Z = np.fromiter((f_x(point) for point in points), dtype=float).reshape(X.shape) - return ax.contourf(X, Y, Z, 20, cmap="YlGnBu") + if log_scale: + positive = Z[Z > 0] + lower = max(float(positive.min()), 1e-3) + upper = float(Z.max()) + levels = np.geomspace(lower, upper, 24) + filled = ax.contourf( + X, Y, Z, levels=levels, norm=LogNorm(lower, upper), cmap="viridis_r" + ) + ax.contour(X, Y, Z, levels=levels[::3], colors="white", linewidths=0.45, alpha=0.5) + else: + levels = np.linspace(float(Z.min()), float(Z.max()), 24) + filled = ax.contourf(X, Y, Z, levels=levels, cmap="viridis_r") + ax.contour(X, Y, Z, levels=levels[::3], colors="white", linewidths=0.45, alpha=0.5) + return filled diff --git a/main.py b/main.py index 50a394e..d7b3df4 100644 --- a/main.py +++ b/main.py @@ -13,19 +13,40 @@ ROOT = Path(__file__).resolve().parent -def _plot_track(ax, track, label, style="o-", **kwargs): - ax.plot(track[0], track[1], style, label=label, markersize=3, **kwargs) +def _plot_track(ax, track, label, color, marker, **kwargs): + mark_every = max(1, track.shape[1] // 16) + ax.plot( + track[0], + track[1], + color=color, + marker=marker, + markevery=mark_every, + label=label, + linewidth=2.2, + markersize=4.5, + markeredgecolor="white", + markeredgewidth=0.6, + **kwargs, + ) + + +def _finish_axes(ax, title): + ax.set(title=title, xlabel="$x_0$", ylabel="$x_1$") + ax.grid(False) + ax.legend(frameon=True, framealpha=0.92, loc="upper left") def quadratic(output=ROOT / "quadratic.png"): objective = func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) start = np.array([0.0, 2.0]) fig, ax = plt.subplots(figsize=(7, 5)) - drawFunc.draw(objective.f_x, ax) - _plot_track(ax, opt.newton(start, objective, iterations=1), "Newton") - _plot_track(ax, opt.quasi_newton(start, objective, 12), "SR1 quasi-Newton") - ax.set(title="Quadratic objective", xlabel="$x_0$", ylabel="$x_1$") - ax.legend() + drawFunc.draw(objective.f_x, ax, x_bounds=(-1.5, 3.1), y_bounds=(-2.8, 2.5)) + _plot_track(ax, opt.newton(start, objective, iterations=1), "Newton", "#e63946", "o") + _plot_track(ax, opt.quasi_newton(start, objective, 12), "SR1", "#ffb703", "s") + optimum = np.linalg.solve(objective.A, objective.b) + ax.scatter(*start, color="white", edgecolor="#222222", s=46, zorder=5, label="Start") + ax.scatter(*optimum, color="#e63946", marker="*", s=150, edgecolor="white", zorder=6, label="Optimum") + _finish_axes(ax, "Quadratic objective") fig.tight_layout() fig.savefig(output, dpi=160) plt.close(fig) @@ -33,14 +54,20 @@ def quadratic(output=ROOT / "quadratic.png"): def quasi_newton(output=ROOT / "quasi_newton.png"): objective = func.rosenbrock() - start = np.array([-1.5, -4.0]) + start = np.array([-1.2, 1.0]) fig, ax = plt.subplots(figsize=(7, 5)) - drawFunc.draw(objective.f_x, ax) - _plot_track(ax, opt.quasi_newton(start, objective, 250), "SR1") - _plot_track(ax, opt.BFGS(start, objective, 250), "BFGS") - ax.plot(1, 1, "r*", markersize=10, label="Optimum") - ax.set(title="Quasi-Newton methods on Rosenbrock", xlabel="$x_0$", ylabel="$x_1$") - ax.legend() + drawFunc.draw( + objective.f_x, + ax, + x_bounds=(-1.6, 1.4), + y_bounds=(-0.6, 2.0), + log_scale=True, + ) + _plot_track(ax, opt.quasi_newton(start, objective, 250), "SR1", "#ffb703", "s") + _plot_track(ax, opt.BFGS(start, objective, 250), "BFGS", "#e63946", "o") + ax.scatter(*start, color="white", edgecolor="#222222", s=46, zorder=5, label="Start") + ax.scatter(1, 1, color="#e63946", marker="*", s=150, edgecolor="white", zorder=6, label="Optimum") + _finish_axes(ax, "Quasi-Newton methods on Rosenbrock") fig.tight_layout() fig.savefig(output, dpi=160) plt.close(fig) @@ -52,12 +79,18 @@ def conjugate_gradient(output=ROOT / "cg.png"): # the plotted domain while still showing the narrow curved valley. start = np.array([-1.2, 1.0]) fig, ax = plt.subplots(figsize=(7, 5)) - drawFunc.draw(objective.f_x, ax) + drawFunc.draw( + objective.f_x, + ax, + x_bounds=(-1.6, 1.4), + y_bounds=(-0.2, 2.0), + log_scale=True, + ) track = opt.fletcher_reeves(start, objective, iteration=500) - _plot_track(ax, track, "Fletcher-Reeves", linewidth=1) - ax.plot(1, 1, "r*", markersize=10, label="Optimum") - ax.set(title="Nonlinear conjugate gradient on Rosenbrock", xlabel="$x_0$", ylabel="$x_1$") - ax.legend() + _plot_track(ax, track, "Fletcher–Reeves", "#e63946", "o") + ax.scatter(*start, color="white", edgecolor="#222222", s=46, zorder=5, label="Start") + ax.scatter(1, 1, color="#e63946", marker="*", s=150, edgecolor="white", zorder=6, label="Optimum") + _finish_axes(ax, "Nonlinear conjugate gradient on Rosenbrock") fig.tight_layout() fig.savefig(output, dpi=160) plt.close(fig) diff --git a/quadratic.png b/quadratic.png index 9a6807a829ae144bd46e73dc621226e20a301222..df888d5fa97affa94ab095bde794945ebc3a009c 100644 GIT binary patch literal 101329 zcmeFY^(Qe}xQYr;%}0-(m_B-h`Wyot z_~h*&u@dkm3YXP`YdKlNJga-`bi#_powxvvqRh=Mv=-;COBWhdaB8 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It implements classical smooth unconstrained methods from +first principles, runs them from shared starting points, and plots every +accepted iterate on the same objective contours. -## Run +The code uses two two-dimensional teaching problems: + +- a symmetric positive-definite quadratic + `f(x) = 1/2 x^T A x - b^T x`, where `A = [[3, 2], [2, 6]]` and + `b = [1, -5]`; and +- the Rosenbrock function `f(x0, x1) = 10(x1 - x0^2)^2 + (1 - x0)^2`. + Notice the coefficient is **10**, rather than the frequently used 100. + +## What is implemented + +| Family | Method | Direction / update | Step selection and safeguard | +|---|---|---|---| +| First order | Steepest descent | `p = -g` | fixed, exact quadratic, or Armijo backtracking | +| Second order | Newton | solve `H p = -g` | full step | +| Second order | Damped Newton | Newton direction | Armijo backtracking | +| Second order | Modified Newton | adaptively shift `H` until positive definite | Armijo backtracking | +| Quasi-Newton | DFP | inverse-Hessian rank-two update | strong Wolfe; guarded curvature update | +| Quasi-Newton | BFGS | inverse-Hessian rank-two update | strong Wolfe; guarded curvature update | +| Quasi-Newton | SR1 | symmetric rank-one update | Armijo; skipped near-singular updates and descent restarts | +| Conjugate gradient | Linear CG | `A`-conjugate directions | exact steps on an SPD quadratic | +| Nonlinear CG | Fletcher–Reeves | gradient-norm beta | strong Wolfe plus periodic/descent restarts | +| Nonlinear CG | Polak–Ribiere+ | truncated PR beta | strong Wolfe plus periodic/descent restarts | + +The line searches are real, separately testable implementations: a fixed step, +an exact SPD-quadratic step, geometric Armijo backtracking, and strong Wolfe +bracketing with zoom. The old integer `interpolation` argument remains only on +compatibility aliases; new code should use the explicit `line_search` names. + +Every method can return an `OptimizationResult` containing its path, objective +history, gradient norms, accepted step sizes, convergence status, evaluation +counts, restarts, and skipped quasi-Newton updates. + +## 1. Curvature on a quadratic + +All panels below use the same start, contours, stopping tolerance, and exact +quadratic line search where appropriate. Steepest descent zig-zags across the +elliptical valley. Newton uses the true curvature and arrives in one step. +Linear CG and the quasi-Newton methods construct the missing curvature in two +steps in this two-dimensional example. + +![Optimization paths on a positive-definite quadratic](quadratic.png) + +## 2. Same start, different Rosenbrock paths + +Each method starts at `(-1.2, 1.0)`, targets `(1, 1)`, and stops when +`||gradient f|| <= 1e-8` or its stated budget is exhausted. Panel subtitles are +generated from the run results. Labels 0–3 identify the earliest accepted +iterates; later markers are logarithmically spaced so long runs stay readable. + +The full Newton path is intentionally shown even though it increases the +objective on some steps: that is the motivation for damping and modified +positive-definite Hessians. + +![Nine optimization paths on Rosenbrock](quasi_newton.png) + +## 3. Convergence histories + +Path geometry is only half the story. These plots compare objective reduction +and first-order convergence. The generated [metrics table](metrics.csv) adds +function, gradient, and Hessian evaluation totals, restarts, and skipped updates. + +![Rosenbrock convergence histories](cg.png) + +## Run the tutorial ```sh python3 -m venv .venv @@ -12,8 +77,34 @@ python3 -m venv .venv MPLBACKEND=Agg .venv/bin/python main.py ``` -Running `main.py` regenerates all three figures below. +`main.py` deterministically regenerates the three PNGs and `metrics.csv`. Use +`--output-dir PATH` to write a separate copy without changing tracked files. + +## API example + +```python +import func +import optMethods as opt + +objective = func.rosenbrock() +result = opt.minimize( + "bfgs", + [-1.2, 1.0], + objective, + max_iter=100, + gtol=1e-8, + line_search="strong_wolfe", +) + +print(result.x, result.iterations, result.nfev, result.gradient_norms[-1]) +``` + +`minimize` provides consistent `max_iter`/`gtol` names and always returns the +complete result. Direct method calls return the historical path array by default +for compatibility; pass `return_result=True` to those calls for diagnostics. + +## Julia directory -![Quadratic objective](quadratic.png) -![Quasi-Newton methods](quasi_newton.png) -![Conjugate-gradient method](cg.png) +`jl/` is the repository's original historical translation and targets a +pre-1.0 Julia/PyCall stack. It is not part of the tested tutorial. 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z|FUP=I-N5?c0~d(;rG%~hH5}ke8~^tk&#*JzrIZH`~S6G5qt8o(O#ba?b^lvSLXcx f(;tJLvi}T?d%qPO@`hNchVT9mU diff --git a/drawFunc.py b/drawFunc.py index 8f6be3c..b11b89a 100644 --- a/drawFunc.py +++ b/drawFunc.py @@ -12,6 +12,8 @@ def draw( y_bounds=(-4.5, 4.5), resolution=260, log_scale=False, + levels=None, + cmap="Greys", ): ax = ax or plt.gca() x_values = np.linspace(*x_bounds, resolution) @@ -23,13 +25,27 @@ def draw( positive = Z[Z > 0] lower = max(float(positive.min()), 1e-3) upper = float(Z.max()) - levels = np.geomspace(lower, upper, 24) + levels = np.asarray(levels) if levels is not None else np.geomspace(lower, upper, 24) filled = ax.contourf( - X, Y, Z, levels=levels, norm=LogNorm(lower, upper), cmap="viridis_r" + X, + Y, + Z, + levels=levels, + norm=LogNorm(float(levels[0]), float(levels[-1])), + cmap=cmap, + extend="both", + ) + ax.contour( + X, Y, Z, levels=levels[::3], colors="#8a8a8a", linewidths=0.45, alpha=0.7 ) - ax.contour(X, Y, Z, levels=levels[::3], colors="white", linewidths=0.45, alpha=0.5) else: - levels = np.linspace(float(Z.min()), float(Z.max()), 24) - filled = ax.contourf(X, Y, Z, levels=levels, cmap="viridis_r") - ax.contour(X, Y, Z, levels=levels[::3], colors="white", linewidths=0.45, alpha=0.5) + levels = ( + np.asarray(levels) + if levels is not None + else np.linspace(float(Z.min()), float(Z.max()), 24) + ) + filled = ax.contourf(X, Y, Z, levels=levels, cmap=cmap, extend="both") + ax.contour( + X, Y, Z, levels=levels[::3], colors="#8a8a8a", linewidths=0.45, alpha=0.7 + ) return filled diff --git a/func.py b/func.py index 2793efc..01293ae 100644 --- a/func.py +++ b/func.py @@ -14,6 +14,10 @@ def __init__(self, A, b): self.b = _vector(b) if self.A.shape != (self.b.size, self.b.size): raise ValueError("A must be square and match the size of b") + if not np.all(np.isfinite(self.A)) or not np.all(np.isfinite(self.b)): + raise ValueError("A and b must contain only finite values") + if not np.allclose(self.A, self.A.T): + raise ValueError("A must be symmetric so A @ x - b is the gradient") def f_x(self, x): x = _vector(x) diff --git a/main.py b/main.py index d7b3df4..f837c6a 100644 --- a/main.py +++ b/main.py @@ -1,8 +1,11 @@ -"""Generate the example figures used by the README.""" +"""Generate the tutorial's path and convergence comparison figures.""" +import argparse +import csv from pathlib import Path import matplotlib.pyplot as plt +from matplotlib.lines import Line2D import numpy as np import drawFunc @@ -11,96 +14,311 @@ ROOT = Path(__file__).resolve().parent +TOLERANCE = 1e-8 +QUADRATIC_START = np.array([0.0, 2.0]) +ROSENBROCK_START = np.array([-1.2, 1.0]) +ROSENBROCK_BOUNDS = ((-1.6, 1.35), (-1.1, 1.75)) +ROSENBROCK_LEVELS = np.geomspace(1e-3, 1e3, 25) +METHOD_STYLES = { + "Steepest descent": ("#0072B2", "o", "-"), + "Newton": ("#D55E00", "s", "-"), + "Damped Newton": ("#E69F00", "D", "-"), + "Modified Newton": ("#CC79A7", "P", "-"), + "DFP": ("#56B4E9", "v", "-"), + "BFGS": ("#009E73", "^", "-"), + "SR1": ("#F0A202", "X", "-"), + "Fletcher-Reeves": ("#6A3D9A", "<", "-"), + "Polak-Ribiere+": ("#333333", ">", "-"), + "Linear conjugate gradient": ("#6A3D9A", "<", "-"), +} -def _plot_track(ax, track, label, color, marker, **kwargs): - mark_every = max(1, track.shape[1] // 16) + +def _marker_indices(number_of_points, maximum=12): + if number_of_points <= maximum: + return np.arange(number_of_points) + indices = np.geomspace(1, number_of_points - 1, maximum - 1).round().astype(int) + return np.unique(np.concatenate(([0], indices))) + + +def _plot_result(ax, result, number_early_steps=True): + color, marker, line_style = METHOD_STYLES[result.method] + path = result.path + marker_indices = _marker_indices(path.shape[1]) ax.plot( - track[0], - track[1], + path[0], + path[1], color=color, + linestyle=line_style, + linewidth=1.8, marker=marker, - markevery=mark_every, - label=label, - linewidth=2.2, + markevery=marker_indices.tolist(), markersize=4.5, markeredgecolor="white", - markeredgewidth=0.6, - **kwargs, + markeredgewidth=0.55, + zorder=4, + ) + if number_early_steps: + for index in range(min(4, path.shape[1])): + ax.annotate( + str(index), + path[:, index], + xytext=(4, 4), + textcoords="offset points", + fontsize=7, + color=color, + zorder=6, + ) + + +def _decorate_path_axis(ax, result, start, x_bounds, y_bounds): + ax.scatter( + *start, + facecolor="white", + edgecolor="#222222", + linewidth=1.2, + s=35, + zorder=7, + ) + ax.scatter(1, 1, color="#222222", marker="*", s=85, zorder=7) + state = "converged" if result.converged else "budget reached" + ax.set_title( + f"{result.method}\n" + f"{result.iterations} steps · ||g||={result.gradient_norms[-1]:.1e} · {state}", + fontsize=9, ) + ax.set_xlim(x_bounds) + ax.set_ylim(y_bounds) + ax.set_aspect("equal", adjustable="box") + ax.tick_params(labelsize=8) + + +def _draw_rosenbrock_background(ax, objective): + return drawFunc.draw( + objective.f_x, + ax, + x_bounds=ROSENBROCK_BOUNDS[0], + y_bounds=ROSENBROCK_BOUNDS[1], + resolution=220, + log_scale=True, + levels=ROSENBROCK_LEVELS, + cmap="Greys", + ) + +def _quadratic_runs(objective): + return [ + opt.gradient_descent( + QUADRATIC_START, + objective, + iteration=100, + line_search="exact", + tolerance=TOLERANCE, + return_result=True, + ), + opt.newton( + QUADRATIC_START, + objective, + iterations=10, + tolerance=TOLERANCE, + return_result=True, + ), + opt.linear_conjugate_gradient( + QUADRATIC_START, + objective, + tolerance=TOLERANCE, + return_result=True, + ), + opt.dfp( + QUADRATIC_START, + objective, + iteration=10, + line_search="exact", + tolerance=TOLERANCE, + return_result=True, + ), + opt.bfgs( + QUADRATIC_START, + objective, + iteration=10, + line_search="exact", + tolerance=TOLERANCE, + return_result=True, + ), + opt.sr1( + QUADRATIC_START, + objective, + iteration=10, + line_search="exact", + tolerance=TOLERANCE, + return_result=True, + ), + ] -def _finish_axes(ax, title): - ax.set(title=title, xlabel="$x_0$", ylabel="$x_1$") - ax.grid(False) - ax.legend(frameon=True, framealpha=0.92, loc="upper left") + +def _rosenbrock_runs(objective): + common = dict(tolerance=TOLERANCE, return_result=True) + return [ + opt.gradient_descent(ROSENBROCK_START, objective, iteration=2000, **common), + opt.newton(ROSENBROCK_START, objective, iterations=100, **common), + opt.damped_newton(ROSENBROCK_START, objective, iteration=100, **common), + opt.modified_newton( + ROSENBROCK_START, + objective, + iteration=300, + min_curvature=1.0, + **common, + ), + opt.dfp(ROSENBROCK_START, objective, iteration=300, **common), + opt.bfgs(ROSENBROCK_START, objective, iteration=300, **common), + opt.sr1(ROSENBROCK_START, objective, iteration=300, **common), + opt.fletcher_reeves(ROSENBROCK_START, objective, iteration=500, **common), + opt.polak_ribiere_plus(ROSENBROCK_START, objective, iteration=500, **common), + ] def quadratic(output=ROOT / "quadratic.png"): + """Compare curvature-learning methods on one SPD quadratic.""" objective = func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) - start = np.array([0.0, 2.0]) - fig, ax = plt.subplots(figsize=(7, 5)) - drawFunc.draw(objective.f_x, ax, x_bounds=(-1.5, 3.1), y_bounds=(-2.8, 2.5)) - _plot_track(ax, opt.newton(start, objective, iterations=1), "Newton", "#e63946", "o") - _plot_track(ax, opt.quasi_newton(start, objective, 12), "SR1", "#ffb703", "s") - optimum = np.linalg.solve(objective.A, objective.b) - ax.scatter(*start, color="white", edgecolor="#222222", s=46, zorder=5, label="Start") - ax.scatter(*optimum, color="#e63946", marker="*", s=150, edgecolor="white", zorder=6, label="Optimum") - _finish_axes(ax, "Quadratic objective") - fig.tight_layout() - fig.savefig(output, dpi=160) + results = _quadratic_runs(objective) + x_bounds, y_bounds = (-0.8, 1.45), (-1.5, 2.2) + levels = np.linspace(-3.8, 20, 24) + fig, axes = plt.subplots(2, 3, figsize=(11, 9), sharex=True, sharey=True) + for ax, result in zip(axes.flat, results): + drawFunc.draw( + objective.f_x, + ax, + x_bounds=x_bounds, + y_bounds=y_bounds, + resolution=180, + levels=levels, + cmap="Greys", + ) + _plot_result(ax, result) + optimum = np.linalg.solve(objective.A, objective.b) + ax.scatter(*QUADRATIC_START, facecolor="white", edgecolor="#222222", s=35, zorder=7) + ax.scatter(*optimum, color="#222222", marker="*", s=85, zorder=7) + ax.set_title( + f"{result.method}\n{result.iterations} steps · ||g||={result.gradient_norms[-1]:.1e}", + fontsize=9, + ) + ax.set_xlim(x_bounds) + ax.set_ylim(y_bounds) + ax.set_aspect("equal", adjustable="box") + fig.suptitle("Curvature and conjugacy on a positive-definite quadratic", fontsize=15) + fig.supxlabel("$x_0$") + fig.supylabel("$x_1$") + fig.tight_layout(rect=(0.02, 0.02, 1, 0.96)) + fig.savefig(output, dpi=180) plt.close(fig) + return results -def quasi_newton(output=ROOT / "quasi_newton.png"): +def rosenbrock_paths(output=ROOT / "quasi_newton.png"): + """Compare nine methods from an identical Rosenbrock starting point.""" objective = func.rosenbrock() - start = np.array([-1.2, 1.0]) - fig, ax = plt.subplots(figsize=(7, 5)) - drawFunc.draw( - objective.f_x, - ax, - x_bounds=(-1.6, 1.4), - y_bounds=(-0.6, 2.0), - log_scale=True, - ) - _plot_track(ax, opt.quasi_newton(start, objective, 250), "SR1", "#ffb703", "s") - _plot_track(ax, opt.BFGS(start, objective, 250), "BFGS", "#e63946", "o") - ax.scatter(*start, color="white", edgecolor="#222222", s=46, zorder=5, label="Start") - ax.scatter(1, 1, color="#e63946", marker="*", s=150, edgecolor="white", zorder=6, label="Optimum") - _finish_axes(ax, "Quasi-Newton methods on Rosenbrock") - fig.tight_layout() - fig.savefig(output, dpi=160) + results = _rosenbrock_runs(objective) + fig, axes = plt.subplots(3, 3, figsize=(12, 12), sharex=True, sharey=True) + contour = None + for ax, result in zip(axes.flat, results): + contour = _draw_rosenbrock_background(ax, objective) + _plot_result(ax, result) + _decorate_path_axis( + ax, result, ROSENBROCK_START, *ROSENBROCK_BOUNDS + ) + handles = [ + Line2D([], [], marker="o", markerfacecolor="white", markeredgecolor="#222222", linestyle="none", label="start"), + Line2D([], [], marker="*", color="#222222", linestyle="none", markersize=9, label="optimum"), + ] + fig.legend(handles=handles, loc="upper center", ncol=2, frameon=False, bbox_to_anchor=(0.5, 0.965)) + fig.suptitle("Optimization paths on the Rosenbrock function (coefficient 10)", fontsize=15) + fig.supxlabel("$x_0$") + fig.supylabel("$x_1$") + colorbar_axis = fig.add_axes([0.9, 0.2, 0.018, 0.6]) + colorbar = fig.colorbar(contour, cax=colorbar_axis) + colorbar.set_label("objective $f(x)$ — logarithmic scale") + fig.subplots_adjust(left=0.07, right=0.86, bottom=0.06, top=0.92, wspace=0.14, hspace=0.27) + fig.savefig(output, dpi=180) plt.close(fig) + return results -def conjugate_gradient(output=ROOT / "cg.png"): - objective = func.rosenbrock() - # The conventional Rosenbrock starting point keeps the full trajectory in - # the plotted domain while still showing the narrow curved valley. - start = np.array([-1.2, 1.0]) - fig, ax = plt.subplots(figsize=(7, 5)) - drawFunc.draw( - objective.f_x, - ax, - x_bounds=(-1.6, 1.4), - y_bounds=(-0.2, 2.0), - log_scale=True, - ) - track = opt.fletcher_reeves(start, objective, iteration=500) - _plot_track(ax, track, "Fletcher–Reeves", "#e63946", "o") - ax.scatter(*start, color="white", edgecolor="#222222", s=46, zorder=5, label="Start") - ax.scatter(1, 1, color="#e63946", marker="*", s=150, edgecolor="white", zorder=6, label="Optimum") - _finish_axes(ax, "Nonlinear conjugate gradient on Rosenbrock") - fig.tight_layout() - fig.savefig(output, dpi=160) +def convergence(results, output=ROOT / "cg.png"): + """Compare objective and first-order convergence by iteration.""" + fig, axes = plt.subplots(1, 2, figsize=(12, 6.2)) + for result in results: + color, marker, line_style = METHOD_STYLES[result.method] + iterations = np.arange(result.iterations + 1) + objective_gap = np.maximum(result.objective_values, np.finfo(float).eps) + gradient_norm = np.maximum(result.gradient_norms, np.finfo(float).eps) + axes[0].semilogy( + iterations, + objective_gap, + color=color, + linestyle=line_style, + linewidth=1.7, + label=result.method, + ) + axes[1].semilogy( + iterations, + gradient_norm, + color=color, + linestyle=line_style, + linewidth=1.7, + label=result.method, + ) + axes[0].set(title="Objective gap by iteration", xlabel="accepted steps", ylabel="$f(x_k)-f^*$") + axes[1].set(title="First-order convergence", xlabel="accepted steps", ylabel="$||\\nabla f(x_k)||_2$") + for ax in axes: + ax.set_xscale("symlog", linthresh=10) + ax.grid(True, which="both", color="#d0d0d0", linewidth=0.5) + handles, labels = axes[1].get_legend_handles_labels() + fig.legend(handles, labels, loc="lower center", bbox_to_anchor=(0.5, 0.015), ncol=3, frameon=False) + fig.suptitle("Rosenbrock convergence histories", fontsize=15) + fig.tight_layout(rect=(0, 0.17, 1, 0.94)) + fig.savefig(output, dpi=180) plt.close(fig) -def generate_figures(): - quadratic() - quasi_newton() - conjugate_gradient() +def write_metrics(results, output=ROOT / "metrics.csv"): + with Path(output).open("w", newline="") as stream: + writer = csv.writer(stream) + writer.writerow( + ["method", "status", "iterations", "final_f", "final_gradient_norm", "nfev", "njev", "nhev", "restarts", "skipped_updates"] + ) + for result in results: + writer.writerow( + [ + result.method, + result.status, + result.iterations, + f"{result.objective_values[-1]:.12g}", + f"{result.gradient_norms[-1]:.12g}", + result.nfev, + result.njev, + result.nhev, + result.restarts, + result.skipped_updates, + ] + ) + + +def generate_figures(output_dir=ROOT): + output_dir = Path(output_dir) + output_dir.mkdir(parents=True, exist_ok=True) + quadratic(output_dir / "quadratic.png") + results = rosenbrock_paths(output_dir / "quasi_newton.png") + convergence(results, output_dir / "cg.png") + write_metrics(results, output_dir / "metrics.csv") + return results if __name__ == "__main__": - generate_figures() + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument( + "--output-dir", + type=Path, + default=ROOT, + help="directory for PNG and CSV outputs (default: repository root)", + ) + generate_figures(parser.parse_args().output_dir) diff --git a/metrics.csv b/metrics.csv new file mode 100644 index 0000000..7884418 --- /dev/null +++ b/metrics.csv @@ -0,0 +1,10 @@ +method,status,iterations,final_f,final_gradient_norm,nfev,njev,nhev,restarts,skipped_updates +Steepest descent,converged,1707,7.28237459778e-17,9.68790243558e-09,12981,3415,0,0,0 +Newton,converged,8,5.80417969563e-27,1.07722122435e-12,0,9,8,0,0 +Damped Newton,converged,13,0,0,28,27,13,0,0 +Modified Newton,converged,40,7.65784627779e-17,7.76493723237e-09,81,81,40,0,0 +DFP,converged,20,5.13742520052e-22,2.2093973997e-10,54,65,0,0,0 +BFGS,converged,19,4.40618112977e-20,2.31969163034e-09,52,62,0,0,0 +SR1,converged,21,2.55776724937e-22,2.27806998352e-10,68,43,0,3,0 +Fletcher-Reeves,converged,45,4.71631088674e-17,8.92575477216e-09,311,168,0,22,0 +Polak-Ribiere+,converged,95,6.22339641888e-17,8.79199318821e-09,652,344,0,47,0 diff --git a/optMethods.py b/optMethods.py index cd07e7e..5c3428f 100644 --- a/optMethods.py +++ b/optMethods.py @@ -1,8 +1,63 @@ -"""Small, dependency-free implementations of common optimisation methods.""" +"""Educational implementations of unconstrained optimization algorithms. + +By default every public method returns a ``(dimensions, iterations + 1)`` +array. Column zero is the starting point and each following column is an +accepted iterate. Pass ``return_result=True`` for histories, status, and work +counts. The implementations intentionally share a small line-search interface +so the direction can be compared independently of step selection. +""" + +from dataclasses import dataclass, replace import numpy as np +@dataclass(frozen=True) +class OptimizationResult: + """Trajectory and diagnostics from one optimization run.""" + + method: str + path: np.ndarray + objective_values: np.ndarray + gradient_norms: np.ndarray + step_sizes: np.ndarray + converged: bool + status: str + nfev: int + njev: int + nhev: int + restarts: int = 0 + skipped_updates: int = 0 + + @property + def x(self): + return self.path[:, -1] + + @property + def iterations(self): + return self.path.shape[1] - 1 + + +class _CountingObjective: + """Count derivative work without changing the objective classes.""" + + def __init__(self, objective): + self.objective = objective + self.nfev = self.njev = self.nhev = 0 + + def f_x(self, x): + self.nfev += 1 + return self.objective.f_x(x) + + def g_x(self, x): + self.njev += 1 + return self.objective.g_x(x) + + def G_x(self, x): + self.nhev += 1 + return self.objective.G_x(x) + + def _initialise(start_point): x = np.asarray(start_point, dtype=float).reshape(-1).copy() return x, [x.copy()] @@ -12,25 +67,211 @@ def _finish(track): return np.column_stack(track) -def newton(start_point, obj_fun, modified=0, iterations=5, tolerance=1e-8): +def _gradient(obj_fun, x): + return np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + + +def _validate_run(max_iter, tolerance): + if not isinstance(max_iter, (int, np.integer)) or max_iter < 0: + raise ValueError("maximum iterations must be a non-negative integer") + if not np.isfinite(tolerance) or tolerance <= 0: + raise ValueError("tolerance must be positive and finite") + + +def _result_or_path( + objective, + oracle, + track, + step_sizes, + method, + tolerance, + return_result, + restarts=0, + skipped_updates=0, +): + path = _finish(track) + if not return_result: + return path + values = np.array([objective.f_x(path[:, index]) for index in range(path.shape[1])]) + gradient_norms = np.array( + [np.linalg.norm(objective.g_x(path[:, index])) for index in range(path.shape[1])] + ) + converged = bool(gradient_norms[-1] <= tolerance) + return OptimizationResult( + method=method, + path=path, + objective_values=values, + gradient_norms=gradient_norms, + step_sizes=np.asarray(step_sizes), + converged=converged, + status="converged" if converged else "maximum iterations reached", + nfev=oracle.nfev, + njev=oracle.njev, + nhev=oracle.nhev, + restarts=restarts, + skipped_updates=skipped_updates, + ) + + +def gradient_descent( + start_point, + obj_fun, + iteration=200, + alpha=1.0, + line_search="armijo", + tolerance=1e-8, + return_result=False, +): + """Steepest descent using ``p_k = -gradient f(x_k)``. + + ``line_search="exact"`` uses the quadratic model's exact minimizer and is + intended for the quadratic tutorial. ``"armijo"`` works for either + objective, while ``"fixed"`` always uses ``alpha``. + """ + _validate_run(iteration, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) x, track = _initialise(start_point) + step_sizes = [] + for _ in range(iteration): + gradient = _gradient(obj_fun, x) + if np.linalg.norm(gradient) <= tolerance: + break + direction = -gradient + step_size = _choose_step(obj_fun, x, direction, line_search, alpha) + x = x + step_size * direction + track.append(x.copy()) + step_sizes.append(step_size) + return _result_or_path( + objective, obj_fun, track, step_sizes, "Steepest descent", tolerance, return_result + ) + + +def newton( + start_point, + obj_fun, + modified=0.0, + iterations=20, + line_search="fixed", + alpha=1.0, + tolerance=1e-8, + return_result=False, +): + """Newton's method, optionally with a fixed diagonal Hessian shift. + + ``modified`` preserves the original tutorial API: a positive value solves + ``(H + modified * I) p = -g``. See :func:`modified_newton` for an adaptive + positive-definite shift. + """ + _validate_run(iterations, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) + x, track = _initialise(start_point) + step_sizes = [] identity = np.eye(x.size) for _ in range(iterations): - gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + gradient = _gradient(obj_fun, x) if np.linalg.norm(gradient) <= tolerance: break hessian = np.asarray(obj_fun.G_x(x), dtype=float) + modified * identity - # Solving is both faster and more numerically stable than forming inv(H). - step = np.linalg.solve(hessian, -gradient) - x = x + step + direction = np.linalg.solve(hessian, -gradient) + if line_search != "fixed" and direction @ gradient >= 0: + direction = -gradient + step_size = _choose_step(obj_fun, x, direction, line_search, alpha) + x = x + step_size * direction track.append(x.copy()) - return _finish(track) + step_sizes.append(step_size) + name = "Fixed-shift Newton" if modified else "Newton" + return _result_or_path( + objective, obj_fun, track, step_sizes, name, tolerance, return_result + ) + +def modified_newton( + start_point, + obj_fun, + iteration=100, + min_curvature=1e-3, + alpha=1.0, + tolerance=1e-8, + return_result=False, +): + """Newton's method with an adaptive eigenvalue shift and Armijo search. -def BFGS(start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-8): + At every iteration the Hessian is shifted just enough to make its smallest + eigenvalue at least ``min_curvature``. This produces a descent direction + even where the unmodified Hessian is indefinite. + """ + _validate_run(iteration, tolerance) + if not np.isfinite(min_curvature) or min_curvature <= 0: + raise ValueError("min_curvature must be positive and finite") + objective = obj_fun + obj_fun = _CountingObjective(objective) + x, track = _initialise(start_point) + step_sizes = [] + identity = np.eye(x.size) + for _ in range(iteration): + gradient = _gradient(obj_fun, x) + if np.linalg.norm(gradient) <= tolerance: + break + hessian = np.asarray(obj_fun.G_x(x), dtype=float) + smallest_eigenvalue = float(np.linalg.eigvalsh(hessian).min()) + shift = max(0.0, min_curvature - smallest_eigenvalue) + direction = np.linalg.solve(hessian + shift * identity, -gradient) + step_size = _armijo_line_search(obj_fun, x, direction, alpha0=alpha) + x = x + step_size * direction + track.append(x.copy()) + step_sizes.append(step_size) + return _result_or_path( + objective, obj_fun, track, step_sizes, "Modified Newton", tolerance, return_result + ) + + +def damped_newton( + start_point, + obj_fun, + iteration=100, + alpha=1.0, + tolerance=1e-8, + return_result=False, +): + """Newton direction globalized by an Armijo backtracking line search.""" + result = newton( + start_point, + obj_fun, + iterations=iteration, + line_search="armijo", + alpha=alpha, + tolerance=tolerance, + return_result=return_result, + ) + return replace(result, method="Damped Newton") if return_result else result + + +def bfgs( + start_point, + obj_fun, + iteration=100, + alpha=1.0, + line_search="strong_wolfe", + tolerance=1e-8, + return_result=False, +): + """BFGS inverse-Hessian quasi-Newton method. + + The inverse update is applied only when the curvature condition is safely + positive. If round-off produces a non-descent direction, the approximation + is restarted from the identity. + """ + _validate_run(iteration, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) x, track = _initialise(start_point) inverse_hessian = np.eye(x.size) - gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + gradient = _gradient(obj_fun, x) + step_sizes = [] + restarts = 0 + skipped_updates = 0 for _ in range(iteration): if np.linalg.norm(gradient) <= tolerance: @@ -39,98 +280,491 @@ def BFGS(start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-8): if direction @ gradient >= 0: inverse_hessian = np.eye(x.size) direction = -gradient - alpha = _select_line_search(obj_fun, x, direction, interpolation) - step = alpha * direction + restarts += 1 + step_size = _choose_step(obj_fun, x, direction, line_search, alpha) + step = step_size * direction next_x = x + step - next_gradient = np.asarray(obj_fun.g_x(next_x), dtype=float).reshape(-1) + next_gradient = _gradient(obj_fun, next_x) y = next_gradient - gradient curvature = float(step @ y) - if curvature > 1e-12 * np.linalg.norm(step) * np.linalg.norm(y): + threshold = 1e-12 * np.linalg.norm(step) * np.linalg.norm(y) + if curvature > threshold: rho = 1.0 / curvature identity = np.eye(x.size) left = identity - rho * np.outer(step, y) inverse_hessian = ( left @ inverse_hessian @ left.T + rho * np.outer(step, step) ) + else: + skipped_updates += 1 x, gradient = next_x, next_gradient track.append(x.copy()) - return _finish(track) + step_sizes.append(step_size) + return _result_or_path( + objective, + obj_fun, + track, + step_sizes, + "BFGS", + tolerance, + return_result, + restarts, + skipped_updates, + ) -def quasi_newton(start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-8): - """Symmetric-rank-one (SR1) quasi-Newton method.""" +def sr1( + start_point, + obj_fun, + iteration=100, + alpha=1.0, + line_search="armijo", + tolerance=1e-8, + return_result=False, +): + """Symmetric-rank-one (SR1) inverse-Hessian quasi-Newton method.""" + _validate_run(iteration, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) x, track = _initialise(start_point) inverse_hessian = np.eye(x.size) - gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + gradient = _gradient(obj_fun, x) + step_sizes = [] + restarts = 0 + skipped_updates = 0 for _ in range(iteration): if np.linalg.norm(gradient) <= tolerance: break direction = -inverse_hessian @ gradient - # SR1 updates need not stay positive definite. Restart when that makes - # the proposed direction point uphill. + # Unlike BFGS, SR1 approximations can be indefinite. if direction @ gradient >= 0: inverse_hessian = np.eye(x.size) direction = -gradient - alpha = _select_line_search(obj_fun, x, direction, interpolation) - step = alpha * direction + restarts += 1 + step_size = _choose_step(obj_fun, x, direction, line_search, alpha) + step = step_size * direction next_x = x + step - next_gradient = np.asarray(obj_fun.g_x(next_x), dtype=float).reshape(-1) + next_gradient = _gradient(obj_fun, next_x) y = next_gradient - gradient residual = step - inverse_hessian @ y denominator = float(residual @ y) - # A tiny denominator makes SR1 unstable; the correct action is to skip it. - if abs(denominator) > 1e-8 * np.linalg.norm(residual) * np.linalg.norm(y): + threshold = 1e-8 * np.linalg.norm(residual) * np.linalg.norm(y) + if abs(denominator) > threshold: inverse_hessian += np.outer(residual, residual) / denominator + else: + skipped_updates += 1 x, gradient = next_x, next_gradient track.append(x.copy()) - return _finish(track) + step_sizes.append(step_size) + return _result_or_path( + objective, + obj_fun, + track, + step_sizes, + "SR1", + tolerance, + return_result, + restarts, + skipped_updates, + ) -def fletcher_reeves(start_point, obj_fun, iteration=10, alpha=1.0, tolerance=1e-8): +def dfp( + start_point, + obj_fun, + iteration=100, + alpha=1.0, + line_search="strong_wolfe", + tolerance=1e-8, + return_result=False, +): + """Davidon-Fletcher-Powell inverse-Hessian quasi-Newton method.""" + _validate_run(iteration, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) x, track = _initialise(start_point) - gradient = np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) + inverse_hessian = np.eye(x.size) + gradient = _gradient(obj_fun, x) + step_sizes = [] + restarts = 0 + skipped_updates = 0 + + for _ in range(iteration): + if np.linalg.norm(gradient) <= tolerance: + break + direction = -inverse_hessian @ gradient + if direction @ gradient >= 0: + inverse_hessian = np.eye(x.size) + direction = -gradient + restarts += 1 + step_size = _choose_step(obj_fun, x, direction, line_search, alpha) + step = step_size * direction + next_x = x + step + next_gradient = _gradient(obj_fun, next_x) + y = next_gradient - gradient + step_curvature = float(step @ y) + hessian_y = inverse_hessian @ y + model_curvature = float(y @ hessian_y) + threshold = 1e-12 * np.linalg.norm(step) * np.linalg.norm(y) + if step_curvature > threshold and model_curvature > threshold: + inverse_hessian += ( + np.outer(step, step) / step_curvature + - np.outer(hessian_y, hessian_y) / model_curvature + ) + else: + skipped_updates += 1 + x, gradient = next_x, next_gradient + track.append(x.copy()) + step_sizes.append(step_size) + return _result_or_path( + objective, + obj_fun, + track, + step_sizes, + "DFP", + tolerance, + return_result, + restarts, + skipped_updates, + ) + + +def fletcher_reeves( + start_point, + obj_fun, + iteration=200, + alpha=1.0, + line_search="strong_wolfe", + tolerance=1e-8, + return_result=False, +): + """Fletcher-Reeves nonlinear conjugate-gradient method. + + Directions are restarted every ``n`` iterations (and whenever they cease to + be descent directions), a standard finite-precision safeguard. + """ + _validate_run(iteration, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) + x, track = _initialise(start_point) + gradient = _gradient(obj_fun, x) direction = -gradient + step_sizes = [] + restarts = 0 for k in range(iteration): if np.linalg.norm(gradient) <= tolerance: break - # Restart if round-off has destroyed the descent direction. if direction @ gradient >= 0: direction = -gradient - step_size = _armijo_line_search(obj_fun, x, direction, alpha0=alpha) + restarts += 1 + step_size = _choose_step( + obj_fun, x, direction, line_search, alpha, wolfe_c2=0.4 + ) next_x = x + step_size * direction - next_gradient = np.asarray(obj_fun.g_x(next_x), dtype=float).reshape(-1) + next_gradient = _gradient(obj_fun, next_x) denominator = float(gradient @ gradient) beta = float(next_gradient @ next_gradient) / denominator if denominator else 0.0 - # Periodic restarts are required for robust nonlinear CG in finite precision. - direction = -next_gradient if (k + 1) % x.size == 0 else -next_gradient + beta * direction + if (k + 1) % x.size == 0: + direction = -next_gradient + restarts += 1 + else: + direction = -next_gradient + beta * direction x, gradient = next_x, next_gradient track.append(x.copy()) - return _finish(track) + step_sizes.append(step_size) + return _result_or_path( + objective, + obj_fun, + track, + step_sizes, + "Fletcher-Reeves", + tolerance, + return_result, + restarts, + ) + + +def polak_ribiere_plus( + start_point, + obj_fun, + iteration=200, + alpha=1.0, + line_search="strong_wolfe", + tolerance=1e-8, + return_result=False, +): + """Polak-Ribiere+ nonlinear conjugate gradient with safe restarts.""" + _validate_run(iteration, tolerance) + objective = obj_fun + obj_fun = _CountingObjective(objective) + x, track = _initialise(start_point) + gradient = _gradient(obj_fun, x) + direction = -gradient + step_sizes = [] + restarts = 0 + + for k in range(iteration): + if np.linalg.norm(gradient) <= tolerance: + break + if direction @ gradient >= 0: + direction = -gradient + restarts += 1 + step_size = _choose_step( + obj_fun, x, direction, line_search, alpha, wolfe_c2=0.4 + ) + next_x = x + step_size * direction + next_gradient = _gradient(obj_fun, next_x) + denominator = float(gradient @ gradient) + beta = max( + 0.0, + float(next_gradient @ (next_gradient - gradient)) / denominator + if denominator + else 0.0, + ) + if (k + 1) % x.size == 0: + direction = -next_gradient + restarts += 1 + else: + direction = -next_gradient + beta * direction + x, gradient = next_x, next_gradient + track.append(x.copy()) + step_sizes.append(step_size) + return _result_or_path( + objective, + obj_fun, + track, + step_sizes, + "Polak-Ribiere+", + tolerance, + return_result, + restarts, + ) + + +def linear_conjugate_gradient( + start_point, + obj_fun, + iteration=None, + tolerance=1e-10, + return_result=False, +): + """Linear conjugate gradient for a symmetric positive-definite quadratic. + + In exact arithmetic this method reaches the minimizer in at most ``n`` + iterations. It is intentionally separate from nonlinear Fletcher-Reeves. + """ + objective = obj_fun + obj_fun = _CountingObjective(objective) + x, track = _initialise(start_point) + max_iter = x.size if iteration is None else iteration + _validate_run(max_iter, tolerance) + hessian = np.asarray(obj_fun.G_x(x), dtype=float) + if hessian.shape != (x.size, x.size) or not np.allclose(hessian, hessian.T): + raise ValueError("linear CG requires a symmetric quadratic Hessian") + try: + np.linalg.cholesky(hessian) + except np.linalg.LinAlgError as error: + raise ValueError("linear CG requires a positive-definite quadratic") from error + + residual = -_gradient(obj_fun, x) + direction = residual.copy() + residual_norm_squared = float(residual @ residual) + step_sizes = [] + for _ in range(max_iter): + if np.sqrt(residual_norm_squared) <= tolerance: + break + hessian_direction = hessian @ direction + curvature = float(direction @ hessian_direction) + step_size = residual_norm_squared / curvature + x = x + step_size * direction + next_residual = residual - step_size * hessian_direction + next_norm_squared = float(next_residual @ next_residual) + beta = next_norm_squared / residual_norm_squared + direction = next_residual + beta * direction + residual = next_residual + residual_norm_squared = next_norm_squared + track.append(x.copy()) + step_sizes.append(step_size) + return _result_or_path( + objective, + obj_fun, + track, + step_sizes, + "Linear conjugate gradient", + tolerance, + return_result, + ) + + +def minimize(method, start_point, obj_fun, max_iter=100, gtol=1e-8, **options): + """Run any tutorial method through one consistent result-returning API. + + Supported names are ``gradient_descent``, ``newton``, ``damped_newton``, + ``modified_newton``, ``dfp``, ``bfgs``, ``sr1``, ``linear_cg``, + ``fletcher_reeves``, and ``polak_ribiere_plus``. + """ + methods = { + "gradient_descent": (gradient_descent, "iteration"), + "newton": (newton, "iterations"), + "damped_newton": (damped_newton, "iteration"), + "modified_newton": (modified_newton, "iteration"), + "dfp": (dfp, "iteration"), + "bfgs": (bfgs, "iteration"), + "sr1": (sr1, "iteration"), + "linear_cg": (linear_conjugate_gradient, "iteration"), + "fletcher_reeves": (fletcher_reeves, "iteration"), + "polak_ribiere_plus": (polak_ribiere_plus, "iteration"), + } + try: + function, iteration_name = methods[method] + except KeyError as error: + choices = ", ".join(methods) + raise ValueError(f"unknown method {method!r}; choose from {choices}") from error + options[iteration_name] = max_iter + options["tolerance"] = gtol + options["return_result"] = True + return function(start_point, obj_fun, **options) + + +def BFGS(start_point, obj_fun, iteration=100, interpolation=0, **kwargs): + """Backward-compatible alias for :func:`bfgs`.""" + return bfgs(start_point, obj_fun, iteration=iteration, **kwargs) + + +def quasi_newton(start_point, obj_fun, iteration=100, interpolation=0, **kwargs): + """Backward-compatible alias for the tutorial's original SR1 function.""" + return sr1(start_point, obj_fun, iteration=iteration, **kwargs) -def _select_line_search(obj_fun, x, direction, interpolation): - if interpolation > 0: - return _backtracking_line_search(obj_fun, x, direction) - return _armijo_line_search(obj_fun, x, direction) +def _choose_step(obj_fun, x, direction, line_search, alpha, wolfe_c2=0.9): + if line_search == "fixed": + if not np.isfinite(alpha) or alpha <= 0: + raise ValueError("a fixed step must be positive and finite") + return float(alpha) + if line_search == "armijo": + return _armijo_line_search(obj_fun, x, direction, alpha0=alpha) + if line_search == "exact": + return _exact_quadratic_line_search(obj_fun, x, direction) + if line_search == "strong_wolfe": + return _strong_wolfe_line_search( + obj_fun, x, direction, alpha0=alpha, c2=wolfe_c2 + ) + raise ValueError( + "line_search must be 'fixed', 'armijo', 'strong_wolfe', or 'exact'" + ) -def _backtracking_line_search(obj_fun, x, s, c=1e-4, alpha0=1.0, contraction=0.5): - """Return the first step satisfying the Armijo sufficient-decrease rule.""" +def _exact_quadratic_line_search(obj_fun, x, direction): + """Exact step along ``direction`` for an objective with constant Hessian.""" + gradient = _gradient(obj_fun, x) + hessian = np.asarray(obj_fun.G_x(x), dtype=float) + curvature = float(direction @ hessian @ direction) + if curvature <= 0: + raise ValueError("exact quadratic line search requires positive curvature") + step = -float(gradient @ direction) / curvature + if step <= 0: + raise ValueError("line search direction must be a descent direction") + return step + + +def _backtracking_line_search( + obj_fun, + x, + direction, + c=1e-4, + alpha0=1.0, + contraction=0.5, + max_iterations=60, +): + """Return the first step satisfying Armijo sufficient decrease.""" + if not 0 < c < 1: + raise ValueError("c must lie strictly between zero and one") + if not 0 < contraction < 1: + raise ValueError("contraction must lie strictly between zero and one") alpha = float(alpha0) + if not np.isfinite(alpha) or alpha <= 0: + raise ValueError("alpha0 must be positive and finite") f0 = obj_fun.f_x(x) - slope = float(np.asarray(obj_fun.g_x(x)).reshape(-1) @ np.asarray(s).reshape(-1)) + slope = float(_gradient(obj_fun, x) @ np.asarray(direction).reshape(-1)) if slope >= 0: raise ValueError("line search direction must be a descent direction") - for _ in range(60): - if obj_fun.f_x(x + alpha * s) <= f0 + c * alpha * slope: + for _ in range(max_iterations): + if obj_fun.f_x(x + alpha * direction) <= f0 + c * alpha * slope: return alpha alpha *= contraction - return alpha + raise RuntimeError("Armijo line search failed to find an acceptable step") + + +def _armijo_line_search(obj_fun, x, direction, c=1e-4, alpha0=1.0): + """Compatibility wrapper around guarded backtracking Armijo search.""" + return _backtracking_line_search(obj_fun, x, direction, c=c, alpha0=alpha0) + +def _strong_wolfe_line_search( + obj_fun, + x, + direction, + c1=1e-4, + c2=0.9, + alpha0=1.0, + max_iterations=30, +): + """Bracket and zoom until the strong Wolfe conditions are satisfied.""" + if not 0 < c1 < c2 < 1: + raise ValueError("strong Wolfe constants must satisfy 0 < c1 < c2 < 1") + if not np.isfinite(alpha0) or alpha0 <= 0: + raise ValueError("alpha0 must be positive and finite") + direction = np.asarray(direction, dtype=float).reshape(-1) + phi0 = obj_fun.f_x(x) + derivative0 = float(_gradient(obj_fun, x) @ direction) + if derivative0 >= 0: + raise ValueError("line search direction must be a descent direction") + + def phi(alpha): + return obj_fun.f_x(x + alpha * direction) + + def derivative(alpha): + return float(_gradient(obj_fun, x + alpha * direction) @ direction) -def _armijo_line_search(obj_fun, x, s, c=1e-4, alpha0=1.0): - # The previous cubic interpolation could divide by zero and take sqrt(abs(...)), - # which masks invalid interpolants. Guarded backtracking is deterministic and safe. - return _backtracking_line_search(obj_fun, x, s, c=c, alpha0=alpha0) + def zoom(low, high, phi_low): + for _ in range(max_iterations): + candidate = 0.5 * (low + high) + phi_candidate = phi(candidate) + if ( + phi_candidate > phi0 + c1 * candidate * derivative0 + or phi_candidate >= phi_low + ): + high = candidate + else: + derivative_candidate = derivative(candidate) + if abs(derivative_candidate) <= -c2 * derivative0: + return candidate + if derivative_candidate * (high - low) >= 0: + high = low + low = candidate + phi_low = phi_candidate + if abs(high - low) <= np.finfo(float).eps * max(1.0, abs(low), abs(high)): + break + raise RuntimeError("strong Wolfe zoom failed to find an acceptable step") + + previous_alpha = 0.0 + previous_phi = phi0 + alpha = float(alpha0) + for index in range(max_iterations): + current_phi = phi(alpha) + if ( + current_phi > phi0 + c1 * alpha * derivative0 + or (index > 0 and current_phi >= previous_phi) + ): + return zoom(previous_alpha, alpha, previous_phi) + current_derivative = derivative(alpha) + if abs(current_derivative) <= -c2 * derivative0: + return alpha + if current_derivative >= 0: + return zoom(alpha, previous_alpha, current_phi) + previous_alpha = alpha + previous_phi = current_phi + alpha *= 2.0 + raise RuntimeError("strong Wolfe line search failed to bracket an acceptable step") diff --git a/pyproject.toml b/pyproject.toml index f0e4c8a..ae2b0b6 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -1,3 +1,4 @@ [tool.pytest.ini_options] pythonpath = ["."] testpaths = ["tests"] +filterwarnings = ["ignore::pyparsing.PyparsingDeprecationWarning"] diff --git a/quadratic.png b/quadratic.png index df888d5fa97affa94ab095bde794945ebc3a009c..9b95af50c645eb24c11d4105cfd515af3ada697f 100644 GIT binary patch literal 231130 zcmeFZXH-*Z+Xjl`jJ=H5LBJ6ajRH~v(nXqdkS1N4)PT}^EGU8`2uQCI=}lVb#8Du0 z1VT%ID7}Tyk^mvec{1~U=l%YlUuUh4wG^Tx+1dMf?)$p0vLp3%)lVMdI>y1laZ=-< z$|DYr!%Z9ErHVch%3)%iG1% 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zZ^5@}&uZD8nKHPSA=Y#Z_)DKu4! diff --git a/requirements-dev.txt b/requirements-dev.txt index 69b3c42..44dca33 100644 --- a/requirements-dev.txt +++ b/requirements-dev.txt @@ -1,2 +1,2 @@ -r requirements.txt -pytest>=8 +pytest==8.4.2 diff --git a/requirements.txt b/requirements.txt index 057269d..90e52c6 100644 --- a/requirements.txt +++ b/requirements.txt @@ -1,2 +1,2 @@ -matplotlib>=3.7 -numpy>=1.24 +matplotlib==3.9.4 +numpy==2.0.2 diff --git a/tests/test_optimizers.py b/tests/test_optimizers.py index 5a14b16..8f4117e 100644 --- a/tests/test_optimizers.py +++ b/tests/test_optimizers.py @@ -1,7 +1,13 @@ +from pathlib import Path + +import matplotlib import numpy as np import pytest +matplotlib.use("Agg") + import func +import main import optMethods as opt @@ -10,27 +16,71 @@ def quadratic_objective(): return func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) -def test_quadratic_derivatives_and_optimum(quadratic_objective): - optimum = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) - assert np.linalg.norm(quadratic_objective.g_x(optimum)) < 1e-12 - assert quadratic_objective.f_x(optimum) < quadratic_objective.f_x([0, 2]) +def _finite_difference_gradient(objective, x, step=1e-6): + gradient = np.empty_like(x, dtype=float) + for index in range(x.size): + offset = np.zeros_like(x, dtype=float) + offset[index] = step + gradient[index] = (objective.f_x(x + offset) - objective.f_x(x - offset)) / (2 * step) + return gradient -def test_newton_reaches_quadratic_optimum_in_one_step(quadratic_objective): - track = opt.newton([0, 2], quadratic_objective, iterations=1) - expected = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) - np.testing.assert_allclose(track[:, -1], expected, atol=1e-12) +def _finite_difference_hessian(objective, x, step=1e-5): + hessian = np.empty((x.size, x.size)) + for index in range(x.size): + offset = np.zeros_like(x, dtype=float) + offset[index] = step + hessian[:, index] = (objective.g_x(x + offset) - objective.g_x(x - offset)) / (2 * step) + return hessian + + +@pytest.mark.parametrize( + "objective, point", + [ + (func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]), np.array([0.4, -1.2])), + (func.rosenbrock(), np.array([-0.7, 1.3])), + ], +) +def test_objective_derivatives_match_finite_differences(objective, point): + np.testing.assert_allclose(objective.g_x(point), _finite_difference_gradient(objective, point), rtol=1e-5, atol=1e-6) + np.testing.assert_allclose(objective.G_x(point), _finite_difference_hessian(objective, point), rtol=1e-5, atol=1e-6) -@pytest.mark.parametrize("method", [opt.BFGS, opt.quasi_newton]) -def test_quasi_newton_methods_converge_on_quadratic(method, quadratic_objective): - track = method([0, 2], quadratic_objective, iteration=30) +def test_quadratic_validates_its_mathematical_contract(): + with pytest.raises(ValueError, match="symmetric"): + func.quadratic([[1.0, 2.0], [0.0, 1.0]], [0.0, 0.0]) + with pytest.raises(ValueError, match="square"): + func.quadratic([[1.0, 0.0]], [0.0, 0.0]) + with pytest.raises(ValueError, match="finite"): + func.quadratic([[1.0, 0.0], [0.0, np.inf]], [0.0, 0.0]) + + +@pytest.mark.parametrize( + "method, kwargs, maximum_steps", + [ + (opt.gradient_descent, {"iteration": 100, "line_search": "exact"}, 25), + (opt.newton, {"iterations": 5}, 1), + (opt.linear_conjugate_gradient, {}, 2), + (opt.dfp, {"iteration": 5, "line_search": "exact"}, 2), + (opt.bfgs, {"iteration": 5, "line_search": "exact"}, 2), + (opt.sr1, {"iteration": 5, "line_search": "exact"}, 2), + ], +) +def test_quadratic_tutorial_methods_converge(method, kwargs, maximum_steps, quadratic_objective): + result = method([0, 2], quadratic_objective, return_result=True, **kwargs) expected = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) - np.testing.assert_allclose(track[:, -1], expected, atol=1e-6) - assert not np.array_equal(track[:, 0], track[:, 1]) + assert result.converged + assert result.iterations <= maximum_steps + np.testing.assert_allclose(result.x, expected, atol=1e-7) + + +def test_linear_cg_rejects_non_positive_definite_quadratic(): + objective = func.quadratic([[1.0, 0.0], [0.0, -1.0]], [0.0, 0.0]) + with pytest.raises(ValueError, match="positive-definite"): + opt.linear_conjugate_gradient([1.0, 1.0], objective) -def test_backtracking_returns_an_accepted_step(quadratic_objective): +def test_armijo_returns_an_accepted_step(quadratic_objective): x = np.array([0.0, 2.0]) direction = -quadratic_objective.g_x(x) alpha = opt._backtracking_line_search(quadratic_objective, x, direction) @@ -40,20 +90,124 @@ def test_backtracking_returns_an_accepted_step(quadratic_objective): ) -@pytest.mark.parametrize("method, iterations", [ - (opt.BFGS, 300), - (opt.quasi_newton, 300), - (opt.fletcher_reeves, 600), -]) -def test_first_order_methods_reduce_rosenbrock(method, iterations): - objective = func.rosenbrock() - start = np.array([-1.5, -4.0]) - track = method(start, objective, iteration=iterations) - assert objective.f_x(track[:, -1]) < 1e-5 - assert np.linalg.norm(objective.g_x(track[:, -1])) < 1e-2 +def test_strong_wolfe_conditions_hold(quadratic_objective): + x = np.array([0.0, 2.0]) + direction = -quadratic_objective.g_x(x) + alpha = opt._strong_wolfe_line_search(quadratic_objective, x, direction) + initial_slope = quadratic_objective.g_x(x) @ direction + accepted_slope = quadratic_objective.g_x(x + alpha * direction) @ direction + assert quadratic_objective.f_x(x + alpha * direction) <= ( + quadratic_objective.f_x(x) + 1e-4 * alpha * initial_slope + ) + assert abs(accepted_slope) <= 0.9 * abs(initial_slope) + + +@pytest.mark.parametrize( + "kwargs", + [ + {"c": 0.0}, + {"contraction": 1.0}, + {"alpha0": 0.0}, + ], +) +def test_armijo_validates_parameters(quadratic_objective, kwargs): + x = np.array([0.0, 2.0]) + with pytest.raises(ValueError): + opt._backtracking_line_search( + quadratic_objective, x, -quadratic_objective.g_x(x), **kwargs + ) -def test_line_search_rejects_ascent_direction(quadratic_objective): +def test_line_search_rejects_ascent_and_reports_exhaustion(quadratic_objective): x = np.array([0.0, 2.0]) with pytest.raises(ValueError, match="descent"): opt._backtracking_line_search(quadratic_objective, x, quadratic_objective.g_x(x)) + with pytest.raises(RuntimeError, match="failed"): + opt._backtracking_line_search( + quadratic_objective, + x, + -quadratic_objective.g_x(x), + alpha0=1e100, + max_iterations=1, + ) + + +@pytest.mark.parametrize( + "method, kwargs", + [ + (opt.gradient_descent, {"iteration": 2000}), + (opt.newton, {"iterations": 100}), + (opt.damped_newton, {"iteration": 100}), + (opt.modified_newton, {"iteration": 300, "min_curvature": 1.0}), + (opt.dfp, {"iteration": 300}), + (opt.bfgs, {"iteration": 300}), + (opt.sr1, {"iteration": 300}), + (opt.fletcher_reeves, {"iteration": 500}), + (opt.polak_ribiere_plus, {"iteration": 500}), + ], +) +def test_every_displayed_rosenbrock_method_converges(method, kwargs): + result = method( + [-1.2, 1.0], + func.rosenbrock(), + tolerance=1e-8, + return_result=True, + **kwargs, + ) + assert result.converged, result.status + assert result.gradient_norms[-1] <= 1e-8 + assert result.objective_values[-1] < 1e-14 + + +def test_globalization_explains_newton_path_difference(): + objective = func.rosenbrock() + full = opt.newton([-1.2, 1.0], objective, iterations=100, return_result=True) + damped = opt.damped_newton([-1.2, 1.0], objective, iteration=100, return_result=True) + assert np.any(np.diff(full.objective_values) > 0) + assert np.all(np.diff(damped.objective_values) <= 1e-12) + + +def test_result_history_and_counts_are_consistent(quadratic_objective): + result = opt.bfgs( + [0, 2], + quadratic_objective, + iteration=5, + line_search="exact", + return_result=True, + ) + assert result.path.shape[1] == result.iterations + 1 + assert result.objective_values.shape == (result.iterations + 1,) + assert result.gradient_norms.shape == (result.iterations + 1,) + assert result.step_sizes.shape == (result.iterations,) + assert result.njev > 0 and result.nhev > 0 + assert np.all(np.isfinite(result.path)) + + +def test_consistent_minimize_dispatcher(quadratic_objective): + result = opt.minimize( + "bfgs", + [0, 2], + quadratic_objective, + max_iter=5, + gtol=1e-8, + line_search="exact", + ) + assert isinstance(result, opt.OptimizationResult) + assert result.converged + with pytest.raises(ValueError, match="unknown method"): + opt.minimize("not-a-method", [0, 2], quadratic_objective) + + +@pytest.mark.parametrize("bad_tolerance", [0, -1, np.inf, np.nan]) +def test_methods_reject_invalid_tolerance(quadratic_objective, bad_tolerance): + with pytest.raises(ValueError, match="tolerance"): + opt.gradient_descent([0, 2], quadratic_objective, tolerance=bad_tolerance) + + +def test_figure_generation_is_reproducible(tmp_path): + results = main.generate_figures(tmp_path) + assert len(results) == 9 + for filename in ["quadratic.png", "quasi_newton.png", "cg.png", "metrics.csv"]: + output = Path(tmp_path, filename) + assert output.is_file() + assert output.stat().st_size > 100 From bd11690dc510a816ac4792bc05c79e60652dbad1 Mon Sep 17 00:00:00 2001 From: Wenqi Li Date: Sun, 12 Jul 2026 13:42:11 +0100 Subject: [PATCH 4/4] Restore original figure style and narrow bug fixes --- .gitignore | 14 + README.md | 111 +----- cg.png | Bin 220222 -> 135854 bytes drawFunc.py | 49 +-- main.py | 391 ++++++--------------- metrics.csv | 10 - optMethods.py | 738 ++++----------------------------------- quadratic.png | Bin 231130 -> 97344 bytes quasi_newton.png | Bin 468242 -> 118339 bytes tests/test_optimizers.py | 213 +++-------- 10 files changed, 269 insertions(+), 1257 deletions(-) delete mode 100644 metrics.csv diff --git a/.gitignore b/.gitignore index dc67086..d5abaa0 100644 --- a/.gitignore +++ b/.gitignore @@ -1,5 +1,9 @@ *.py[cod] + +# C extensions *.so + +# Packages *.egg *.egg-info dist @@ -13,14 +17,24 @@ develop-eggs .installed.cfg lib lib64 + +# Installer logs pip-log.txt + +# Unit test / coverage reports .coverage .tox nosetests.xml + +# Translations *.mo + +# Mr Developer .mr.developer.cfg .project .pydevproject + +# Local development .venv/ __pycache__/ .pytest_cache/ diff --git a/README.md b/README.md index 30ef919..81e2d3f 100644 --- a/README.md +++ b/README.md @@ -1,74 +1,15 @@ -# Optimization paths +# gradient -This repository is a small, executable tutorial about **how optimization -algorithms move**. It implements classical smooth unconstrained methods from -first principles, runs them from shared starting points, and plots every -accepted iterate on the same objective contours. +Optimization path comparisons: -The code uses two two-dimensional teaching problems: +- Newton and modified Newton +- symmetric-rank-one quasi-Newton +- BFGS +- Fletcher–Reeves nonlinear conjugate gradient +- backtracking and interpolating Armijo line searches -- a symmetric positive-definite quadratic - `f(x) = 1/2 x^T A x - b^T x`, where `A = [[3, 2], [2, 6]]` and - `b = [1, -5]`; and -- the Rosenbrock function `f(x0, x1) = 10(x1 - x0^2)^2 + (1 - x0)^2`. - Notice the coefficient is **10**, rather than the frequently used 100. - -## What is implemented - -| Family | Method | Direction / update | Step selection and safeguard | -|---|---|---|---| -| First order | Steepest descent | `p = -g` | fixed, exact quadratic, or Armijo backtracking | -| Second order | Newton | solve `H p = -g` | full step | -| Second order | Damped Newton | Newton direction | Armijo backtracking | -| Second order | Modified Newton | adaptively shift `H` until positive definite | Armijo backtracking | -| Quasi-Newton | DFP | inverse-Hessian rank-two update | strong Wolfe; guarded curvature update | -| Quasi-Newton | BFGS | inverse-Hessian rank-two update | strong Wolfe; guarded curvature update | -| Quasi-Newton | SR1 | symmetric rank-one update | Armijo; skipped near-singular updates and descent restarts | -| Conjugate gradient | Linear CG | `A`-conjugate directions | exact steps on an SPD quadratic | -| Nonlinear CG | Fletcher–Reeves | gradient-norm beta | strong Wolfe plus periodic/descent restarts | -| Nonlinear CG | Polak–Ribiere+ | truncated PR beta | strong Wolfe plus periodic/descent restarts | - -The line searches are real, separately testable implementations: a fixed step, -an exact SPD-quadratic step, geometric Armijo backtracking, and strong Wolfe -bracketing with zoom. The old integer `interpolation` argument remains only on -compatibility aliases; new code should use the explicit `line_search` names. - -Every method can return an `OptimizationResult` containing its path, objective -history, gradient norms, accepted step sizes, convergence status, evaluation -counts, restarts, and skipped quasi-Newton updates. - -## 1. Curvature on a quadratic - -All panels below use the same start, contours, stopping tolerance, and exact -quadratic line search where appropriate. Steepest descent zig-zags across the -elliptical valley. Newton uses the true curvature and arrives in one step. -Linear CG and the quasi-Newton methods construct the missing curvature in two -steps in this two-dimensional example. - -![Optimization paths on a positive-definite quadratic](quadratic.png) - -## 2. Same start, different Rosenbrock paths - -Each method starts at `(-1.2, 1.0)`, targets `(1, 1)`, and stops when -`||gradient f|| <= 1e-8` or its stated budget is exhausted. Panel subtitles are -generated from the run results. Labels 0–3 identify the earliest accepted -iterates; later markers are logarithmically spaced so long runs stay readable. - -The full Newton path is intentionally shown even though it increases the -objective on some steps: that is the motivation for damping and modified -positive-definite Hessians. - -![Nine optimization paths on Rosenbrock](quasi_newton.png) - -## 3. Convergence histories - -Path geometry is only half the story. These plots compare objective reduction -and first-order convergence. The generated [metrics table](metrics.csv) adds -function, gradient, and Hessian evaluation totals, restarts, and skipped updates. - -![Rosenbrock convergence histories](cg.png) - -## Run the tutorial +The original figure layout and plotting style are preserved. Running `main.py` +regenerates the three figures with the corrected numerical paths. ```sh python3 -m venv .venv @@ -77,34 +18,6 @@ python3 -m venv .venv MPLBACKEND=Agg .venv/bin/python main.py ``` -`main.py` deterministically regenerates the three PNGs and `metrics.csv`. Use -`--output-dir PATH` to write a separate copy without changing tracked files. - -## API example - -```python -import func -import optMethods as opt - -objective = func.rosenbrock() -result = opt.minimize( - "bfgs", - [-1.2, 1.0], - objective, - max_iter=100, - gtol=1e-8, - line_search="strong_wolfe", -) - -print(result.x, result.iterations, result.nfev, result.gradient_norms[-1]) -``` - -`minimize` provides consistent `max_iter`/`gtol` names and always returns the -complete result. Direct method calls return the historical path array by default -for compatibility; pass `return_result=True` to those calls for diagnostics. - -## Julia directory - -`jl/` is the repository's original historical translation and targets a -pre-1.0 Julia/PyCall stack. It is not part of the tested tutorial. The Python -implementation and figures are the supported, reproducible version. +![Quadratic comparison](quadratic.png) +![Newton and quasi-Newton comparison](quasi_newton.png) +![Rosenbrock method comparison](cg.png) diff --git a/cg.png b/cg.png index 46a8f21a561c37c431e5cf9bb8fcb7985981f1f1..66ffb0e5f4a78e98ad3b0bfad0e4a59f597d2826 100644 GIT binary patch literal 135854 zcmd43bx_pt|L=<+4H8OsBi%>{3MeI#OLup7iqa`v0)l|B#L`{T(vs2*N_WS7uix`K z_nbL%=G>Wk|F}Ej!0fL3`Ru!1ujk|Sd=aknUK$&N6axVP0b5o^N(BMoaXA74(k&Va zct)LES{?ila($!es_J0w>S5$;hM-{N>S*iWYHMXo?QZ7mV&!1Z%OS+U$3|`G>gwns z%*ko@-`~LD;B3K3`FYV2d@J|XCb$uy z95Vg2x3Ze?IcPC;m_eh(2>XbLfx%o@NZqHRl!i!4jYh@HjL3rc@xyM;i$||#KJKBAind(b^Lw}Ge$7ZFJds;=VTs4*}3vp#-Wu}oc7OR}Td3&lfqYCF!*nb{_ZqB z6-b3$dOc$)mi1_RwKVVO5h#W?EcTc-mz*fWb#5l~)jFQ>(}NB&&yE96sbRI|S58t( z<7AvfKZ&B`An{+k-$|{D`SS6P*;e&>GJ)F+3+t7PrN9YowdN(jL(;g26`X+^XiF07k7WlHfb|A@snjzGEUL z{X!suKvQbnS6?r1( z>EYSHJ?g~G#aUSRFIrI#`)4ePBSuMNQl5wdKEgrXXe(cpQpsvCi=HFIE5(2c<+^2C{mwh|es1QkBc 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zn)uHU<6!)dQEeQaxVQ88okEc5mKkVw^KCFEk1j&cNV-)V+V3E~AHwV3?2Jf;Nigqx zo7vvnD_59zzxnTrDUVySK=%`*={Cx9*bL@xR)HF-4yL%)Rys^mV9&c%h(N)C1tG1O zBgvR{C{c4fX}5cYeU2;sWF|q)Bx`=Eno0bmWRfw4`r6D?Iq0Ykap2dVqq&%dYR0Tc zjsM*l$)$?%BLnP5F9nxV3D5=JN9yv`&uH!j*t{Ce`wJ+5aSI>Y`+j zfv_R0=!C^ycfO&uXE2XAso0@nne_TPFiEz$`NQAl;^K~drJ!iDI6%7-kaHw^#CfM| z9-*(uzZyNPVU`v!cbof#rc2ciCBFo;n+iJ^h#NFeT-P>DYcR%2CvV=CGyQJSSDe<& z_p 0] - lower = max(float(positive.min()), 1e-3) - upper = float(Z.max()) - levels = np.asarray(levels) if levels is not None else np.geomspace(lower, upper, 24) - filled = ax.contourf( - X, - Y, - Z, - levels=levels, - norm=LogNorm(float(levels[0]), float(levels[-1])), - cmap=cmap, - extend="both", - ) - ax.contour( - X, Y, Z, levels=levels[::3], colors="#8a8a8a", linewidths=0.45, alpha=0.7 - ) - else: - levels = ( - np.asarray(levels) - if levels is not None - else np.linspace(float(Z.min()), float(Z.max()), 24) - ) - filled = ax.contourf(X, Y, Z, levels=levels, cmap=cmap, extend="both") - ax.contour( - X, Y, Z, levels=levels[::3], colors="#8a8a8a", linewidths=0.45, alpha=0.7 - ) - return filled + + if filled: + return plt.contourf(X, Y, Z, 20, cmap=plt.cm.YlGnBu) + contours = plt.contour(X, Y, Z, levels=levels, colors="black") + plt.clabel(contours, fmt="%.3f", inline=True) + return contours diff --git a/main.py b/main.py index f837c6a..4a060b1 100644 --- a/main.py +++ b/main.py @@ -1,11 +1,8 @@ -"""Generate the tutorial's path and convergence comparison figures.""" +"""Regenerate the repository's three original-style path figures.""" -import argparse -import csv from pathlib import Path import matplotlib.pyplot as plt -from matplotlib.lines import Line2D import numpy as np import drawFunc @@ -14,311 +11,137 @@ ROOT = Path(__file__).resolve().parent -TOLERANCE = 1e-8 -QUADRATIC_START = np.array([0.0, 2.0]) -ROSENBROCK_START = np.array([-1.2, 1.0]) -ROSENBROCK_BOUNDS = ((-1.6, 1.35), (-1.1, 1.75)) -ROSENBROCK_LEVELS = np.geomspace(1e-3, 1e3, 25) -METHOD_STYLES = { - "Steepest descent": ("#0072B2", "o", "-"), - "Newton": ("#D55E00", "s", "-"), - "Damped Newton": ("#E69F00", "D", "-"), - "Modified Newton": ("#CC79A7", "P", "-"), - "DFP": ("#56B4E9", "v", "-"), - "BFGS": ("#009E73", "^", "-"), - "SR1": ("#F0A202", "X", "-"), - "Fletcher-Reeves": ("#6A3D9A", "<", "-"), - "Polak-Ribiere+": ("#333333", ">", "-"), - "Linear conjugate gradient": ("#6A3D9A", "<", "-"), -} +def _plot(track, *args, **kwargs): + return plt.plot(track[0], track[1], *args, **kwargs)[0] -def _marker_indices(number_of_points, maximum=12): - if number_of_points <= maximum: - return np.arange(number_of_points) - indices = np.geomspace(1, number_of_points - 1, maximum - 1).round().astype(int) - return np.unique(np.concatenate(([0], indices))) +def quadratic(output=ROOT / "quadratic.png"): + objective = func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) + start = np.array([0.0, 2.0]) -def _plot_result(ax, result, number_early_steps=True): - color, marker, line_style = METHOD_STYLES[result.method] - path = result.path - marker_indices = _marker_indices(path.shape[1]) - ax.plot( - path[0], - path[1], - color=color, - linestyle=line_style, - linewidth=1.8, - marker=marker, - markevery=marker_indices.tolist(), - markersize=4.5, - markeredgecolor="white", - markeredgewidth=0.55, - zorder=4, + plt.figure(figsize=(8, 6)) + drawFunc.draw( + objective.f_x, + filled=False, + x_bounds=(-1.5, 2.3), + y_bounds=(-3.0, 3.5), + levels=np.arange(0, 68, 4), ) - if number_early_steps: - for index in range(min(4, path.shape[1])): - ax.annotate( - str(index), - path[:, index], - xytext=(4, 4), - textcoords="offset points", - fontsize=7, - color=color, - zorder=6, - ) - - -def _decorate_path_axis(ax, result, start, x_bounds, y_bounds): - ax.scatter( - *start, - facecolor="white", - edgecolor="#222222", - linewidth=1.2, - s=35, - zorder=7, + newton_path = opt.newton(start, objective, iterations=1) + sr1_path = opt.quasi_newton(start, objective, iteration=100) + newton_line = _plot(newton_path, "b-") + sr1_line = _plot( + sr1_path, + "go-", + linewidth=2.0, + markeredgecolor="black", + markeredgewidth=0.5, ) - ax.scatter(1, 1, color="#222222", marker="*", s=85, zorder=7) - state = "converged" if result.converged else "budget reached" - ax.set_title( - f"{result.method}\n" - f"{result.iterations} steps · ||g||={result.gradient_norms[-1]:.1e} · {state}", - fontsize=9, + plt.legend( + [newton_line, sr1_line], + ["Newton", "Quasi_Newton_Rank_One"], + loc="upper right", ) - ax.set_xlim(x_bounds) - ax.set_ylim(y_bounds) - ax.set_aspect("equal", adjustable="box") - ax.tick_params(labelsize=8) + plt.xlim(-1.5, 2.3) + plt.ylim(-3.0, 3.5) + plt.tight_layout() + plt.savefig(output, dpi=100) + plt.close() + +def quasi_newton(output=ROOT / "quasi_newton.png"): + objective = func.rosenbrock() + start = np.array([-1.0, 2.0]) -def _draw_rosenbrock_background(ax, objective): - return drawFunc.draw( + plt.figure(figsize=(8, 6)) + drawFunc.draw( objective.f_x, - ax, - x_bounds=ROSENBROCK_BOUNDS[0], - y_bounds=ROSENBROCK_BOUNDS[1], - resolution=220, - log_scale=True, - levels=ROSENBROCK_LEVELS, - cmap="Greys", + filled=False, + x_bounds=(-1.5, 2.0), + y_bounds=(-3.0, 3.5), + levels=[0, 10, 25, 50, 100, 150, 200, 250], + ) + plt.annotate( + "Start", + xy=start, + xytext=(-1.2, 2.2), + arrowprops=dict(facecolor="black", shrink=0.02), + ) + plt.annotate( + "Optimal", + xy=(1, 1), + xytext=(0.5, 1.2), + arrowprops=dict(facecolor="black", shrink=0.02), ) - -def _quadratic_runs(objective): - return [ - opt.gradient_descent( - QUADRATIC_START, - objective, - iteration=100, - line_search="exact", - tolerance=TOLERANCE, - return_result=True, - ), - opt.newton( - QUADRATIC_START, - objective, - iterations=10, - tolerance=TOLERANCE, - return_result=True, - ), - opt.linear_conjugate_gradient( - QUADRATIC_START, - objective, - tolerance=TOLERANCE, - return_result=True, - ), - opt.dfp( - QUADRATIC_START, - objective, - iteration=10, - line_search="exact", - tolerance=TOLERANCE, - return_result=True, - ), - opt.bfgs( - QUADRATIC_START, - objective, - iteration=10, - line_search="exact", - tolerance=TOLERANCE, - return_result=True, - ), - opt.sr1( - QUADRATIC_START, - objective, - iteration=10, - line_search="exact", - tolerance=TOLERANCE, - return_result=True, - ), + paths = [ + (opt.newton(start, objective, iterations=20), "bo-", "Newton"), + (opt.newton(start, objective, modified=0.1, iterations=80), "go-", "Modified_Newton 0.10"), + (opt.newton(start, objective, modified=1.5, iterations=100), "ro-", "Modified_Newton 1.50"), + (opt.quasi_newton(start, objective, iteration=250), "co-", "Quasi_Newton_Rank_One"), ] - - -def _rosenbrock_runs(objective): - common = dict(tolerance=TOLERANCE, return_result=True) - return [ - opt.gradient_descent(ROSENBROCK_START, objective, iteration=2000, **common), - opt.newton(ROSENBROCK_START, objective, iterations=100, **common), - opt.damped_newton(ROSENBROCK_START, objective, iteration=100, **common), - opt.modified_newton( - ROSENBROCK_START, - objective, - iteration=300, - min_curvature=1.0, - **common, - ), - opt.dfp(ROSENBROCK_START, objective, iteration=300, **common), - opt.bfgs(ROSENBROCK_START, objective, iteration=300, **common), - opt.sr1(ROSENBROCK_START, objective, iteration=300, **common), - opt.fletcher_reeves(ROSENBROCK_START, objective, iteration=500, **common), - opt.polak_ribiere_plus(ROSENBROCK_START, objective, iteration=500, **common), + lines = [ + _plot(path, style, linewidth=2.0, markersize=4, markeredgecolor="black", markeredgewidth=0.4) + for path, style, _ in paths ] + plt.legend(lines, [label for _, _, label in paths], loc="upper right") + plt.xlim(-1.5, 2.0) + plt.ylim(-3.0, 3.5) + plt.tight_layout() + plt.savefig(output, dpi=100) + plt.close() -def quadratic(output=ROOT / "quadratic.png"): - """Compare curvature-learning methods on one SPD quadratic.""" - objective = func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) - results = _quadratic_runs(objective) - x_bounds, y_bounds = (-0.8, 1.45), (-1.5, 2.2) - levels = np.linspace(-3.8, 20, 24) - fig, axes = plt.subplots(2, 3, figsize=(11, 9), sharex=True, sharey=True) - for ax, result in zip(axes.flat, results): - drawFunc.draw( - objective.f_x, - ax, - x_bounds=x_bounds, - y_bounds=y_bounds, - resolution=180, - levels=levels, - cmap="Greys", - ) - _plot_result(ax, result) - optimum = np.linalg.solve(objective.A, objective.b) - ax.scatter(*QUADRATIC_START, facecolor="white", edgecolor="#222222", s=35, zorder=7) - ax.scatter(*optimum, color="#222222", marker="*", s=85, zorder=7) - ax.set_title( - f"{result.method}\n{result.iterations} steps · ||g||={result.gradient_norms[-1]:.1e}", - fontsize=9, - ) - ax.set_xlim(x_bounds) - ax.set_ylim(y_bounds) - ax.set_aspect("equal", adjustable="box") - fig.suptitle("Curvature and conjugacy on a positive-definite quadratic", fontsize=15) - fig.supxlabel("$x_0$") - fig.supylabel("$x_1$") - fig.tight_layout(rect=(0.02, 0.02, 1, 0.96)) - fig.savefig(output, dpi=180) - plt.close(fig) - return results - - -def rosenbrock_paths(output=ROOT / "quasi_newton.png"): - """Compare nine methods from an identical Rosenbrock starting point.""" +def comparison(output=ROOT / "cg.png"): objective = func.rosenbrock() - results = _rosenbrock_runs(objective) - fig, axes = plt.subplots(3, 3, figsize=(12, 12), sharex=True, sharey=True) - contour = None - for ax, result in zip(axes.flat, results): - contour = _draw_rosenbrock_background(ax, objective) - _plot_result(ax, result) - _decorate_path_axis( - ax, result, ROSENBROCK_START, *ROSENBROCK_BOUNDS - ) - handles = [ - Line2D([], [], marker="o", markerfacecolor="white", markeredgecolor="#222222", linestyle="none", label="start"), - Line2D([], [], marker="*", color="#222222", linestyle="none", markersize=9, label="optimum"), - ] - fig.legend(handles=handles, loc="upper center", ncol=2, frameon=False, bbox_to_anchor=(0.5, 0.965)) - fig.suptitle("Optimization paths on the Rosenbrock function (coefficient 10)", fontsize=15) - fig.supxlabel("$x_0$") - fig.supylabel("$x_1$") - colorbar_axis = fig.add_axes([0.9, 0.2, 0.018, 0.6]) - colorbar = fig.colorbar(contour, cax=colorbar_axis) - colorbar.set_label("objective $f(x)$ — logarithmic scale") - fig.subplots_adjust(left=0.07, right=0.86, bottom=0.06, top=0.92, wspace=0.14, hspace=0.27) - fig.savefig(output, dpi=180) - plt.close(fig) - return results - - -def convergence(results, output=ROOT / "cg.png"): - """Compare objective and first-order convergence by iteration.""" - fig, axes = plt.subplots(1, 2, figsize=(12, 6.2)) - for result in results: - color, marker, line_style = METHOD_STYLES[result.method] - iterations = np.arange(result.iterations + 1) - objective_gap = np.maximum(result.objective_values, np.finfo(float).eps) - gradient_norm = np.maximum(result.gradient_norms, np.finfo(float).eps) - axes[0].semilogy( - iterations, - objective_gap, - color=color, - linestyle=line_style, - linewidth=1.7, - label=result.method, - ) - axes[1].semilogy( - iterations, - gradient_norm, - color=color, - linestyle=line_style, - linewidth=1.7, - label=result.method, - ) - axes[0].set(title="Objective gap by iteration", xlabel="accepted steps", ylabel="$f(x_k)-f^*$") - axes[1].set(title="First-order convergence", xlabel="accepted steps", ylabel="$||\\nabla f(x_k)||_2$") - for ax in axes: - ax.set_xscale("symlog", linthresh=10) - ax.grid(True, which="both", color="#d0d0d0", linewidth=0.5) - handles, labels = axes[1].get_legend_handles_labels() - fig.legend(handles, labels, loc="lower center", bbox_to_anchor=(0.5, 0.015), ncol=3, frameon=False) - fig.suptitle("Rosenbrock convergence histories", fontsize=15) - fig.tight_layout(rect=(0, 0.17, 1, 0.94)) - fig.savefig(output, dpi=180) - plt.close(fig) - + start = np.array([-1.5, -4.0]) + + plt.figure(figsize=(10, 8.5)) + drawFunc.draw(objective.f_x, filled=True) + plt.annotate( + "Start", + xy=start, + xytext=(-1.8, -3.5), + arrowprops=dict(facecolor="black", shrink=0.02), + ) + plt.annotate( + "Optimal", + xy=(1, 1), + xytext=(1.1, 1.4), + arrowprops=dict(facecolor="black", shrink=0.02), + ) -def write_metrics(results, output=ROOT / "metrics.csv"): - with Path(output).open("w", newline="") as stream: - writer = csv.writer(stream) - writer.writerow( - ["method", "status", "iterations", "final_f", "final_gradient_norm", "nfev", "njev", "nhev", "restarts", "skipped_updates"] - ) - for result in results: - writer.writerow( - [ - result.method, - result.status, - result.iterations, - f"{result.objective_values[-1]:.12g}", - f"{result.gradient_norms[-1]:.12g}", - result.nfev, - result.njev, - result.nhev, - result.restarts, - result.skipped_updates, - ] - ) + paths = [ + (opt.newton(start, objective, modified=0.1, iterations=80), "Modified_Newton 0.10"), + (opt.newton(start, objective, modified=1.5, iterations=100), "Modified_Newton 1.50"), + (opt.quasi_newton(start, objective, iteration=250, interpolation=1), "Quasi_Newton_trivial"), + (opt.quasi_newton(start, objective, iteration=450), "Quasi_Newton_Armijo"), + (opt.BFGS(start, objective, iteration=350, interpolation=1), "BFGS_trivial"), + (opt.BFGS(start, objective, iteration=350), "BFGS_Armijo"), + (opt.fletcher_reeves(start, objective, iteration=80, alpha=0.001), "CG_FR_0.00100"), + (opt.fletcher_reeves(start, objective, iteration=80, alpha=0.01), "CG_FR_0.01000"), + ] + lines = [ + _plot(path, "o-", linewidth=2.0, markersize=4, markeredgecolor="black", markeredgewidth=0.4) + for path, _ in paths + ] + plt.legend(lines, [label for _, label in paths], loc="upper right") + plt.xlim(-3.0, 2.5) + plt.ylim(-4.5, 4.5) + plt.tight_layout() + plt.savefig(output, dpi=120) + plt.close() def generate_figures(output_dir=ROOT): output_dir = Path(output_dir) output_dir.mkdir(parents=True, exist_ok=True) quadratic(output_dir / "quadratic.png") - results = rosenbrock_paths(output_dir / "quasi_newton.png") - convergence(results, output_dir / "cg.png") - write_metrics(results, output_dir / "metrics.csv") - return results + quasi_newton(output_dir / "quasi_newton.png") + comparison(output_dir / "cg.png") if __name__ == "__main__": - parser = argparse.ArgumentParser(description=__doc__) - parser.add_argument( - "--output-dir", - type=Path, - default=ROOT, - help="directory for PNG and CSV outputs (default: repository root)", - ) - generate_figures(parser.parse_args().output_dir) + generate_figures() diff --git a/metrics.csv b/metrics.csv deleted file mode 100644 index 7884418..0000000 --- a/metrics.csv +++ /dev/null @@ -1,10 +0,0 @@ -method,status,iterations,final_f,final_gradient_norm,nfev,njev,nhev,restarts,skipped_updates -Steepest descent,converged,1707,7.28237459778e-17,9.68790243558e-09,12981,3415,0,0,0 -Newton,converged,8,5.80417969563e-27,1.07722122435e-12,0,9,8,0,0 -Damped Newton,converged,13,0,0,28,27,13,0,0 -Modified Newton,converged,40,7.65784627779e-17,7.76493723237e-09,81,81,40,0,0 -DFP,converged,20,5.13742520052e-22,2.2093973997e-10,54,65,0,0,0 -BFGS,converged,19,4.40618112977e-20,2.31969163034e-09,52,62,0,0,0 -SR1,converged,21,2.55776724937e-22,2.27806998352e-10,68,43,0,3,0 -Fletcher-Reeves,converged,45,4.71631088674e-17,8.92575477216e-09,311,168,0,22,0 -Polak-Ribiere+,converged,95,6.22339641888e-17,8.79199318821e-09,652,344,0,47,0 diff --git a/optMethods.py b/optMethods.py index 5c3428f..0ce5bdd 100644 --- a/optMethods.py +++ b/optMethods.py @@ -1,277 +1,40 @@ -"""Educational implementations of unconstrained optimization algorithms. - -By default every public method returns a ``(dimensions, iterations + 1)`` -array. Column zero is the starting point and each following column is an -accepted iterate. Pass ``return_result=True`` for histories, status, and work -counts. The implementations intentionally share a small line-search interface -so the direction can be compared independently of step selection. -""" - -from dataclasses import dataclass, replace +"""Optimization methods used by the path-comparison figures.""" import numpy as np -@dataclass(frozen=True) -class OptimizationResult: - """Trajectory and diagnostics from one optimization run.""" - - method: str - path: np.ndarray - objective_values: np.ndarray - gradient_norms: np.ndarray - step_sizes: np.ndarray - converged: bool - status: str - nfev: int - njev: int - nhev: int - restarts: int = 0 - skipped_updates: int = 0 - - @property - def x(self): - return self.path[:, -1] - - @property - def iterations(self): - return self.path.shape[1] - 1 - - -class _CountingObjective: - """Count derivative work without changing the objective classes.""" - - def __init__(self, objective): - self.objective = objective - self.nfev = self.njev = self.nhev = 0 - - def f_x(self, x): - self.nfev += 1 - return self.objective.f_x(x) - - def g_x(self, x): - self.njev += 1 - return self.objective.g_x(x) - - def G_x(self, x): - self.nhev += 1 - return self.objective.G_x(x) - - -def _initialise(start_point): +def _start(start_point): x = np.asarray(start_point, dtype=float).reshape(-1).copy() return x, [x.copy()] -def _finish(track): - return np.column_stack(track) +def _track(points): + return np.column_stack(points) def _gradient(obj_fun, x): return np.asarray(obj_fun.g_x(x), dtype=float).reshape(-1) -def _validate_run(max_iter, tolerance): - if not isinstance(max_iter, (int, np.integer)) or max_iter < 0: - raise ValueError("maximum iterations must be a non-negative integer") - if not np.isfinite(tolerance) or tolerance <= 0: - raise ValueError("tolerance must be positive and finite") - - -def _result_or_path( - objective, - oracle, - track, - step_sizes, - method, - tolerance, - return_result, - restarts=0, - skipped_updates=0, -): - path = _finish(track) - if not return_result: - return path - values = np.array([objective.f_x(path[:, index]) for index in range(path.shape[1])]) - gradient_norms = np.array( - [np.linalg.norm(objective.g_x(path[:, index])) for index in range(path.shape[1])] - ) - converged = bool(gradient_norms[-1] <= tolerance) - return OptimizationResult( - method=method, - path=path, - objective_values=values, - gradient_norms=gradient_norms, - step_sizes=np.asarray(step_sizes), - converged=converged, - status="converged" if converged else "maximum iterations reached", - nfev=oracle.nfev, - njev=oracle.njev, - nhev=oracle.nhev, - restarts=restarts, - skipped_updates=skipped_updates, - ) - - -def gradient_descent( - start_point, - obj_fun, - iteration=200, - alpha=1.0, - line_search="armijo", - tolerance=1e-8, - return_result=False, -): - """Steepest descent using ``p_k = -gradient f(x_k)``. - - ``line_search="exact"`` uses the quadratic model's exact minimizer and is - intended for the quadratic tutorial. ``"armijo"`` works for either - objective, while ``"fixed"`` always uses ``alpha``. - """ - _validate_run(iteration, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) - step_sizes = [] - for _ in range(iteration): - gradient = _gradient(obj_fun, x) - if np.linalg.norm(gradient) <= tolerance: - break - direction = -gradient - step_size = _choose_step(obj_fun, x, direction, line_search, alpha) - x = x + step_size * direction - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, obj_fun, track, step_sizes, "Steepest descent", tolerance, return_result - ) - - -def newton( - start_point, - obj_fun, - modified=0.0, - iterations=20, - line_search="fixed", - alpha=1.0, - tolerance=1e-8, - return_result=False, -): - """Newton's method, optionally with a fixed diagonal Hessian shift. - - ``modified`` preserves the original tutorial API: a positive value solves - ``(H + modified * I) p = -g``. See :func:`modified_newton` for an adaptive - positive-definite shift. - """ - _validate_run(iterations, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) - step_sizes = [] +def newton(start_point, obj_fun, modified=0, iterations=5, tolerance=1e-10): + """Newton's method with the tutorial's optional fixed Hessian shift.""" + x, points = _start(start_point) identity = np.eye(x.size) for _ in range(iterations): gradient = _gradient(obj_fun, x) if np.linalg.norm(gradient) <= tolerance: break hessian = np.asarray(obj_fun.G_x(x), dtype=float) + modified * identity - direction = np.linalg.solve(hessian, -gradient) - if line_search != "fixed" and direction @ gradient >= 0: - direction = -gradient - step_size = _choose_step(obj_fun, x, direction, line_search, alpha) - x = x + step_size * direction - track.append(x.copy()) - step_sizes.append(step_size) - name = "Fixed-shift Newton" if modified else "Newton" - return _result_or_path( - objective, obj_fun, track, step_sizes, name, tolerance, return_result - ) - - -def modified_newton( - start_point, - obj_fun, - iteration=100, - min_curvature=1e-3, - alpha=1.0, - tolerance=1e-8, - return_result=False, -): - """Newton's method with an adaptive eigenvalue shift and Armijo search. - - At every iteration the Hessian is shifted just enough to make its smallest - eigenvalue at least ``min_curvature``. This produces a descent direction - even where the unmodified Hessian is indefinite. - """ - _validate_run(iteration, tolerance) - if not np.isfinite(min_curvature) or min_curvature <= 0: - raise ValueError("min_curvature must be positive and finite") - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) - step_sizes = [] - identity = np.eye(x.size) - for _ in range(iteration): - gradient = _gradient(obj_fun, x) - if np.linalg.norm(gradient) <= tolerance: - break - hessian = np.asarray(obj_fun.G_x(x), dtype=float) - smallest_eigenvalue = float(np.linalg.eigvalsh(hessian).min()) - shift = max(0.0, min_curvature - smallest_eigenvalue) - direction = np.linalg.solve(hessian + shift * identity, -gradient) - step_size = _armijo_line_search(obj_fun, x, direction, alpha0=alpha) - x = x + step_size * direction - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, obj_fun, track, step_sizes, "Modified Newton", tolerance, return_result - ) - + x = x + np.linalg.solve(hessian, -gradient) + points.append(x.copy()) + return _track(points) -def damped_newton( - start_point, - obj_fun, - iteration=100, - alpha=1.0, - tolerance=1e-8, - return_result=False, -): - """Newton direction globalized by an Armijo backtracking line search.""" - result = newton( - start_point, - obj_fun, - iterations=iteration, - line_search="armijo", - alpha=alpha, - tolerance=tolerance, - return_result=return_result, - ) - return replace(result, method="Damped Newton") if return_result else result - -def bfgs( - start_point, - obj_fun, - iteration=100, - alpha=1.0, - line_search="strong_wolfe", - tolerance=1e-8, - return_result=False, -): - """BFGS inverse-Hessian quasi-Newton method. - - The inverse update is applied only when the curvature condition is safely - positive. If round-off produces a non-descent direction, the approximation - is restarted from the identity. - """ - _validate_run(iteration, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) +def BFGS(start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-10): + """Inverse-BFGS with a guarded curvature update.""" + x, points = _start(start_point) inverse_hessian = np.eye(x.size) gradient = _gradient(obj_fun, x) - step_sizes = [] - restarts = 0 - skipped_updates = 0 for _ in range(iteration): if np.linalg.norm(gradient) <= tolerance: @@ -280,70 +43,41 @@ def bfgs( if direction @ gradient >= 0: inverse_hessian = np.eye(x.size) direction = -gradient - restarts += 1 - step_size = _choose_step(obj_fun, x, direction, line_search, alpha) - step = step_size * direction + alpha = _line_search(obj_fun, x, direction, interpolation) + step = alpha * direction next_x = x + step next_gradient = _gradient(obj_fun, next_x) y = next_gradient - gradient curvature = float(step @ y) - threshold = 1e-12 * np.linalg.norm(step) * np.linalg.norm(y) - if curvature > threshold: + if curvature > 1e-12 * np.linalg.norm(step) * np.linalg.norm(y): rho = 1.0 / curvature identity = np.eye(x.size) left = identity - rho * np.outer(step, y) inverse_hessian = ( left @ inverse_hessian @ left.T + rho * np.outer(step, step) ) - else: - skipped_updates += 1 x, gradient = next_x, next_gradient - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, - obj_fun, - track, - step_sizes, - "BFGS", - tolerance, - return_result, - restarts, - skipped_updates, - ) + points.append(x.copy()) + return _track(points) -def sr1( - start_point, - obj_fun, - iteration=100, - alpha=1.0, - line_search="armijo", - tolerance=1e-8, - return_result=False, +def quasi_newton( + start_point, obj_fun, iteration=10, interpolation=0, tolerance=1e-10 ): - """Symmetric-rank-one (SR1) inverse-Hessian quasi-Newton method.""" - _validate_run(iteration, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) + """Symmetric-rank-one inverse-Hessian quasi-Newton method.""" + x, points = _start(start_point) inverse_hessian = np.eye(x.size) gradient = _gradient(obj_fun, x) - step_sizes = [] - restarts = 0 - skipped_updates = 0 for _ in range(iteration): if np.linalg.norm(gradient) <= tolerance: break direction = -inverse_hessian @ gradient - # Unlike BFGS, SR1 approximations can be indefinite. if direction @ gradient >= 0: inverse_hessian = np.eye(x.size) direction = -gradient - restarts += 1 - step_size = _choose_step(obj_fun, x, direction, line_search, alpha) - step = step_size * direction + alpha = _line_search(obj_fun, x, direction, interpolation) + step = alpha * direction next_x = x + step next_gradient = _gradient(obj_fun, next_x) y = next_gradient - gradient @@ -352,419 +86,81 @@ def sr1( threshold = 1e-8 * np.linalg.norm(residual) * np.linalg.norm(y) if abs(denominator) > threshold: inverse_hessian += np.outer(residual, residual) / denominator - else: - skipped_updates += 1 x, gradient = next_x, next_gradient - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, - obj_fun, - track, - step_sizes, - "SR1", - tolerance, - return_result, - restarts, - skipped_updates, - ) - - -def dfp( - start_point, - obj_fun, - iteration=100, - alpha=1.0, - line_search="strong_wolfe", - tolerance=1e-8, - return_result=False, -): - """Davidon-Fletcher-Powell inverse-Hessian quasi-Newton method.""" - _validate_run(iteration, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) - inverse_hessian = np.eye(x.size) - gradient = _gradient(obj_fun, x) - step_sizes = [] - restarts = 0 - skipped_updates = 0 - - for _ in range(iteration): - if np.linalg.norm(gradient) <= tolerance: - break - direction = -inverse_hessian @ gradient - if direction @ gradient >= 0: - inverse_hessian = np.eye(x.size) - direction = -gradient - restarts += 1 - step_size = _choose_step(obj_fun, x, direction, line_search, alpha) - step = step_size * direction - next_x = x + step - next_gradient = _gradient(obj_fun, next_x) - y = next_gradient - gradient - step_curvature = float(step @ y) - hessian_y = inverse_hessian @ y - model_curvature = float(y @ hessian_y) - threshold = 1e-12 * np.linalg.norm(step) * np.linalg.norm(y) - if step_curvature > threshold and model_curvature > threshold: - inverse_hessian += ( - np.outer(step, step) / step_curvature - - np.outer(hessian_y, hessian_y) / model_curvature - ) - else: - skipped_updates += 1 - x, gradient = next_x, next_gradient - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, - obj_fun, - track, - step_sizes, - "DFP", - tolerance, - return_result, - restarts, - skipped_updates, - ) + # Record the accepted point, not the previous iterate. + points.append(x.copy()) + return _track(points) def fletcher_reeves( - start_point, - obj_fun, - iteration=200, - alpha=1.0, - line_search="strong_wolfe", - tolerance=1e-8, - return_result=False, + start_point, obj_fun, iteration=10, alpha=0.1, tolerance=1e-10 ): - """Fletcher-Reeves nonlinear conjugate-gradient method. - - Directions are restarted every ``n`` iterations (and whenever they cease to - be descent directions), a standard finite-precision safeguard. - """ - _validate_run(iteration, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) + """Fletcher-Reeves nonlinear conjugate gradient with safe restarts.""" + x, points = _start(start_point) gradient = _gradient(obj_fun, x) direction = -gradient - step_sizes = [] - restarts = 0 - for k in range(iteration): + for _ in range(iteration): if np.linalg.norm(gradient) <= tolerance: break if direction @ gradient >= 0: direction = -gradient - restarts += 1 - step_size = _choose_step( - obj_fun, x, direction, line_search, alpha, wolfe_c2=0.4 - ) + step_size = _armijo_line_search(obj_fun, x, direction, alpha0=alpha) next_x = x + step_size * direction next_gradient = _gradient(obj_fun, next_x) denominator = float(gradient @ gradient) beta = float(next_gradient @ next_gradient) / denominator if denominator else 0.0 - if (k + 1) % x.size == 0: - direction = -next_gradient - restarts += 1 - else: - direction = -next_gradient + beta * direction + direction = -next_gradient + beta * direction x, gradient = next_x, next_gradient - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, - obj_fun, - track, - step_sizes, - "Fletcher-Reeves", - tolerance, - return_result, - restarts, - ) + points.append(x.copy()) + return _track(points) -def polak_ribiere_plus( - start_point, - obj_fun, - iteration=200, - alpha=1.0, - line_search="strong_wolfe", - tolerance=1e-8, - return_result=False, -): - """Polak-Ribiere+ nonlinear conjugate gradient with safe restarts.""" - _validate_run(iteration, tolerance) - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) - gradient = _gradient(obj_fun, x) - direction = -gradient - step_sizes = [] - restarts = 0 - - for k in range(iteration): - if np.linalg.norm(gradient) <= tolerance: - break - if direction @ gradient >= 0: - direction = -gradient - restarts += 1 - step_size = _choose_step( - obj_fun, x, direction, line_search, alpha, wolfe_c2=0.4 - ) - next_x = x + step_size * direction - next_gradient = _gradient(obj_fun, next_x) - denominator = float(gradient @ gradient) - beta = max( - 0.0, - float(next_gradient @ (next_gradient - gradient)) / denominator - if denominator - else 0.0, - ) - if (k + 1) % x.size == 0: - direction = -next_gradient - restarts += 1 - else: - direction = -next_gradient + beta * direction - x, gradient = next_x, next_gradient - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, - obj_fun, - track, - step_sizes, - "Polak-Ribiere+", - tolerance, - return_result, - restarts, - ) - - -def linear_conjugate_gradient( - start_point, - obj_fun, - iteration=None, - tolerance=1e-10, - return_result=False, -): - """Linear conjugate gradient for a symmetric positive-definite quadratic. - - In exact arithmetic this method reaches the minimizer in at most ``n`` - iterations. It is intentionally separate from nonlinear Fletcher-Reeves. - """ - objective = obj_fun - obj_fun = _CountingObjective(objective) - x, track = _initialise(start_point) - max_iter = x.size if iteration is None else iteration - _validate_run(max_iter, tolerance) - hessian = np.asarray(obj_fun.G_x(x), dtype=float) - if hessian.shape != (x.size, x.size) or not np.allclose(hessian, hessian.T): - raise ValueError("linear CG requires a symmetric quadratic Hessian") - try: - np.linalg.cholesky(hessian) - except np.linalg.LinAlgError as error: - raise ValueError("linear CG requires a positive-definite quadratic") from error - - residual = -_gradient(obj_fun, x) - direction = residual.copy() - residual_norm_squared = float(residual @ residual) - step_sizes = [] - for _ in range(max_iter): - if np.sqrt(residual_norm_squared) <= tolerance: - break - hessian_direction = hessian @ direction - curvature = float(direction @ hessian_direction) - step_size = residual_norm_squared / curvature - x = x + step_size * direction - next_residual = residual - step_size * hessian_direction - next_norm_squared = float(next_residual @ next_residual) - beta = next_norm_squared / residual_norm_squared - direction = next_residual + beta * direction - residual = next_residual - residual_norm_squared = next_norm_squared - track.append(x.copy()) - step_sizes.append(step_size) - return _result_or_path( - objective, - obj_fun, - track, - step_sizes, - "Linear conjugate gradient", - tolerance, - return_result, - ) - - -def minimize(method, start_point, obj_fun, max_iter=100, gtol=1e-8, **options): - """Run any tutorial method through one consistent result-returning API. - - Supported names are ``gradient_descent``, ``newton``, ``damped_newton``, - ``modified_newton``, ``dfp``, ``bfgs``, ``sr1``, ``linear_cg``, - ``fletcher_reeves``, and ``polak_ribiere_plus``. - """ - methods = { - "gradient_descent": (gradient_descent, "iteration"), - "newton": (newton, "iterations"), - "damped_newton": (damped_newton, "iteration"), - "modified_newton": (modified_newton, "iteration"), - "dfp": (dfp, "iteration"), - "bfgs": (bfgs, "iteration"), - "sr1": (sr1, "iteration"), - "linear_cg": (linear_conjugate_gradient, "iteration"), - "fletcher_reeves": (fletcher_reeves, "iteration"), - "polak_ribiere_plus": (polak_ribiere_plus, "iteration"), - } - try: - function, iteration_name = methods[method] - except KeyError as error: - choices = ", ".join(methods) - raise ValueError(f"unknown method {method!r}; choose from {choices}") from error - options[iteration_name] = max_iter - options["tolerance"] = gtol - options["return_result"] = True - return function(start_point, obj_fun, **options) - - -def BFGS(start_point, obj_fun, iteration=100, interpolation=0, **kwargs): - """Backward-compatible alias for :func:`bfgs`.""" - return bfgs(start_point, obj_fun, iteration=iteration, **kwargs) - - -def quasi_newton(start_point, obj_fun, iteration=100, interpolation=0, **kwargs): - """Backward-compatible alias for the tutorial's original SR1 function.""" - return sr1(start_point, obj_fun, iteration=iteration, **kwargs) - - -def _choose_step(obj_fun, x, direction, line_search, alpha, wolfe_c2=0.9): - if line_search == "fixed": - if not np.isfinite(alpha) or alpha <= 0: - raise ValueError("a fixed step must be positive and finite") - return float(alpha) - if line_search == "armijo": - return _armijo_line_search(obj_fun, x, direction, alpha0=alpha) - if line_search == "exact": - return _exact_quadratic_line_search(obj_fun, x, direction) - if line_search == "strong_wolfe": - return _strong_wolfe_line_search( - obj_fun, x, direction, alpha0=alpha, c2=wolfe_c2 - ) - raise ValueError( - "line_search must be 'fixed', 'armijo', 'strong_wolfe', or 'exact'" - ) - - -def _exact_quadratic_line_search(obj_fun, x, direction): - """Exact step along ``direction`` for an objective with constant Hessian.""" - gradient = _gradient(obj_fun, x) - hessian = np.asarray(obj_fun.G_x(x), dtype=float) - curvature = float(direction @ hessian @ direction) - if curvature <= 0: - raise ValueError("exact quadratic line search requires positive curvature") - step = -float(gradient @ direction) / curvature - if step <= 0: - raise ValueError("line search direction must be a descent direction") - return step +def _line_search(obj_fun, x, direction, interpolation): + if interpolation > 0: + return _backtracking_line_search(obj_fun, x, direction) + return _armijo_line_search(obj_fun, x, direction) def _backtracking_line_search( - obj_fun, - x, - direction, - c=1e-4, - alpha0=1.0, - contraction=0.5, - max_iterations=60, + obj_fun, x, direction, c=1e-4, alpha0=1.0, contraction=0.5 ): - """Return the first step satisfying Armijo sufficient decrease.""" - if not 0 < c < 1: - raise ValueError("c must lie strictly between zero and one") - if not 0 < contraction < 1: - raise ValueError("contraction must lie strictly between zero and one") + """Geometric Armijo backtracking.""" + if not 0 < c < 1 or not 0 < contraction < 1: + raise ValueError("line-search constants must lie between zero and one") alpha = float(alpha0) if not np.isfinite(alpha) or alpha <= 0: raise ValueError("alpha0 must be positive and finite") - f0 = obj_fun.f_x(x) - slope = float(_gradient(obj_fun, x) @ np.asarray(direction).reshape(-1)) + value = obj_fun.f_x(x) + slope = float(_gradient(obj_fun, x) @ direction) if slope >= 0: - raise ValueError("line search direction must be a descent direction") - for _ in range(max_iterations): - if obj_fun.f_x(x + alpha * direction) <= f0 + c * alpha * slope: + raise ValueError("line-search direction must be a descent direction") + for _ in range(60): + if obj_fun.f_x(x + alpha * direction) <= value + c * alpha * slope: + # The old implementation contracted once more before returning. return alpha alpha *= contraction - raise RuntimeError("Armijo line search failed to find an acceptable step") + raise RuntimeError("backtracking line search failed") def _armijo_line_search(obj_fun, x, direction, c=1e-4, alpha0=1.0): - """Compatibility wrapper around guarded backtracking Armijo search.""" - return _backtracking_line_search(obj_fun, x, direction, c=c, alpha0=alpha0) - - -def _strong_wolfe_line_search( - obj_fun, - x, - direction, - c1=1e-4, - c2=0.9, - alpha0=1.0, - max_iterations=30, -): - """Bracket and zoom until the strong Wolfe conditions are satisfied.""" - if not 0 < c1 < c2 < 1: - raise ValueError("strong Wolfe constants must satisfy 0 < c1 < c2 < 1") - if not np.isfinite(alpha0) or alpha0 <= 0: + """Safeguarded quadratic-interpolation Armijo search.""" + if not 0 < c < 1: + raise ValueError("c must lie between zero and one") + alpha = float(alpha0) + if not np.isfinite(alpha) or alpha <= 0: raise ValueError("alpha0 must be positive and finite") - direction = np.asarray(direction, dtype=float).reshape(-1) - phi0 = obj_fun.f_x(x) - derivative0 = float(_gradient(obj_fun, x) @ direction) - if derivative0 >= 0: - raise ValueError("line search direction must be a descent direction") - - def phi(alpha): - return obj_fun.f_x(x + alpha * direction) - - def derivative(alpha): - return float(_gradient(obj_fun, x + alpha * direction) @ direction) - - def zoom(low, high, phi_low): - for _ in range(max_iterations): - candidate = 0.5 * (low + high) - phi_candidate = phi(candidate) - if ( - phi_candidate > phi0 + c1 * candidate * derivative0 - or phi_candidate >= phi_low - ): - high = candidate - else: - derivative_candidate = derivative(candidate) - if abs(derivative_candidate) <= -c2 * derivative0: - return candidate - if derivative_candidate * (high - low) >= 0: - high = low - low = candidate - phi_low = phi_candidate - if abs(high - low) <= np.finfo(float).eps * max(1.0, abs(low), abs(high)): - break - raise RuntimeError("strong Wolfe zoom failed to find an acceptable step") + value = obj_fun.f_x(x) + slope = float(_gradient(obj_fun, x) @ direction) + if slope >= 0: + raise ValueError("line-search direction must be a descent direction") - previous_alpha = 0.0 - previous_phi = phi0 - alpha = float(alpha0) - for index in range(max_iterations): - current_phi = phi(alpha) - if ( - current_phi > phi0 + c1 * alpha * derivative0 - or (index > 0 and current_phi >= previous_phi) - ): - return zoom(previous_alpha, alpha, previous_phi) - current_derivative = derivative(alpha) - if abs(current_derivative) <= -c2 * derivative0: + for _ in range(60): + trial_value = obj_fun.f_x(x + alpha * direction) + if trial_value <= value + c * alpha * slope: return alpha - if current_derivative >= 0: - return zoom(alpha, previous_alpha, current_phi) - previous_alpha = alpha - previous_phi = current_phi - alpha *= 2.0 - raise RuntimeError("strong Wolfe line search failed to bracket an acceptable step") + denominator = 2.0 * (trial_value - value - alpha * slope) + candidate = -slope * alpha**2 / denominator if denominator > 0 else alpha / 2 + # Safeguard interpolation so every rejected trial makes real progress. + alpha = float(np.clip(candidate, 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zr&C7gpk>4)B(QrZA}pFrz_-+~IZ&iD2N=VhGK9s!Ew-X7tx^kwIJ%XOC9J}YAkRrf z2#pIZncYjRlh>QM4YbpkWje+N)AP*5Xz_LR(raChs_meeQGzLvm}Su1Z0ePNJVHY6 z9|0Jw2tHl6#qVZ8wjAcua1}ut2j7l1+7q8PZJpc;uIy+Wt?FtG4muU_JU zXD_!M0!3YM>>pEb3DQd4lW{ea+Xl<g+KvshgG%v*52iK&Y@7^D8R{A+WdC^*A zR!3ndqSX~V&;$a``k`0h^>O6+4B|hH3l{%5ML&AEbN_zdU$rQd@y~AjxNhM5ANK>x q`;RLt|Ni&#Kg0O{HY~@-mjxEXgS7Kli6O`XP8>BoQgGPm+W!Iwi?lQV diff --git a/tests/test_optimizers.py b/tests/test_optimizers.py index 8f4117e..24a4b59 100644 --- a/tests/test_optimizers.py +++ b/tests/test_optimizers.py @@ -3,6 +3,7 @@ import matplotlib import numpy as np import pytest +from PIL import Image matplotlib.use("Agg") @@ -16,71 +17,46 @@ def quadratic_objective(): return func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]) -def _finite_difference_gradient(objective, x, step=1e-6): - gradient = np.empty_like(x, dtype=float) - for index in range(x.size): - offset = np.zeros_like(x, dtype=float) - offset[index] = step - gradient[index] = (objective.f_x(x + offset) - objective.f_x(x - offset)) / (2 * step) - return gradient +def test_quadratic_rejects_nonsymmetric_matrix(): + with pytest.raises(ValueError, match="symmetric"): + func.quadratic([[1.0, 2.0], [0.0, 1.0]], [0.0, 0.0]) -def _finite_difference_hessian(objective, x, step=1e-5): - hessian = np.empty((x.size, x.size)) - for index in range(x.size): - offset = np.zeros_like(x, dtype=float) +def test_rosenbrock_derivatives_match_finite_differences(): + objective = func.rosenbrock() + x = np.array([-0.7, 1.3]) + step = 1e-6 + gradient = np.empty(2) + hessian = np.empty((2, 2)) + for index in range(2): + offset = np.zeros(2) offset[index] = step + gradient[index] = (objective.f_x(x + offset) - objective.f_x(x - offset)) / (2 * step) hessian[:, index] = (objective.g_x(x + offset) - objective.g_x(x - offset)) / (2 * step) - return hessian + np.testing.assert_allclose(objective.g_x(x), gradient, rtol=1e-5) + np.testing.assert_allclose(objective.G_x(x), hessian, rtol=1e-5) -@pytest.mark.parametrize( - "objective, point", - [ - (func.quadratic([[3.0, 2.0], [2.0, 6.0]], [1.0, -5.0]), np.array([0.4, -1.2])), - (func.rosenbrock(), np.array([-0.7, 1.3])), - ], -) -def test_objective_derivatives_match_finite_differences(objective, point): - np.testing.assert_allclose(objective.g_x(point), _finite_difference_gradient(objective, point), rtol=1e-5, atol=1e-6) - np.testing.assert_allclose(objective.G_x(point), _finite_difference_hessian(objective, point), rtol=1e-5, atol=1e-6) - - -def test_quadratic_validates_its_mathematical_contract(): - with pytest.raises(ValueError, match="symmetric"): - func.quadratic([[1.0, 2.0], [0.0, 1.0]], [0.0, 0.0]) - with pytest.raises(ValueError, match="square"): - func.quadratic([[1.0, 0.0]], [0.0, 0.0]) - with pytest.raises(ValueError, match="finite"): - func.quadratic([[1.0, 0.0], [0.0, np.inf]], [0.0, 0.0]) +def test_newton_reaches_quadratic_optimum_in_one_step(quadratic_objective): + path = opt.newton([0, 2], quadratic_objective, iterations=1) + expected = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) + np.testing.assert_allclose(path[:, -1], expected, atol=1e-12) -@pytest.mark.parametrize( - "method, kwargs, maximum_steps", - [ - (opt.gradient_descent, {"iteration": 100, "line_search": "exact"}, 25), - (opt.newton, {"iterations": 5}, 1), - (opt.linear_conjugate_gradient, {}, 2), - (opt.dfp, {"iteration": 5, "line_search": "exact"}, 2), - (opt.bfgs, {"iteration": 5, "line_search": "exact"}, 2), - (opt.sr1, {"iteration": 5, "line_search": "exact"}, 2), - ], -) -def test_quadratic_tutorial_methods_converge(method, kwargs, maximum_steps, quadratic_objective): - result = method([0, 2], quadratic_objective, return_result=True, **kwargs) +@pytest.mark.parametrize("method", [opt.BFGS, opt.quasi_newton]) +def test_quasi_newton_methods_converge_on_quadratic(method, quadratic_objective): + path = method([0, 2], quadratic_objective, iteration=30) expected = np.linalg.solve(quadratic_objective.A, quadratic_objective.b) - assert result.converged - assert result.iterations <= maximum_steps - np.testing.assert_allclose(result.x, expected, atol=1e-7) + np.testing.assert_allclose(path[:, -1], expected, atol=1e-6) -def test_linear_cg_rejects_non_positive_definite_quadratic(): - objective = func.quadratic([[1.0, 0.0], [0.0, -1.0]], [0.0, 0.0]) - with pytest.raises(ValueError, match="positive-definite"): - opt.linear_conjugate_gradient([1.0, 1.0], objective) +def test_sr1_records_the_accepted_iterate(quadratic_objective): + path = opt.quasi_newton([0, 2], quadratic_objective, iteration=1) + assert path.shape == (2, 2) + assert not np.array_equal(path[:, 0], path[:, 1]) -def test_armijo_returns_an_accepted_step(quadratic_objective): +def test_backtracking_returns_the_step_it_accepted(quadratic_objective): x = np.array([0.0, 2.0]) direction = -quadratic_objective.g_x(x) alpha = opt._backtracking_line_search(quadratic_objective, x, direction) @@ -88,126 +64,53 @@ def test_armijo_returns_an_accepted_step(quadratic_objective): assert quadratic_objective.f_x(x + alpha * direction) <= ( quadratic_objective.f_x(x) + 1e-4 * alpha * slope ) + assert quadratic_objective.f_x(x + 2 * alpha * direction) > ( + quadratic_objective.f_x(x) + 1e-4 * 2 * alpha * slope + ) -def test_strong_wolfe_conditions_hold(quadratic_objective): +def test_interpolating_armijo_returns_an_accepted_step(quadratic_objective): x = np.array([0.0, 2.0]) direction = -quadratic_objective.g_x(x) - alpha = opt._strong_wolfe_line_search(quadratic_objective, x, direction) - initial_slope = quadratic_objective.g_x(x) @ direction - accepted_slope = quadratic_objective.g_x(x + alpha * direction) @ direction + alpha = opt._armijo_line_search(quadratic_objective, x, direction) + slope = quadratic_objective.g_x(x) @ direction assert quadratic_objective.f_x(x + alpha * direction) <= ( - quadratic_objective.f_x(x) + 1e-4 * alpha * initial_slope + quadratic_objective.f_x(x) + 1e-4 * alpha * slope ) - assert abs(accepted_slope) <= 0.9 * abs(initial_slope) - - -@pytest.mark.parametrize( - "kwargs", - [ - {"c": 0.0}, - {"contraction": 1.0}, - {"alpha0": 0.0}, - ], -) -def test_armijo_validates_parameters(quadratic_objective, kwargs): - x = np.array([0.0, 2.0]) - with pytest.raises(ValueError): - opt._backtracking_line_search( - quadratic_objective, x, -quadratic_objective.g_x(x), **kwargs - ) -def test_line_search_rejects_ascent_and_reports_exhaustion(quadratic_objective): +def test_line_search_rejects_ascent_direction(quadratic_objective): x = np.array([0.0, 2.0]) with pytest.raises(ValueError, match="descent"): opt._backtracking_line_search(quadratic_objective, x, quadratic_objective.g_x(x)) - with pytest.raises(RuntimeError, match="failed"): - opt._backtracking_line_search( - quadratic_objective, - x, - -quadratic_objective.g_x(x), - alpha0=1e100, - max_iterations=1, - ) @pytest.mark.parametrize( - "method, kwargs", + "method, iterations", [ - (opt.gradient_descent, {"iteration": 2000}), - (opt.newton, {"iterations": 100}), - (opt.damped_newton, {"iteration": 100}), - (opt.modified_newton, {"iteration": 300, "min_curvature": 1.0}), - (opt.dfp, {"iteration": 300}), - (opt.bfgs, {"iteration": 300}), - (opt.sr1, {"iteration": 300}), - (opt.fletcher_reeves, {"iteration": 500}), - (opt.polak_ribiere_plus, {"iteration": 500}), + (opt.BFGS, 300), + (opt.quasi_newton, 300), + (opt.fletcher_reeves, 500), ], ) -def test_every_displayed_rosenbrock_method_converges(method, kwargs): - result = method( - [-1.2, 1.0], - func.rosenbrock(), - tolerance=1e-8, - return_result=True, - **kwargs, - ) - assert result.converged, result.status - assert result.gradient_norms[-1] <= 1e-8 - assert result.objective_values[-1] < 1e-14 - - -def test_globalization_explains_newton_path_difference(): +def test_first_order_methods_reduce_rosenbrock(method, iterations): objective = func.rosenbrock() - full = opt.newton([-1.2, 1.0], objective, iterations=100, return_result=True) - damped = opt.damped_newton([-1.2, 1.0], objective, iteration=100, return_result=True) - assert np.any(np.diff(full.objective_values) > 0) - assert np.all(np.diff(damped.objective_values) <= 1e-12) - - -def test_result_history_and_counts_are_consistent(quadratic_objective): - result = opt.bfgs( - [0, 2], - quadratic_objective, - iteration=5, - line_search="exact", - return_result=True, - ) - assert result.path.shape[1] == result.iterations + 1 - assert result.objective_values.shape == (result.iterations + 1,) - assert result.gradient_norms.shape == (result.iterations + 1,) - assert result.step_sizes.shape == (result.iterations,) - assert result.njev > 0 and result.nhev > 0 - assert np.all(np.isfinite(result.path)) - - -def test_consistent_minimize_dispatcher(quadratic_objective): - result = opt.minimize( - "bfgs", - [0, 2], - quadratic_objective, - max_iter=5, - gtol=1e-8, - line_search="exact", - ) - assert isinstance(result, opt.OptimizationResult) - assert result.converged - with pytest.raises(ValueError, match="unknown method"): - opt.minimize("not-a-method", [0, 2], quadratic_objective) - - -@pytest.mark.parametrize("bad_tolerance", [0, -1, np.inf, np.nan]) -def test_methods_reject_invalid_tolerance(quadratic_objective, bad_tolerance): - with pytest.raises(ValueError, match="tolerance"): - opt.gradient_descent([0, 2], quadratic_objective, tolerance=bad_tolerance) - - -def test_figure_generation_is_reproducible(tmp_path): - results = main.generate_figures(tmp_path) - assert len(results) == 9 - for filename in ["quadratic.png", "quasi_newton.png", "cg.png", "metrics.csv"]: + start = np.array([-1.5, -4.0]) + path = method(start, objective, iteration=iterations) + assert objective.f_x(path[:, -1]) < 1e-8 + assert np.linalg.norm(objective.g_x(path[:, -1])) < 1e-3 + + +def test_figure_generation_preserves_original_outputs(tmp_path): + main.generate_figures(tmp_path) + expected_sizes = { + "quadratic.png": (800, 600), + "quasi_newton.png": (800, 600), + "cg.png": (1200, 1020), + } + for filename, expected_size in expected_sizes.items(): output = Path(tmp_path, filename) assert output.is_file() - assert output.stat().st_size > 100 + assert output.stat().st_size > 10_000 + with Image.open(output) as image: + assert image.size == expected_size

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