367 lines
12 KiB
Julia
367 lines
12 KiB
Julia
using JuMP
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import MathProgBase: AbstractMathProgSolver
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import Base: rationalize
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using GroupAlgebras
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using ProgressMeter
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using ValidatedNumerics
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function create_product_matrix(basis, limit)
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product_matrix = zeros(Int, (limit,limit))
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basis_dict = Dict{Array, Int}(x => i
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for (i,x) in enumerate(basis))
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for i in 1:limit
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x_inv::eltype(basis) = inv(basis[i])
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for j in 1:limit
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w = x_inv*basis[j]
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product_matrix[i,j] = basis_dict[w]
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# index = findfirst(basis, w)
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# index ≠ 0 || throw(ArgumentError("Product is not supported on basis: $w"))
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# product_matrix[i,j] = index
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end
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end
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return product_matrix
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end
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function constraints_from_pm(pm, total_length=maximum(pm))
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n = size(pm,1)
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constraints = constraints = [Array{Int,1}[] for x in 1:total_length]
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for j in 1:n
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Threads.@threads for i in 1:n
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idx = pm[i,j]
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push!(constraints[idx], [i,j])
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end
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end
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return constraints
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end
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function splaplacian_coeff(S, basis, n=length(basis))
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result = spzeros(n)
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result[1] = float(length(S))
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for s in S
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ind = findfirst(basis, s)
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result[ind] += -1.0
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end
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return result
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end
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function laplacian_coeff(S, basis)
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return full(splaplacian_coeff(S,basis))
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end
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function create_SDP_problem(matrix_constraints, Δ::GroupAlgebraElement; upper_bound=Inf)
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N = size(Δ.product_matrix,1)
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const Δ² = Δ*Δ
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@assert length(Δ) == length(matrix_constraints)
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m = JuMP.Model();
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JuMP.@variable(m, A[1:N, 1:N], SDP)
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JuMP.@SDconstraint(m, A >= 0)
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JuMP.@constraint(m, sum(A[i] for i in eachindex(A)) == 0)
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JuMP.@variable(m, κ >= 0.0)
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if upper_bound < Inf
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JuMP.@constraint(m, κ <= upper_bound)
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end
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JuMP.@objective(m, Max, κ)
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for (pairs, δ², δ) in zip(matrix_constraints, Δ².coefficients, Δ.coefficients)
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JuMP.@constraint(m, sum(A[i,j] for (i,j) in pairs) == δ² - κ*δ)
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end
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return m
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end
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function solve_SDP(SDP_problem, solver)
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@show SDP_problem
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@show solver
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JuMP.setsolver(SDP_problem, solver);
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# @time MathProgBase.writeproblem(SDP_problem, "/tmp/SDP_problem")
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solution_status = JuMP.solve(SDP_problem);
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if solution_status != :Optimal
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warn("The solver did not solve the problem successfully!")
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end
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@show solution_status
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κ = JuMP.getvalue(JuMP.getvariable(SDP_problem, :κ))
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A = JuMP.getvalue(JuMP.getvariable(SDP_problem, :A))
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return κ, A
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end
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function EOI{T<:Number}(Δ::GroupAlgebraElement{T}, κ::T)
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return Δ*Δ - κ*Δ
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end
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function square_as_elt(vector, elt)
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zzz = zeros(elt.coefficients)
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zzz[1:length(vector)] = vector
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# new_base_elt = GroupAlgebraElement(zzz, elt.product_matrix)
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# return (new_base_elt*new_base_elt).coefficients
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return GroupAlgebras.algebra_multiplication(zzz, zzz, elt.product_matrix)
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end
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function compute_SOS{T<:Number}(sqrt_matrix::Array{T,2},
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elt::GroupAlgebraElement{T})
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n = size(sqrt_matrix,2)
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result = zeros(T, length(elt.coefficients))
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p = Progress(n, 1, "Checking SOS decomposition...", 50)
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for i in 1:n
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result .+= square_as_elt(sqrt_matrix[:,i], elt)
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next!(p)
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end
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return GroupAlgebraElement{T}(result, elt.product_matrix)
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end
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function correct_to_augmentation_ideal{T<:Rational}(sqrt_matrix::Array{T,2})
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sqrt_corrected = similar(sqrt_matrix)
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l = size(sqrt_matrix,2)
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for i in 1:l
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col = view(sqrt_matrix,:,i)
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sqrt_corrected[:,i] = col - sum(col)//l
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# @assert sum(sqrt_corrected[:,i]) == 0
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end
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return sqrt_corrected
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end
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function check_solution{T<:Number}(κ::T, sqrt_matrix::Array{T,2}, Δ::GroupAlgebraElement{T}; verbose=true, augmented=false)
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result = compute_SOS(sqrt_matrix, Δ)
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if augmented
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epsilon = GroupAlgebras.ɛ(result)
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if isa(epsilon, Interval)
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@assert 0 in epsilon
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elseif isa(epsilon, Rational)
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@assert epsilon == 0//1
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else
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warn("Does checking for augmentation has meaning for $(typeof(epsilon))?")
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end
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end
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SOS_diff = EOI(Δ, κ) - result
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eoi_SOS_L₁_dist = norm(SOS_diff,1)
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if verbose
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@show κ
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if augmented
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println("ɛ(Δ² - κΔ - ∑ξᵢ*ξᵢ) = ", GroupAlgebras.ɛ(SOS_diff))
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else
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ɛ_dist = GroupAlgebras.ɛ(SOS_diff)
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if typeof(ɛ_dist) <: Interval
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ɛ_dist = ɛ_dist.lo
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end
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@printf("ɛ(Δ² - κΔ - ∑ξᵢ*ξᵢ) ≈ %.10f\n", ɛ_dist)
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end
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L₁_dist = eoi_SOS_L₁_dist
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if typeof(L₁_dist) <: Interval
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L₁_dist = L₁_dist.lo
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end
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@printf("‖Δ² - κΔ - ∑ξᵢ*ξᵢ‖₁ ≈ %.10f\n", L₁_dist)
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end
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distance_to_cone = κ - 2^3*eoi_SOS_L₁_dist
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return distance_to_cone
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end
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import ValidatedNumerics.±
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function (±)(X::AbstractArray, tol::Real)
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r{T}(x::T) = ( x==zero(T) ? @interval(x) : x ± tol)
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return r.(X)
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end
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(±)(X::GroupAlgebraElement, tol::Real) = GroupAlgebraElement(X.coefficients ± tol, X.product_matrix)
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function Base.rationalize{T<:Integer, S<:Real}(::Type{T},
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X::AbstractArray{S}; tol::Real=eps(eltype(X)))
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r(x) = rationalize(T, x, tol=tol)
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return r.(X)
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end
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ℚ(x, tol::Real) = rationalize(BigInt, x, tol=tol)
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function ℚ_distance_to_positive_cone(Δ::GroupAlgebraElement, κ, A;
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tol=1e-7, verbose=true, rational=false)
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isapprox(eigvals(A), abs(eigvals(A)), atol=tol) ||
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warn("The solution matrix doesn't seem to be positive definite!")
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@assert A == Symmetric(A)
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A_sqrt = real(sqrtm(A))
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# println("")
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# println("Checking in floating-point arithmetic...")
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# @time fp_distance = check_solution(κ, A_sqrt, Δ, verbose=verbose)
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# println("Floating point distance (to positive cone) ≈ $(Float64(trunc(fp_distance,8)))")
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# println("-------------------------------------------------------------")
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# println("")
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#
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# if fp_distance ≤ 0
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# return fp_distance
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# end
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println("Checking in interval arithmetic...")
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A_sqrtᴵ = A_sqrt ± tol
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κᴵ = κ ± tol
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Δᴵ = Δ ± tol
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@time Interval_distance = check_solution(κᴵ, A_sqrtᴵ, Δᴵ, verbose=verbose)
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# @assert isa(ℚ_distance, Rational)
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println("The actual distance (to positive cone) is contained in $Interval_distance")
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println("-------------------------------------------------------------")
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println("")
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if Interval_distance.lo ≤ 0
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return Interval_distance.lo
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end
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println("Projecting columns of A_sqrt to the augmentation ideal...")
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A_sqrt_ℚ = ℚ(A_sqrt, tol)
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A_sqrt_ℚ_aug = correct_to_augmentation_ideal(A_sqrt_ℚ)
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κ_ℚ = ℚ(κ, tol)
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Δ_ℚ = ℚ(Δ, tol)
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A_sqrt_ℚ_augᴵ = A_sqrt_ℚ_aug ± tol
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κᴵ = κ_ℚ ± tol
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Δᴵ = Δ_ℚ ± tol
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@time Interval_dist_to_Σ² = check_solution(κᴵ, A_sqrt_ℚ_augᴵ, Δᴵ, verbose=verbose, augmented=true)
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println("The Augmentation-projected actual distance (to positive cone) is contained in $Interval_dist_to_Σ²")
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println("-------------------------------------------------------------")
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println("")
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if Interval_dist_to_Σ².lo ≤ 0 || !rational
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return Interval_dist_to_Σ².lo
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else
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println("Checking Projected SOS decomposition in exact rational arithmetic...")
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@time ℚ_dist_to_Σ² = check_solution(κ_ℚ, A_sqrt_ℚ_aug, Δ_ℚ, verbose=verbose, augmented=true)
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@assert isa(ℚ_dist_to_Σ², Rational)
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println("Augmentation-projected rational distance (to positive cone) ≥ $(Float64(trunc(ℚ_dist_to_Σ²,8)))")
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println("-------------------------------------------------------------")
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return ℚ_dist_to_Σ²
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end
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end
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function pmΔfilenames(name::String)
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if !isdir(name)
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mkdir(name)
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end
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prefix = name
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pm_filename = joinpath(prefix, "product_matrix.jld")
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Δ_coeff_filename = joinpath(prefix, "delta.coefficients.jld")
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return pm_filename, Δ_coeff_filename
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end
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function κSDPfilenames(name::String)
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if !isdir(name)
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mkdir(name)
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end
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prefix = name
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κ_filename = joinpath(prefix, "kappa.jld")
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SDP_filename = joinpath(prefix, "SDPmatrixA.jld")
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return κ_filename, SDP_filename
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end
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function ΔandSDPconstraints(name::String)
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pm_fname, Δ_fname = pmΔfilenames(name)
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f₁ = isfile(pm_fname)
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f₂ = isfile(Δ_fname)
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if f₁ && f₂
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println("Loading precomputed pm, Δ, sdp_constraints...")
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product_matrix = load(pm_fname, "pm")
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L = load(Δ_fname, "Δ")[:, 1]
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Δ = GroupAlgebraElement(L, Array{Int,2}(product_matrix))
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sdp_constraints = constraints_from_pm(product_matrix)
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else
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throw(ArgumentError("You need to precompute pm and Δ to load it!"))
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end
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return Δ, sdp_constraints
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end
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function ΔandSDPconstraints(name::String, ID, generating_func::Function)
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pm_fname, Δ_fname = pmΔfilenames(name)
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Δ, sdp_constraints = ΔandSDPconstraints(ID, generating_func())
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save(pm_fname, "pm", Δ.product_matrix)
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save(Δ_fname, "Δ", Δ.coefficients)
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return Δ, sdp_constraints
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end
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function κandA(name::String)
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κ_fname, SDP_fname = κSDPfilenames(name)
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f₁ = isfile(κ_fname)
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f₂ = isfile(SDP_fname)
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if f₁ && f₂
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println("Loading precomputed κ, A...")
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κ = load(κ_fname, "κ")
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A = load(SDP_fname, "A")
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else
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throw(ArgumentError("You need to precompute κ and SDP matrix A to load it!"))
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end
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return κ, A
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end
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function κandA(name::String, sdp_constraints, Δ::GroupAlgebraElement, solver::AbstractMathProgSolver; upper_bound=Inf)
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println("Creating SDP problem...")
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@time SDP_problem = create_SDP_problem(sdp_constraints, Δ; upper_bound=upper_bound)
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println("Solving SDP problem maximizing κ...")
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κ, A = solve_SDP(SDP_problem, solver)
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κ_fname, A_fname = κSDPfilenames(name)
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if κ > 0
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save(κ_fname, "κ", κ)
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save(A_fname, "A", A)
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else
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throw(ErrorException("Solver $solver did not produce a valid solution!: κ = $κ"))
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end
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return κ, A
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end
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function check_property_T(name::String, ID, generate_B₄::Function;
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verbose=true, tol=1e-6, upper_bound=Inf)
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# solver = MosekSolver(INTPNT_CO_TOL_REL_GAP=tol, QUIET=!verbose)
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solver = SCSSolver(eps=tol, max_iters=100000, verbose=verbose)
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@show name
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@show verbose
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@show tol
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Δ, sdp_constraints = try
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ΔandSDPconstraints(name)
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catch err
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if isa(err, ArgumentError)
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ΔandSDPconstraints(name, ID, generate_B₄)
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else
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throw(err)
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end
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end
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println("|S| = $(countnz(Δ.coefficients) -1)")
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@show length(Δ)
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@show size(Δ.product_matrix)
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κ, A = try
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κandA(name)
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catch err
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if isa(err, ArgumentError)
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κandA(name, sdp_constraints, Δ, solver; upper_bound=upper_bound)
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else
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throw(err)
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end
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end
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@show κ
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@show sum(A)
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@show maximum(A)
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@show minimum(A)
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if κ > 0
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true_kappa = ℚ_distance_to_positive_cone(Δ, κ, A, tol=tol, verbose=verbose, rational=true)
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true_kappa = Float64(trunc(true_kappa,12))
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if true_kappa > 0
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println("κ($name, S) ≥ $true_kappa: Group HAS property (T)!")
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else
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println("κ($name, S) ≥ $true_kappa: Group may NOT HAVE property (T)!")
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end
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else
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println("κ($name, S) ≥ $κ < 0: Tells us nothing about property (T)")
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end
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end
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