Speed up Expr * Expr

[Zeroto521]

This PR is at least faster 1.2x than before.

• Before optimized
  • First time running
    • Time taken for 50x50 matrix: 1.0127 seconds
    • Time taken for 100x100 matrix: 6.8916 second
  • Second time running
    • Time taken for 50x50 matrix: 0.8220 seconds
    • Time taken for 100x100 matrix: 5.8312 seconds
  • Third time running
    • Time taken for 50x50 matrix: 0.9372 seconds
    • Time taken for 100x100 matrix: 5.2823 seconds
• After optimized Expr.__mul__(Expr):
  • First time running
    • Time taken for 50x50 matrix: 0.5335 seconds
    • Time taken for 100x100 matrix: 6.5986 seconds
  • Second time running
    • Time taken for 50x50 matrix: 0.7765 seconds
    • Time taken for 100x100 matrix: 5.2647 seconds
  • Third time running
    • Time taken for 50x50 matrix: 0.6112 seconds
    • Time taken for 100x100 matrix: 6.7011 seconds
• After optimized Expr.__mul__(Expr) and Term.__mul__(Term)
  • First time running
    • Time taken for 50x50 matrix: 0.6092 seconds
    • Time taken for 100x100 matrix: 5.7285 seconds
  • Second time running
    • Time taken for 50x50 matrix: 0.5105 seconds
    • Time taken for 100x100 matrix: 5.0206 seconds
  • Third time running
    • Time taken for 50x50 matrix: 0.5816 seconds
    • Time taken for 100x100 matrix: 4.3978 seconds

from timeit import timeit

from pyscipopt import Model

m = Model()

n = 50
x = m.addMatrixVar((n, n))

cost = timeit(lambda: x @ x, number=5) / 5
print(f"Time taken for {n}x{n} matrix: {cost:.4f} seconds")

n = 100
x = m.addMatrixVar((n, n))
cost = timeit(lambda: x @ x, number=5) / 5
print(f"Time taken for {n}x{n} matrix: {cost:.4f} seconds")

[Zeroto521]

__mul__ is better. We call this function actually to multiply

[Zeroto521]

Here is the Merge Sort Algorithm, with a time complexity of $O(n)$.
The time complexity of sorted is $O(nlog(n))$.

[Copilot]

Pull request overview

This PR optimizes the multiplication of Expr objects (polynomial expressions) by using low-level C API calls and introducing a more efficient Term.__mul__ method. According to the benchmarks provided, this results in at least 1.2x speedup for matrix multiplication operations.

Changes:

• Introduced Term.__mul__ method using an efficient merge algorithm for combining sorted variable tuples
• Optimized Expr.__mul__ to use C-level Python dict iteration APIs (PyDict_Next, PyDict_GetItem) and skip zero coefficients
• Updated CHANGELOG to document the performance improvement

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated no comments.

File	Description
src/pyscipopt/expr.pxi	Implements optimized Term.__mul__ method and refactors Expr.__mul__ to use low-level C APIs for faster dictionary iteration and term multiplication
CHANGELOG.md	Adds entry documenting the Expr * Expr performance improvement in the Changed section

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[Zeroto521]

Use Cython API to speed up.