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add symmetrized sos_problem_dual
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@ -70,6 +70,59 @@ function decompose(v::AbstractVector, invariant_vecs)
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return res, norm(current - v)
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return res, norm(current - v)
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end
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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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wd::WedderburnDecomposition,
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lower_bound = -Inf,
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supp::AbstractVector{<:Integer} = axes(parent(elt).mstructure, 1),
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)
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@assert parent(elt) == parent(order_unit)
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inv_vecs = invariant_vectors(wd)
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model = Model()
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# 1 dual variable per orbit of G acting on basis
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JuMP.@variable(model, y_orb[1:length(inv_vecs)])
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# the value of y on order_unit is 1 (y is normalized)
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let unit_orbit_cfs = decompose(order_unit, wd)
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JuMP.@constraint(model, λ_dual, dot(unit_orbit_cfs, y_orb) == 1)
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end
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# here we reconstruct the original y;
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# TODO: this is **BAD**; do something about it
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# y = sum(y .* iv for (y, iv) in zip(y_orb, invariant_vectors(wd)))
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y = let y = [JuMP.AffExpr() for _ in 1:length(first(invariant_vectors(wd)))]
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for (y_o, iv) in zip(y_orb, invariant_vectors(wd))
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for i in SparseArrays.nonzeroinds(iv)
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JuMP.add_to_expression!(y[i], nnz(iv) * iv[i], y_o)
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end
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end
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y
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end
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Ps = let mstr = parent(elt).mstructure, y = y
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moment_matrix = [mstr[-i, j] for i in supp, j in supp]
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# JuMP.@constraint(model, psd, y[moment_matrix] in PSDCone())
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Ps = SymbolicWedderburn.diagonalize(y[moment_matrix], wd)
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for P in Ps
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JuMP.@constraint(model, P in PSDCone())
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end
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Ps
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end
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if !isinf(lower_bound)
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throw("Not Implemented yet")
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JuMP.@variable(model, λ_ub_dual)
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# JuMP.@objective(model, Min, _dot(elt, y_orb, wd) + lower_bound * λ_ub_dual)
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else
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let elt_orbit_cfs = decompose(elt, wd)
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JuMP.@objective(model, Min, dot(elt_orbit_cfs, y_orb, wd))
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end
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end
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return model, Ps
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end
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"""
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"""
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sos_problem_primal(X, [u = zero(X); upper_bound=Inf])
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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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Formulate sum of squares decomposition problem for `X - λ·u`.
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