rewrite sos_spds.jl
This includes changes related to: * SymbolicWedderburn * new formulation of constraints * general warmstart for JuMP-^1.3
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@ -5,11 +5,12 @@ using LinearAlgebra
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using SparseArrays
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using Dates
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using Groups
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using SymbolicWedderburn
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using JuMP
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using Groups
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using StarAlgebras
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using SymbolicWedderburn
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include("laplacians.jl")
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include("sos_sdps.jl")
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include("checksolution.jl")
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525
src/sos_sdps.jl
525
src/sos_sdps.jl
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@ -1,260 +1,359 @@
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###############################################################################
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#
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# Constraints
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#
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###############################################################################
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"""
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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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function constraints(pm::Matrix{I}, total_length=maximum(pm)) where {I<:Integer}
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cnstrs = [Vector{I}() for _ in 1:total_length]
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for i in eachindex(pm)
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push!(cnstrs[pm[i]], i)
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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::AlgebraElement,
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order_unit::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 orbit_constraint!(result::SparseMatrixCSC, cnstrs, orbit; val=1.0/length(orbit))
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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[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::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::AlgebraElement,
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order_unit::AlgebraElement=zero(elt);
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upper_bound=Inf,
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augmented::Bool=iszero(aug(elt)) && iszero(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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dropzeros!(result)
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for constraint in cnstrs[orbit]
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for idx in constraint
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result[idx] = val
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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 orbit_spvector(vect::AbstractVector, orbits)
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orb_vector = spzeros(length(orbits))
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for (i,o) in enumerate(orbits)
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k = vect[collect(o)]
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val = k[1]
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@assert all(k .== val)
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orb_vector[i] = val
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end
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return orb_vector
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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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###############################################################################
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#
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# Naive SDP
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#
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###############################################################################
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sos_problem_primal(
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elt::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_dual(elt::GroupRingElem, order_unit::GroupRingElem;
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lower_bound::Float64=Inf)
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@assert parent(elt) == parent(order_unit)
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function sos_problem_primal(
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elt::AlgebraElement,
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orderunit::AlgebraElement,
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wedderburn::WedderburnDecomposition;
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upper_bound=Inf,
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augmented=iszero(aug(elt)) && iszero(aug(orderunit))
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)
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RG = parent(elt)
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m = Model()
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y = @variable(m, y[1:length(elt.coeffs)])
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@constraint(m, λ_dual, dot(order_unit.coeffs, y) == 1)
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@constraint(m, psd, [y[i] for i in RG.pm] in PSDCone())
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if !isinf(lower_bound)
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@variable(m, λ_ub_dual)
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expr = dot(elt.coeffs, y) + lower_bound*λ_ub_dual
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# @constraint m expr >= lower_bound
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@objective m Min expr
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else
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@objective m Min dot(elt.coeffs, y)
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@assert parent(elt) === parent(orderunit)
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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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return m
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end
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feasibility_problem = iszero(orderunit)
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function SOS_problem_primal(X::GroupRingElem, orderunit::GroupRingElem;
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upper_bound::Float64=Inf)
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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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N = size(parent(X).pm, 1)
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m = JuMP.Model();
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JuMP.@variable(m, P[1:N, 1:N])
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# SP = Symmetric(P)
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JuMP.@SDconstraint(m, sdp, P >= 0)
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if iszero(aug(X)) && iszero(aug(orderunit))
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JuMP.@constraint(m, augmentation, sum(P) == 0)
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end
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if upper_bound < Inf
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λ = JuMP.@variable(m, λ <= upper_bound)
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else
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λ = JuMP.@variable(m, λ)
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end
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cnstrs = constraints(parent(X).pm)
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@assert length(cnstrs) == length(X.coeffs) == length(orderunit.coeffs)
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x, u = X.coeffs, orderunit.coeffs
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JuMP.@constraint(m, lincnstr[i=1:length(cnstrs)],
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x[i] - λ*u[i] == sum(P[cnstrs[i]]))
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JuMP.@objective(m, Max, λ)
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return m
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end
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###############################################################################
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#
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# Symmetrized SDP
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#
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###############################################################################
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function SOS_problem_primal(X::GroupRingElem, orderunit::GroupRingElem, data::BlockDecomposition; upper_bound::Float64=Inf)
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Ns = size.(data.Uπs, 2)
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m = JuMP.Model();
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Ps = Vector{Matrix{JuMP.VariableRef}}(undef, length(Ns))
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for (k,s) in enumerate(Ns)
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Ps[k] = JuMP.@variable(m, [1:s, 1:s])
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JuMP.@SDconstraint(m, Ps[k] >= 0)
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end
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if upper_bound < Inf
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λ = JuMP.@variable(m, λ <= upper_bound)
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else
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λ = JuMP.@variable(m, λ)
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end
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@info "Adding $(length(data.orbits)) constraints..."
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@time addconstraints!(m, Ps, X, orderunit, data)
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JuMP.@objective(m, Max, λ)
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return m, Ps
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end
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function constraintLHS!(M, cnstr, Us, Ust, dims, eps=1000*eps(eltype(first(M))))
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for π in 1:lastindex(Us)
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M[π] = dims[π].*PropertyT.clamp_small!(Ust[π]*(cnstr*Us[π]), eps)
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end
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end
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function addconstraints!(m::JuMP.Model,
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P::Vector{Matrix{JuMP.VariableRef}},
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X::GroupRingElem, orderunit::GroupRingElem, data::BlockDecomposition)
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orderunit_orb = orbit_spvector(orderunit.coeffs, data.orbits)
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X_orb = orbit_spvector(X.coeffs, data.orbits)
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UπsT = [U' for U in data.Uπs]
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cnstrs = constraints(parent(X).pm)
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orb_cnstr = spzeros(Float64, size(parent(X).pm)...)
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M = [Array{Float64}(undef, n,n) for n in size.(UπsT,1)]
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λ = m[:λ]
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for (t, orbit) in enumerate(data.orbits)
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orbit_constraint!(orb_cnstr, cnstrs, orbit)
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constraintLHS!(M, orb_cnstr, data.Uπs, UπsT, data.dims)
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x, u = X_orb[t], orderunit_orb[t]
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JuMP.@constraints m begin
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x - λ*u == sum(dot(M[π], P[π]) for π in eachindex(data.Uπs))
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end
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end
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return m
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end
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function reconstruct(Ps::Vector{Matrix{F}}, data::BlockDecomposition) where F
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return reconstruct(Ps, data.preps, data.Uπs, data.dims)
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end
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function reconstruct(Ps::Vector{M},
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preps::Dict{GEl, P}, Uπs::Vector{U}, dims::Vector{Int}) where
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{M<:AbstractMatrix, GEl<:GroupElem, P<:Generic.Perm, U<:AbstractMatrix}
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lU = length(Uπs)
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transfP = [dims[π].*Uπs[π]*Ps[π]*Uπs[π]' for π in 1:lU]
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tmp = [zeros(Float64, size(first(transfP))) for _ in 1:lU]
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Threads.@threads for π in 1:lU
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tmp[π] = perm_avg!(tmp[π], transfP[π], values(preps))
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end
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recP = sum(tmp)./length(preps)
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return recP
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end
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function perm_avg!(result, P, perms)
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lp = length(first(perms).d)
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for p in perms
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# result .+= view(P, p.d, p.d)
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@inbounds for j in 1:lp
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k = p[j]
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for i in 1:lp
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result[i,j] += P[p[i], k]
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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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return result
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end
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###############################################################################
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#
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# Low-level solve
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#
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###############################################################################
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function setwarmstart!(m::JuMP.Model, warmstart)
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if solver_name(m) == "SCS"
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primal, dual, slack = warmstart
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m.moi_backend.optimizer.model.optimizer.data.primal = primal
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m.moi_backend.optimizer.model.optimizer.data.dual = dual
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m.moi_backend.optimizer.model.optimizer.data.slack = slack
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else
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@warn "Setting warmstart for $(solver_name(m)) is not implemented! Ignoring..."
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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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return m
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end
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function getwarmstart(m::JuMP.Model)
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if solver_name(m) == "SCS"
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warmstart = (
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primal = m.moi_backend.optimizer.model.optimizer.data.primal,
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dual = m.moi_backend.optimizer.model.optimizer.data.dual,
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slack = m.moi_backend.optimizer.model.optimizer.data.slack
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)
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else
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@warn "Saving warmstart for $(solver_name(m)) is not implemented!"
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return (primal=Float64[], dual=Float64[], slack=Float64[])
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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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return warmstart
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X = convert(Vector{T}, coeffs(elt))
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U = convert(Vector{T}, 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 solve(m::JuMP.Model, with_optimizer::JuMP.OptimizerFactory, warmstart=nothing)
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function reconstruct(Ps, wd::WedderburnDecomposition)
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N = size(first(direct_summands(wd)), 2)
|
||||
P = zeros(eltype(wd), N, N)
|
||||
return reconstruct!(P, Ps, wd)
|
||||
end
|
||||
|
||||
set_optimizer(m, with_optimizer)
|
||||
function group_of(wd::WedderburnDecomposition)
|
||||
# this is veeeery hacky... ;)
|
||||
return parent(first(keys(wd.hom.cache)))
|
||||
end
|
||||
|
||||
# TODO: move to SymbolicWedderburn
|
||||
SymbolicWedderburn.action(wd::WedderburnDecomposition) =
|
||||
SymbolicWedderburn.action(wd.hom)
|
||||
|
||||
function reconstruct!(
|
||||
res::AbstractMatrix,
|
||||
Ps,
|
||||
wedderburn::WedderburnDecomposition,
|
||||
)
|
||||
G = group_of(wedderburn)
|
||||
|
||||
act = SymbolicWedderburn.action(wedderburn)
|
||||
|
||||
@assert act isa SymbolicWedderburn.ByPermutations
|
||||
|
||||
for (π, ds) in pairs(direct_summands(wedderburn))
|
||||
Uπ = SymbolicWedderburn.image_basis(ds)
|
||||
|
||||
# LinearAlgebra.mul!(tmp, Uπ', P[π])
|
||||
# LinearAlgebra.mul!(tmp2, tmp, Uπ)
|
||||
tmp2 = Uπ' * Ps[π] * Uπ
|
||||
if eltype(res) <: AbstractFloat
|
||||
SymbolicWedderburn.zerotol!(tmp2, atol=1e-12)
|
||||
end
|
||||
tmp2 .*= degree(ds)
|
||||
|
||||
@assert size(tmp2) == size(res)
|
||||
|
||||
for g in G
|
||||
p = SymbolicWedderburn.induce(wedderburn.hom, g)
|
||||
for c in axes(res, 2)
|
||||
for r in axes(res, 1)
|
||||
res[r, c] += tmp2[r^p, c^p]
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
res ./= order(Int, G)
|
||||
|
||||
return res
|
||||
end
|
||||
|
||||
##
|
||||
# Low-level solve
|
||||
|
||||
setwarmstart!(model::JuMP.Model, ::Nothing) = model
|
||||
|
||||
function setwarmstart!(model::JuMP.Model, warmstart)
|
||||
constraint_map = Dict(
|
||||
ct => JuMP.all_constraints(model, ct...) for
|
||||
ct in JuMP.list_of_constraint_types(model)
|
||||
)
|
||||
|
||||
JuMP.set_start_value.(JuMP.all_variables(model), warmstart.primal)
|
||||
|
||||
for (ct, idx) in pairs(constraint_map)
|
||||
JuMP.set_start_value.(idx, warmstart.slack[ct])
|
||||
JuMP.set_dual_start_value.(idx, warmstart.dual[ct])
|
||||
end
|
||||
return model
|
||||
end
|
||||
|
||||
function getwarmstart(model::JuMP.Model)
|
||||
constraint_map = Dict(
|
||||
ct => JuMP.all_constraints(model, ct...) for
|
||||
ct in JuMP.list_of_constraint_types(model)
|
||||
)
|
||||
|
||||
primal = value.(JuMP.all_variables(model))
|
||||
|
||||
slack = Dict(k => value.(v) for (k, v) in constraint_map)
|
||||
duals = Dict(k => JuMP.dual.(v) for (k, v) in constraint_map)
|
||||
|
||||
return (primal=primal, dual=duals, slack=slack)
|
||||
end
|
||||
|
||||
function solve(m::JuMP.Model, optimizer, warmstart=nothing)
|
||||
|
||||
JuMP.set_optimizer(m, optimizer)
|
||||
MOIU.attach_optimizer(m)
|
||||
|
||||
if warmstart != nothing
|
||||
setwarmstart!(m, warmstart)
|
||||
end
|
||||
m = setwarmstart!(m, warmstart)
|
||||
|
||||
optimize!(m)
|
||||
JuMP.optimize!(m)
|
||||
Base.Libc.flush_cstdio()
|
||||
|
||||
status = termination_status(m)
|
||||
status = JuMP.termination_status(m)
|
||||
|
||||
return status, getwarmstart(m)
|
||||
end
|
||||
|
||||
function solve(solverlog::String, m::JuMP.Model, with_optimizer::JuMP.OptimizerFactory, warmstart=nothing)
|
||||
function solve(solverlog::String, m::JuMP.Model, optimizer, warmstart=nothing)
|
||||
|
||||
isdir(dirname(solverlog)) || mkpath(dirname(solverlog))
|
||||
|
||||
Base.flush(Base.stdout)
|
||||
Base.Libc.flush_cstdio()
|
||||
status, warmstart = open(solverlog, "a+") do logfile
|
||||
redirect_stdout(logfile) do
|
||||
status, warmstart = PropertyT.solve(m, with_optimizer, warmstart)
|
||||
status, warmstart = solve(m, optimizer, warmstart)
|
||||
status, warmstart
|
||||
end
|
||||
end
|
||||
|
|
|
|||
Loading…
Reference in New Issue