366 lines
10 KiB
Julia
366 lines
10 KiB
Julia
"""
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sos_problem_dual(X, [u = zero(X); upper_bound=Inf])
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Formulate the dual to the sum of squares decomposition problem for `X - λ·u`.
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See also [sos_problem_primal](@ref).
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"""
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function sos_problem_dual(
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elt::StarAlgebras.AlgebraElement,
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order_unit::StarAlgebras.AlgebraElement=zero(elt);
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lower_bound=-Inf
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)
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@assert parent(elt) == parent(order_unit)
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algebra = parent(elt)
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mstructure = if StarAlgebras._istwisted(algebra.mstructure)
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algebra.mstructure
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else
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StarAlgebras.MTable{true}(basis(algebra), table_size=size(algebra.mstructure))
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end
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# 1 variable for every primal constraint
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# 1 dual variable for every element of basis
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# Symmetrized:
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# 1 dual variable for every orbit of G acting on basis
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model = Model()
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@variable model y[1:length(basis(algebra))]
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@constraint model λ_dual dot(order_unit, y) == 1
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@constraint(model, psd, y[mstructure] in PSDCone())
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if !isinf(lower_bound)
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throw("Not Implemented yet")
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@variable model λ_ub_dual
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@objective model Min dot(elt, y) + lower_bound * λ_ub_dual
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else
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@objective model Min dot(elt, y)
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end
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return model
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end
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function constraints(
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mstr::AbstractMatrix{<:Integer},
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total_length=maximum(mstr),
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)
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cnstrs = [Vector{eltype(mstr)}() for _ = 1:total_length]
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li = LinearIndices(mstr)
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for (idx, val) in pairs(mstr)
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push!(cnstrs[val], li[idx])
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end
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return cnstrs
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end
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function constraints(
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basis::StarAlgebras.AbstractBasis,
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mstr::AbstractMatrix{<:Integer};
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augmented::Bool=false,
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table_size=size(mstr)
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)
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cnstrs = [signed(eltype(mstr))[] for _ in basis]
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LI = LinearIndices(table_size)
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for ci in CartesianIndices(table_size)
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k = LI[ci]
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a_star_b = basis[mstr[k]]
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push!(cnstrs[basis[a_star_b]], k)
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if augmented
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# (1-a_star)(1-b) = 1 - a_star - b + a_star_b
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i, j = Tuple(ci)
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a, b = basis[i], basis[j]
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push!(cnstrs[basis[one(a)]], k)
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push!(cnstrs[basis[StarAlgebras.star(a)]], -k)
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push!(cnstrs[basis[b]], -k)
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end
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end
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return Dict(
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basis[i] => ConstraintMatrix(c, table_size..., 1) for (i, c) in pairs(cnstrs)
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)
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end
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function constraints(A::StarAlgebras.StarAlgebra; augmented::Bool, twisted::Bool)
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mstructure = if StarAlgebras._istwisted(A.mstructure) == twisted
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A.mstructure
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else
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StarAlgebras.MTable{twisted}(basis(A), table_size=size(A.mstructure))
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end
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return constraints(basis(A), mstructure, augmented=augmented)
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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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Given element `X` of a star-algebra `A` and an order unit `u` of `Σ²A` the sum
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of squares cone in `A`, fomulate sum of squares decomposition problem:
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```
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max: λ
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subject to: X - λ·u ∈ Σ²A
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```
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If `upper_bound` is given a finite value, additionally `λ ≤ upper_bound` will
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be added to the model. This may improve the accuracy of the solution if
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`upper_bound` is less than the optimal `λ`.
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The default `u = zero(X)` formulates a simple feasibility problem.
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"""
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function sos_problem_primal(
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elt::StarAlgebras.AlgebraElement,
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order_unit::StarAlgebras.AlgebraElement=zero(elt);
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upper_bound=Inf,
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augmented::Bool=iszero(StarAlgebras.aug(elt)) && iszero(StarAlgebras.aug(order_unit))
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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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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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if iszero(order_unit) && isfinite(upper_bound)
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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, twisted=true)
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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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JuMP.@constraint model λ <= upper_bound
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end
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JuMP.@objective(model, Max, λ)
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for b in basis(parent(elt))
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JuMP.@constraint(model, elt(b) - λ * order_unit(b) == dot(A[b], P))
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end
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else
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for b in basis(parent(elt))
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JuMP.@constraint(model, elt(b) == dot(A[b], P))
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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 invariant_constraint!(
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result::AbstractMatrix,
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basis::StarAlgebras.AbstractBasis,
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cnstrs::AbstractDict{K,CM},
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invariant_vec::SparseVector,
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) where {K,CM<:ConstraintMatrix}
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result .= zero(eltype(result))
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for i in SparseArrays.nonzeroinds(invariant_vec)
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g = basis[i]
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A = cnstrs[g]
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for (idx, v) in nzpairs(A)
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result[idx] += invariant_vec[i] * v
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end
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end
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return result
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end
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function isorth_projection(ds::SymbolicWedderburn.DirectSummand)
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U = SymbolicWedderburn.image_basis(ds)
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return isapprox(U * U', I)
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end
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sos_problem_primal(
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elt::StarAlgebras.AlgebraElement,
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wedderburn::WedderburnDecomposition;
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kwargs...
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) = sos_problem_primal(elt, zero(elt), wedderburn; kwargs...)
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function sos_problem_primal(
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elt::StarAlgebras.AlgebraElement,
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orderunit::StarAlgebras.AlgebraElement,
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wedderburn::WedderburnDecomposition;
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upper_bound=Inf,
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augmented=iszero(StarAlgebras.aug(elt)) && iszero(StarAlgebras.aug(orderunit)),
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check_orthogonality=true
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)
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@assert parent(elt) === parent(orderunit)
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if check_orthogonality
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if any(!isorth_projection, direct_summands(wedderburn))
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error("Wedderburn decomposition contains a non-orthogonal projection")
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end
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end
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feasibility_problem = iszero(orderunit)
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model = JuMP.Model()
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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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end
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P = map(direct_summands(wedderburn)) do ds
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dim = size(ds, 1)
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P = JuMP.@variable(model, [1:dim, 1:dim], Symmetric)
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@constraint(model, P in PSDCone())
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P
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end
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begin # preallocating
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T = eltype(wedderburn)
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M = zeros.(T, size.(P))
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M_orb = zeros(T, size(parent(elt).mstructure)...)
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tmps = SymbolicWedderburn._tmps(wedderburn)
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end
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X = convert(Vector{T}, StarAlgebras.coeffs(elt))
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U = convert(Vector{T}, StarAlgebras.coeffs(orderunit))
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# defining constraints based on the multiplicative structure
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cnstrs = constraints(parent(elt), augmented=augmented, twisted=true)
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@info "Adding $(length(invariant_vectors(wedderburn))) constraints"
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for iv in invariant_vectors(wedderburn)
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x = dot(X, iv)
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u = dot(U, iv)
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M_orb = invariant_constraint!(M_orb, basis(parent(elt)), cnstrs, iv)
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M = SymbolicWedderburn.diagonalize!(M, M_orb, wedderburn, tmps)
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SymbolicWedderburn.zerotol!.(M, atol=1e-12)
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M_dot_P = sum(dot(M[π], P[π]) for π in eachindex(M) if !iszero(M[π]))
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if feasibility_problem
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JuMP.@constraint(model, x == M_dot_P)
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else
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JuMP.@constraint(model, x - λ * u == M_dot_P)
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end
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end
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return model, P
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end
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function reconstruct(Ps, wd::WedderburnDecomposition)
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N = size(first(direct_summands(wd)), 2)
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P = zeros(eltype(wd), N, N)
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return reconstruct!(P, Ps, wd)
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end
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function group_of(wd::WedderburnDecomposition)
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# this is veeeery hacky... ;)
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return parent(first(keys(wd.hom.cache)))
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end
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# TODO: move to SymbolicWedderburn
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SymbolicWedderburn.action(wd::WedderburnDecomposition) =
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SymbolicWedderburn.action(wd.hom)
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function reconstruct!(
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res::AbstractMatrix,
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Ps,
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wedderburn::WedderburnDecomposition,
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)
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G = group_of(wedderburn)
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act = SymbolicWedderburn.action(wedderburn)
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@assert act isa SymbolicWedderburn.ByPermutations
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for (π, ds) in pairs(direct_summands(wedderburn))
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Uπ = SymbolicWedderburn.image_basis(ds)
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# LinearAlgebra.mul!(tmp, Uπ', P[π])
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# LinearAlgebra.mul!(tmp2, tmp, Uπ)
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tmp2 = Uπ' * Ps[π] * Uπ
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if eltype(res) <: AbstractFloat
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SymbolicWedderburn.zerotol!(tmp2, atol=1e-12)
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end
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tmp2 .*= SymbolicWedderburn.degree(ds)
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@assert size(tmp2) == size(res)
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for g in G
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p = SymbolicWedderburn.induce(wedderburn.hom, g)
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for c in axes(res, 2)
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for r in axes(res, 1)
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res[r, c] += tmp2[r^p, c^p]
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end
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end
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end
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end
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res ./= Groups.order(Int, G)
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return res
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end
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##
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# Low-level solve
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setwarmstart!(model::JuMP.Model, ::Nothing) = model
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function setwarmstart!(model::JuMP.Model, warmstart)
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constraint_map = Dict(
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ct => JuMP.all_constraints(model, ct...) for
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ct in JuMP.list_of_constraint_types(model)
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)
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JuMP.set_start_value.(JuMP.all_variables(model), warmstart.primal)
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for (ct, idx) in pairs(constraint_map)
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JuMP.set_start_value.(idx, warmstart.slack[ct])
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JuMP.set_dual_start_value.(idx, warmstart.dual[ct])
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end
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return model
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end
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function getwarmstart(model::JuMP.Model)
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constraint_map = Dict(
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ct => JuMP.all_constraints(model, ct...) for
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ct in JuMP.list_of_constraint_types(model)
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)
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primal = value.(JuMP.all_variables(model))
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slack = Dict(k => value.(v) for (k, v) in constraint_map)
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duals = Dict(k => JuMP.dual.(v) for (k, v) in constraint_map)
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return (primal=primal, dual=duals, slack=slack)
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end
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function solve(m::JuMP.Model, optimizer, warmstart=nothing)
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JuMP.set_optimizer(m, optimizer)
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MOIU.attach_optimizer(m)
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m = setwarmstart!(m, warmstart)
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JuMP.optimize!(m)
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Base.Libc.flush_cstdio()
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status = JuMP.termination_status(m)
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return status, getwarmstart(m)
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end
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function solve(solverlog::String, m::JuMP.Model, optimizer, warmstart=nothing)
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isdir(dirname(solverlog)) || mkpath(dirname(solverlog))
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Base.flush(Base.stdout)
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Base.Libc.flush_cstdio()
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status, warmstart = open(solverlog, "a+") do logfile
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redirect_stdout(logfile) do
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status, warmstart = solve(m, optimizer, warmstart)
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status, warmstart
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end
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end
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return status, warmstart
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end
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