Sampling basis

Contributed by: Benoît Legat

using DynamicPolynomials
using SumOfSquares
import MultivariateBases as MB

using Dualization
import Hypatia
import SCS

In this tutorial, we show how to use a sampling basis for enforcing the equality between the polynomial and its Sum-of-Squares decomposition. As introduced in Zero basis, we can set the zero basis to a sampling basis using the zero_basis keyword.

@polyvar x
p = x^4 - 4x^3 - 2x^2 + 12x + 3

model = Model(dual_optimizer(SCS.Optimizer))
set_silent(model)
@variable(model, γ)
@objective(model, Max, γ)
@constraint(model, p - γ in SOSCone(), zero_basis = BoxSampling([-1], [1]))
optimize!(model)
solution_summary(model)

solution_summary(; result = 1, verbose = false)
├ solver_name          : Dual model with SCS attached
├ Termination
│ ├ termination_status : OPTIMAL
│ ├ result_count       : 1
│ └ raw_status         : solved
├ Solution (result = 1)
│ ├ primal_status        : FEASIBLE_POINT
│ ├ dual_status          : FEASIBLE_POINT
│ ├ objective_value      : -5.99998e+00
│ └ dual_objective_value : -6.00058e+00
└ Work counters
  └ solve_time (sec)   : 1.23385e-03

We can see that the SOS constraint is converted into a PSD constraint.

print_active_bridges(model)

 * Supported objective: MOI.VariableIndex
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-SumOfSquares.SOSPolynomialSet{FullSpace, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, SumOfSquares.Certificate.Newton{SOSCone, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, MultivariateBases.ImplicitLagrangeBasis{Float64, Vector{Float64}, BoxSampling{Float64, Vector{Float64}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, SumOfSquares.Certificate.NewtonFilter{SumOfSquares.Certificate.NewtonDegreeBounds{Tuple{}}}}}
 |  bridged by:
 |   SumOfSquares.Bridges.Constraint.SOSPolynomialBridge{Float64, MOI.VectorAffineFunction{Float64}, FullSpace, MOI.PositiveSemidefiniteConeTriangle, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, SumOfSquares.Certificate.Newton{SOSCone, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, MultivariateBases.ImplicitLagrangeBasis{Float64, Vector{Float64}, BoxSampling{Float64, Vector{Float64}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, SumOfSquares.Certificate.NewtonFilter{SumOfSquares.Certificate.NewtonDegreeBounds{Tuple{}}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-SumOfSquares.WeightedSOSCone{MOI.PositiveSemidefiniteConeTriangle, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorSlackBridge{Float64, MOI.VectorAffineFunction{Float64}, SumOfSquares.WeightedSOSCone{MOI.PositiveSemidefiniteConeTriangle, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros
 |   |   * Unsupported variable: SumOfSquares.WeightedSOSCone{MOI.PositiveSemidefiniteConeTriangle, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}
 |   |   |  bridged by:
 |   |   |    SumOfSquares.Bridges.Variable.LowRankBridge{Float64, MOI.PositiveSemidefiniteConeTriangle}
 |   |   |  may introduce:
 |   |   |   * Unsupported variable: LowRankOpt.SetDotProducts{LowRankOpt.WITHOUT_SET, MOI.PositiveSemidefiniteConeTriangle, LowRankOpt.TriangleVectorization{Float64, LowRankOpt.Factorization{Float64, Vector{Float64}, Array{Float64, 0}}}}
 |   |   |   |  bridged by:
 |   |   |   |    LowRankOpt.Bridges.Variable.AppendSetBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, LowRankOpt.TriangleVectorization{Float64, LowRankOpt.Factorization{Float64, Vector{Float64}, Array{Float64, 0}}}}
 |   |   |   |  may introduce:
 |   |   |   |   * Unsupported variable: LowRankOpt.SetDotProducts{LowRankOpt.WITH_SET, MOI.PositiveSemidefiniteConeTriangle, LowRankOpt.TriangleVectorization{Float64, LowRankOpt.Factorization{Float64, Vector{Float64}, Array{Float64, 0}}}}
 |   |   |   |   |  bridged by:
 |   |   |   |   |    LowRankOpt.Bridges.Variable.DotProductsBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, LowRankOpt.TriangleVectorization{Float64, LowRankOpt.Factorization{Float64, Vector{Float64}, Array{Float64, 0}}}}
 |   |   |   |   |  may introduce:
 |   |   |   |   |   * Supported variable: MOI.PositiveSemidefiniteConeTriangle

Let's try with Hypatia now:

set_optimizer(model, dual_optimizer(Hypatia.Optimizer))
optimize!(model)
solution_summary(model)

solution_summary(; result = 1, verbose = false)
├ solver_name          : Dual model with Hypatia attached
├ Termination
│ ├ termination_status : OPTIMAL
│ ├ result_count       : 1
│ └ raw_status         : Optimal
├ Solution (result = 1)
│ ├ primal_status        : FEASIBLE_POINT
│ ├ dual_status          : FEASIBLE_POINT
│ ├ objective_value      : -6.00000e+00
│ └ dual_objective_value : -6.00000e+00
└ Work counters
  ├ solve_time (sec)   : 9.16516e-01
  └ barrier_iterations : 7

We can see that the SOS constraint is passed as a LowRankOpt.SetDotProducts{LowRankOpt.WITHOUT_SET}. This maps into Hypatia's native WSOSInterpNonnegativeCone.

print_active_bridges(model)

 * Supported objective: MOI.VariableIndex
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-SumOfSquares.SOSPolynomialSet{FullSpace, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, SumOfSquares.Certificate.Newton{SOSCone, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, MultivariateBases.ImplicitLagrangeBasis{Float64, Vector{Float64}, BoxSampling{Float64, Vector{Float64}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, SumOfSquares.Certificate.NewtonFilter{SumOfSquares.Certificate.NewtonDegreeBounds{Tuple{}}}}}
 |  bridged by:
 |   SumOfSquares.Bridges.Constraint.SOSPolynomialBridge{Float64, MOI.VectorAffineFunction{Float64}, FullSpace, MOI.PositiveSemidefiniteConeTriangle, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, SumOfSquares.Certificate.Newton{SOSCone, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, MultivariateBases.ImplicitLagrangeBasis{Float64, Vector{Float64}, BoxSampling{Float64, Vector{Float64}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, SumOfSquares.Certificate.NewtonFilter{SumOfSquares.Certificate.NewtonDegreeBounds{Tuple{}}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-SumOfSquares.WeightedSOSCone{MOI.PositiveSemidefiniteConeTriangle, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorSlackBridge{Float64, MOI.VectorAffineFunction{Float64}, SumOfSquares.WeightedSOSCone{MOI.PositiveSemidefiniteConeTriangle, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros
 |   |   * Unsupported variable: SumOfSquares.WeightedSOSCone{MOI.PositiveSemidefiniteConeTriangle, MultivariateBases.LagrangeBasis{Float64, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}, ColumnSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, StarAlgebras.SubBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Int64, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}, Vector{Vector{Int64}}}, StarAlgebras.AlgebraElement{Float64, StarAlgebras.StarAlgebra{MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, MultivariateBases.MStruct{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}, Vector{Int64}, StarAlgebras.MappedBasis{MultivariateBases.Polynomial{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}, Vector{Int64}}, Vector{Int64}, MultivariatePolynomials.ExponentsIterator{MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}, Nothing, Vector{Int64}}, MultivariateBases.Variables{Monomial, Vector{DynamicPolynomials.Variable{DynamicPolynomials.Commutative{DynamicPolynomials.CreationOrder}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}, typeof(MultivariatePolynomials.exponents)}}}, StarAlgebras.SparseCoefficients{Vector{Int64}, Float64, Tuple{Vector{Int64}}, Tuple{Float64}, MultivariatePolynomials.Graded{MultivariatePolynomials.LexOrder}}}}
 |   |   |  bridged by:
 |   |   |    SumOfSquares.Bridges.Variable.LowRankBridge{Float64, MOI.PositiveSemidefiniteConeTriangle}
 |   |   |  may introduce:
 |   |   |   * Unsupported variable: LowRankOpt.SetDotProducts{LowRankOpt.WITHOUT_SET, MOI.PositiveSemidefiniteConeTriangle, LowRankOpt.TriangleVectorization{Float64, LowRankOpt.Factorization{Float64, Vector{Float64}, Array{Float64, 0}}}}
 |   |   |   |  bridged by:
 |   |   |   |    LowRankOpt.Bridges.Variable.ToPositiveBridge{Float64, LowRankOpt.WITHOUT_SET, MOI.PositiveSemidefiniteConeTriangle, Vector{Float64}, Array{Float64, 0}}
 |   |   |   |  may introduce:
 |   |   |   |   * Supported variable: LowRankOpt.SetDotProducts{LowRankOpt.WITHOUT_SET, MOI.PositiveSemidefiniteConeTriangle, LowRankOpt.TriangleVectorization{Float64, LowRankOpt.Factorization{Float64, Vector{Float64}, FillArrays.Ones{Float64, 0, Tuple{}}}}}

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