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add cocycle constraints from SDP duality paper by M.Nitsche
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111
src/sos_sdps.jl
111
src/sos_sdps.jl
@ -36,6 +36,31 @@ function sos_problem_dual(
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return model
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end
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function geometric_constraints!(
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model::JuMP.Model,
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elt::StarAlgebras.AlgebraElement,
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)
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A = parent(elt)
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G = parent(A)
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mstr = A.mstructure
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b = basis(A)
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y = model[:y]
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for g in gens(G)
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for h in gens(G)
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gh = mstr[b[g], b[h]]
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if elt[gh] > 0
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for γ in axes(mstr, 1)
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γgh = mstr[γ, gh]
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γg = mstr[γ, b[g]]
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γh = mstr[γ, b[h]]
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JuMP.@constraint model y[γgh] + y[γ] == y[γg] + y[γh]
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end
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end
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end
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end
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return model
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end
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function decompose(
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elt::StarAlgebras.AlgebraElement,
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wd::WedderburnDecomposition,
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@ -70,6 +95,29 @@ function decompose(v::AbstractVector, invariant_vecs)
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return res, norm(current - v)
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end
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function _dot(elt::StarAlgebras.AlgebraElement, Y, wd::WedderburnDecomposition)
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inv_vecs = invariant_vectors(wd)
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v = StarAlgebras.coeffs(elt)
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res, error = _dot(v, Y, inv_vecs)
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_eps = length(v) * eps(typeof(error))
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error < _eps || @warn "elt does not seem to be invariant" error
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return res
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end
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function _dot(v::AbstractVector, Y::AbstractVector{<:JuMP.AbstractVariableRef})
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@assert length(inv_vecs) == length(Y)
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@assert length(v) == length(first(inv_vecs))
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res = JuMP.AffExpr()
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cfs, error = decompose(v, inv_vecs)
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for i in SparseArrays.nonzeroinds(cfs)
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(c, y) = cfs[i], Y[i]
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JuMP.add_to_expression!(res, c, y)
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end
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return res, error
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end
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function sos_problem_dual(
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elt::StarAlgebras.AlgebraElement,
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order_unit::StarAlgebras.AlgebraElement,
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@ -123,6 +171,44 @@ function sos_problem_dual(
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return model, Ps
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end
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function __find_firstnz(i, inv_vecs)
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for (idx, iv) in enumerate(inv_vecs)
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iv[i] ≠ 0 && return idx
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end
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return nothing
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end
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function geometric_constraints!(
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model::JuMP.Model,
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elt::StarAlgebras.AlgebraElement,
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wd::WedderburnDecomposition,
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)
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A = parent(elt)
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G = parent(A)
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mstr = A.mstructure
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b = basis(A)
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y = model[:y_orb]
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# cfs = PropertyT.decompose(elt, wd)
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inv_vecs = invariant_vectors(wd)
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for g in gens(G)
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for h in gens(G)
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gh = mstr[b[g], b[h]]
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if elt[gh] > 0
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for (γ, iv) in enumerate(inv_vecs)
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γ_basis_idx = first(SparseArrays.nonzeroinds(iv))
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γ_basis_idx > size(mstr, 1) && break
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γgh = __find_firstnz(mstr[γ_basis_idx, gh], inv_vecs)
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γg = __find_firstnz(mstr[γ_basis_idx, b[g]], inv_vecs)
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γh = __find_firstnz(mstr[γ_basis_idx, b[h]], inv_vecs)
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JuMP.@constraint model y[γgh] + y[γ] == y[γg] + y[γh]
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end
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end
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end
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end
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end
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"""
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sos_problem_primal(X, [u = zero(X); upper_bound=Inf])
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Formulate sum of squares decomposition problem for `X - λ·u`.
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@ -147,10 +233,11 @@ function sos_problem_primal(
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upper_bound = Inf,
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augmented::Bool = iszero(StarAlgebras.aug(elt)) &&
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iszero(StarAlgebras.aug(order_unit)),
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support = 1:size(parent(elt).mstructure, 1),
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)
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@assert parent(elt) === parent(order_unit)
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N = LinearAlgebra.checksquare(parent(elt).mstructure)
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N = length(support)
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model = JuMP.Model()
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P = JuMP.@variable(model, P[1:N, 1:N], Symmetric)
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JuMP.@constraint(model, psd, P in PSDCone())
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@ -159,11 +246,11 @@ function sos_problem_primal(
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@warn "Setting `upper_bound` together with zero `order_unit` has no effect"
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end
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A = constraints(parent(elt); augmented = augmented)
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A = constraints(parent(elt), support; augmented = augmented)
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if !iszero(order_unit)
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λ = JuMP.@variable(model, λ)
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if isfinite(upper_bound)
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if !isfinite(upper_bound)
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JuMP.@constraint model λ <= upper_bound
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end
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JuMP.@objective(model, Max, λ)
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@ -282,9 +369,9 @@ function sos_problem_primal(
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end
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end
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id_one = findfirst(invariant_vectors(wedderburn)) do v
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b = basis(parent(elt))
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return sparsevec([b[one(first(b))]], [1 // 1], length(v)) == v
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id_one = let b = basis(parent(elt)), iv = invariant_vectors(wedderburn)
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id_v = sparsevec([b[one(first(b))]], [1 // 1], length(first(iv)))
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findfirst(==(id_v), iv)
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end
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prog = ProgressMeter.Progress(
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@ -303,12 +390,12 @@ function sos_problem_primal(
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if !feasibility_problem # add λ or not?
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λ = JuMP.@variable(model, λ)
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JuMP.@objective(model, Max, λ)
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if isfinite(upper_bound)
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JuMP.@constraint(model, λ <= upper_bound)
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if feasibility_problem
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@warn "setting `upper_bound` with zero `orderunit` has no effect"
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end
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end
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if isfinite(upper_bound)
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if feasibility_problem
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@warn "setting `upper_bound` with zero `orderunit` has no effect"
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else
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JuMP.@constraint(model, ub, λ <= upper_bound)
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end
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end
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