TODO list

[blegat]

This is just the comment of jump-dev/MathOptInterface.jl#2198 moved here

Note that once this is done, you can write an SDP in SDPA format directly in JuMP with something like

C = rand(d, d)
A = [rand(d, d) for i in 1:m]
model = Model()
@variable(model, x[1:m+1] in MOI.SetWithDotProducts(
    MOI.PositiveSemidefiniteConeTriangle(d),
    MOI.TriangleVectorization.([A; [C]]),
))
@constraint(model, x[1:m] == b)
@objective(model, Min, x[end])

or

model = Model()
@variable(model, y[1:m])
@constraint(model, [y; -1] in MOI.LinearCombinationInSet([A; [C]]))
@objective(model, Max, dot(y, b))

And Dualization.dualize should take you from one to the other one!

Solvers implementing this:

• Hypatia.LinMatrixIneqCone : A1 PSD and A2, ..., A[m] arbitrary
• Hypatia.WSOSInterpNonnegativeCone : A = u u' (Rank-1 PSD) Add support for SumOfSquares cone jump-dev/Hypatia.jl#844
• DSDP : A = a * u * u' (Rank-1) Add support for rank-1 constraints jump-dev/DSDP.jl#37
• SDPLR : A = U * Diagonal(d) * U' (Low-Rank) Add support for low rank constraint jump-dev/SDPLR.jl#26

Basic

• Add a new AbstractScalarSet or AbstractVectorSet to src/sets.jl
• If isbitstype(S) == false, implement Base.copy(set::S)
• If isbitstype(S) == false, implement Base.:(==)(x::S, y::S)
• If an AbstractVectorSet, implement dimension(set::S), unless the
  dimension is given by set.dimension.

Utilities

• If an AbstractVectorSet, implement Utilities.set_dot,
  unless the dot product between two vectors in the set is equivalent to
  LinearAlgebra.dot
• If an AbstractVectorSet, implement Utilities.set_with_dimension in
  src/Utilities/matrix_of_constraints.jl
• Add the set to the @model macro at the bottom of src/Utilities.model.jl

Documentation

• Add a docstring, which gives the mathematical definition of the set,
  along with an ## Example block containing a jldoctest
• Add the docstring to docs/src/reference/standard_form.md
• Add the set to the relevant table in docs/src/manual/standard_form.md

Tests

• Define a new _set(::Type{S}) method in src/Test/test_basic_constraint.jl
  and add the name of the set to the list at the bottom of that files
• If the set has any checks in its constructor, add tests to test/sets.jl

MathOptFormat

• Open an issue at https://github.com/jump-dev/MathOptFormat to add
  support for the new set {{ replace with link to the issue }} I don't think we should add any set specialization of that set or at least not yet

Optional

• Implement dual_set(::S) and dual_set_type(::Type{S})
• Add new tests to the Test submodule exercising your new set
• Add new bridges to convert your set into more commonly used sets