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Add WeightedSOSCone (#351)
* Add WeightedSOSCone * Fix * Fix format * Fix * fix * Add docstrings in docs * Fix * Fix * rename * Add tests * Fix format * Update ci.yml * include tests * MOI master
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‎.github/workflows/ci.yml‎

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version: ${{ matrix.version }}
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arch: ${{ matrix.arch }}
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- uses: julia-actions/cache@v1
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- name: MOI
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shell: julia --project=@. {0}
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run: |
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using Pkg
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Pkg.add([
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PackageSpec(name="MultivariateBases", rev="master"),
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PackageSpec(name="MathOptInterface", rev="master"),
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])
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- uses: julia-actions/julia-buildpkg@v1
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- uses: julia-actions/julia-runtest@v1
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with:

‎docs/src/reference/standard_form.md‎

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@@ -43,6 +43,12 @@ SumOfSquares.PolyJuMP.SAGE.AGEBridge
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## MOI Sets
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Sum-of-Squares cones:
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```@docs
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SumOfSquares.SOSPolynomialSet
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SumOfSquares.WeightedSOSCone
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```
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Special cases of positive semidefinite cones:
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```@docs
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SumOfSquares.EmptyCone

‎docs/src/references.bib‎

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@@ -61,6 +61,19 @@ @Article{Hesse1973
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publisher = {Institute for Operations Research and the Management Sciences (INFORMS)},
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}
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@article{Kapelevich2023,
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title = {Sum of Squares Generalizations for Conic Sets},
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author = {Kapelevich, Lea and Coey, Chris and Vielma, Juan Pablo},
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year = {2023},
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month = may,
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journal = {Mathematical Programming},
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volume = {199},
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number = {1},
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pages = {1417--1429},
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issn = {1436-4646},
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doi = {10.1007/s10107-022-01831-6},
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}
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@Book{Lasserre2009,
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author = {Lasserre, Jean Bernard},
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publisher = {Imperial College Press},
@@ -98,6 +111,18 @@ @Article{Murty1987
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publisher = {Springer-Verlag Berlin, Heidelberg}
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}
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@Article{Papp2017,
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title = {Sum-of-Squares Optimization without Semidefinite Programming},
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author = {Papp, D. and Y{\i}ld{\i}z, S.},
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year = {2017},
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month = dec,
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journal = {ArXiv e-prints},
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eprint = {1712.01792},
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primaryclass = {math.OC},
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adsnote = {Provided by the SAO/NASA Astrophysics Data System},
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archiveprefix = {arxiv},
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}
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@Article{Parrilo2003,
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title = {Semidefinite Programming Relaxations for Semialgebraic Problems},
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author = {Parrilo, Pablo A},

‎src/Bridges/Constraint/utilities.jl‎

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@@ -14,7 +14,7 @@ function union_set_types(MCT)
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}
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end
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function constrained_variable_types(MCT)
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return [
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return Tuple{Type}[
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(typeof(SOS.matrix_cone(MCT, 0)),),
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(typeof(SOS.matrix_cone(MCT, 1)),),
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(typeof(SOS.matrix_cone(MCT, 2)),),

‎src/Bridges/Variable/Variable.jl‎

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module Variable
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import MutableArithmetics as MA
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import MathOptInterface as MOI
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import MultivariatePolynomials as MP
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import SumOfSquares as SOS
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include("psd2x2.jl")
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include("scaled_diagonally_dominant.jl")
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include("copositive_inner.jl")
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include("kernel.jl")
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end

‎src/Bridges/Variable/kernel.jl‎

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struct KernelBridge{T,M} <: MOI.Bridges.Variable.AbstractBridge
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affine::Vector{MOI.ScalarAffineFunction{T}}
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variables::Vector{Vector{MOI.VariableIndex}}
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constraints::Vector{MOI.ConstraintIndex{MOI.VectorOfVariables}}
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set::SOS.WeightedSOSCone{M}
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end
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function MOI.Bridges.Variable.bridge_constrained_variable(
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::Type{KernelBridge{T}},
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model::MOI.ModelLike,
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set::SOS.WeightedSOSCone{M},
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) where {T,M}
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variables = Vector{MOI.VariableIndex}[]
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constraints = MOI.ConstraintIndex{MOI.VectorOfVariables}[]
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acc = MA.Zero()
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for (gram_basis, weight) in zip(set.gram_bases, set.weights)
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gram, vars, con = SOS.add_gram_matrix(model, M, gram_basis, T)
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push!(variables, vars)
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push!(constraints, con)
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acc = MA.add_mul!!(acc, weight, gram)
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end
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affine = MP.coefficients(acc, set.basis)
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return KernelBridge{T,M}(affine, variables, constraints, set)
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end
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function MOI.Bridges.Variable.supports_constrained_variable(
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::Type{<:KernelBridge},
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::Type{<:SOS.WeightedSOSCone},
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)
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return true
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end
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function MOI.Bridges.added_constrained_variable_types(
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::Type{KernelBridge{T,M}},
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) where {T,M}
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return SOS.Bridges.Constraint.constrained_variable_types(M)
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end
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function MOI.Bridges.added_constraint_types(::Type{<:KernelBridge})
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return Tuple{Type,Type}[]
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end
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# Attributes, Bridge acting as a model
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function MOI.get(bridge::KernelBridge, ::MOI.NumberOfVariables)
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return sum(length, bridge.variables)
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end
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function MOI.get(bridge::KernelBridge, ::MOI.ListOfVariableIndices)
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return reduce(vcat, bridge.variables)
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end
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function MOI.get(
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bridge::KernelBridge,
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::MOI.NumberOfConstraints{MOI.VectorOfVariables,S},
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) where {S<:MOI.AbstractVectorSet}
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return count(bridge.constraints) do ci
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return ci isa MOI.ConstraintIndex{MOI.VectorOfVariables,S}
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end
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end
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function MOI.get(
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bridge::KernelBridge,
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::MOI.ListOfConstraintIndices{MOI.VectorOfVariables,S},
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) where {S}
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return [
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ci for ci in bridge.constraints if
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ci isa MOI.ConstraintIndex{MOI.VectorOfVariables,S}
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]
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end
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# Indices
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function MOI.delete(model::MOI.ModelLike, bridge::KernelBridge)
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for vars in bridge.variables
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MOI.delete(model, vars)
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end
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return
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end
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# Attributes, Bridge acting as a constraint
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function MOI.get(::MOI.ModelLike, ::MOI.ConstraintSet, bridge::KernelBridge)
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return bridge.set
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end
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function MOI.get(
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model::MOI.ModelLike,
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attr::MOI.ConstraintPrimal,
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bridge::KernelBridge,
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)
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return [
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MOI.get(
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model,
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MOI.VariablePrimal(attr.result_index),
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bridge,
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MOI.Bridges.IndexInVector(i),
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) for i in eachindex(bridge.affine)
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]
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end
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function MOI.get(
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model::MOI.ModelLike,
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attr::MOI.VariablePrimal,
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bridge::KernelBridge,
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i::MOI.Bridges.IndexInVector,
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)
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return MOI.Utilities.eval_variable(bridge.affine[i.value]) do
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return vi -> MOI.get(model, MOI.VariablePrimal(attr.result_index), vi)
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end
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end
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function MOI.Bridges.bridged_function(
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bridge::KernelBridge,
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i::MOI.Bridges.IndexInVector,
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)
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return bridge.affine[i.value]
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end
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function MOI.Bridges.Variable.unbridged_map(
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bridge::KernelBridge{T},
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coefs::Vector{MOI.VariableIndex},
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) where {T}
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F = MOI.ScalarAffineFunction{T}
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map = Pair{MOI.VariableIndex,F}[]
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return nothing
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end

‎src/sets.jl‎

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return set
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end
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"""
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struct WeightedSOSCone{
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M,
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B<:AbstractPolynomialBasis,
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G<:AbstractPolynomialBasis,
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W<:MP.AbstractPolynomialLike,
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} <: MOI.AbstractVectorSet
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basis::B
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gram_bases::Vector{G}
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weights::Vector{W}
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end
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The weighted sum-of-squares cone is the set of vectors of coefficients `a` in `basis`
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that are the sum of `weights[i]` multiplied by a gram matrix with basis `gram_bases[i]`.
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The matrix cone type `M` is used to decide in which cone the gram matrix is constrained,
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e.g., `MOI.PositiveSemidefiniteConeTriangle` or
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[`SumOfSquares.ScaledDiagonallyDominantConeTriangle`](@ref).
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See [Papp2017; Section 1.1](@cite) and [Kapelevich2023; Section 1](@cite).
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"""
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struct WeightedSOSCone{
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M,
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B<:AbstractPolynomialBasis,
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G<:AbstractPolynomialBasis,
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W<:MP.AbstractPolynomialLike,
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} <: MOI.AbstractVectorSet
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basis::B
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gram_bases::Vector{G}
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weights::Vector{W}
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end
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function WeightedSOSCone{M}(
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basis::AbstractPolynomialBasis,
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gram_bases::Vector{G},
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weights::Vector{W},
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) where {M,G<:AbstractPolynomialBasis,W<:MP.AbstractPolynomialLike}
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return WeightedSOSCone{M,typeof(basis),G,W}(basis, gram_bases, weights)
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end
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MOI.dimension(set::WeightedSOSCone) = length(set.basis)
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Base.copy(set::WeightedSOSCone) = set
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function Base.:(==)(a::WeightedSOSCone, b::WeightedSOSCone)
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return a.basis == b.basis &&
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a.gram_bases == b.gram_bases &&
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a.weights == b.weights
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end
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"""
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struct SOSPolynomialSet{
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DT<:AbstractSemialgebraicSet,
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MT<:MP.AbstractMonomial,
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MVT<:AbstractVector{MT},
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CT<:Certificate.AbstractCertificate,
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} <: MOI.AbstractVectorSet
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domain::DT
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monomials::MVT
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certificate::CT
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end
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The sum-of-squares cone is the set of vectors of coefficients `a` for monomials `monomials`
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for which the polynomial is a sum-of-squares `certificate` over `domain`.
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"""
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struct SOSPolynomialSet{
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DT<:AbstractSemialgebraicSet,
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MT<:MP.AbstractMonomial,

‎test/Bridges/Bridges.jl‎

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using Test
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@testset "$(dir)" for dir in ["Variable"]
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@testset "$(file)" for file in readdir(joinpath(@__DIR__, dir))
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if !endswith(file, ".jl")
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continue
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elseif dir == "." && file == "Bridges.jl"
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continue
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end
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include(joinpath(@__DIR__, dir, file))
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end
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end

‎test/Bridges/Variable/kernel.jl‎

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module TestVariableKernel
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using Test
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using DynamicPolynomials
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using SumOfSquares
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function runtests()
8+
for name in names(@__MODULE__; all = true)
9+
if startswith("$(name)", "test_")
10+
@testset "$(name)" begin
11+
getfield(@__MODULE__, name)()
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end
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end
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end
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return
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end
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function test_runtests()
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@polyvar x y
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MOI.Bridges.runtests(
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SumOfSquares.Bridges.Variable.KernelBridge,
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model -> begin
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MOI.add_constrained_variables(
24+
model,
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SumOfSquares.WeightedSOSCone{
26+
MOI.PositiveSemidefiniteConeTriangle,
27+
}(
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MonomialBasis([x^4, x^3 * y, x^2 * y^2, y^4]),
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[MonomialBasis([x^2, y^2, x * y])],
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[1.0 * x^0 * y^0],
31+
),
32+
)
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end,
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model -> begin
35+
Q, _ = MOI.add_constrained_variables(
36+
model,
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MOI.PositiveSemidefiniteConeTriangle(3),
38+
)
39+
end,
40+
)
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return
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end
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end # module
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TestVariableKernel.runtests()

‎test/runtests.jl‎

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include("variable.jl")
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include("constraint.jl")
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include("Bridges/Bridges.jl")
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include("Mock/mock_tests.jl")
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