add paper version of check_positivity script
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paper_data/check_positivity.jl
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142
paper_data/check_positivity.jl
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using AbstractAlgebra
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using Groups
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using GroupRings
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using PropertyT
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using IntervalArithmetic
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using SCS
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using JLD
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include("sqadjop.jl")
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invalid_use_message = """You need to call this script in the parent folder of oSAutF5_r2 folder.
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Provide also the two parameters: `-k` and `-lambda`"""
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if !(iseven(length(ARGS)))
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throw(invalid_use_message)
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end
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K = LAMBDA = nothing
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for i in 1:2:length(ARGS)
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if ARGS[i] == "-k"
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K = parse(Float64, ARGS[i+1])
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elseif ARGS[i] == "-lambda"
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LAMBDA = parse(Float64, ARGS[i+1])
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end
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end
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if K == nothing || LAMBDA == nothing
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throw(invalid_use_message)
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end
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info("Running checks for Adj₅ + $K·Op₅ - $LAMBDA·Δ₅")
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G = AutGroup(FreeGroup(5), special=true)
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S = generating_set(G)
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const prefix = "oSAutF5_r2"
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isdir(prefix) || mkpath(prefix)
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const DELTA_FILE = joinpath(prefix,"delta.jld")
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const SQADJOP_FILE = joinpath(prefix, "SqAdjOp_coeffs.jld")
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const ORBITDATA_FILE = joinpath(prefix, "OrbitData.jld")
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const fullpath = joinpath(prefix, string(LAMBDA))
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isdir(fullpath) || mkpath(fullpath)
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const SOLUTION_FILE = joinpath(fullpath, "solution.jld")
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info("Looking for delta.jld, SqAdjOp_coeffs.jld and OrbitData.jld in $prefix")
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if isfile(DELTA_FILE) && isfile(SQADJOP_FILE) && isfile(ORBITDATA_FILE)
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# cached
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Δ = PropertyT.loadGRElem(DELTA_FILE, G)
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RG = parent(Δ)
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orbit_data = load(ORBITDATA_FILE, "OrbitData")
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sq_c, adj_c, op_c = load(SQADJOP_FILE, "Sq", "Adj", "Op")
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sq = GroupRingElem(sq_c, RG)
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adj = GroupRingElem(adj_c, RG)
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op = GroupRingElem(op_c, RG);
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else
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info("Computing Laplacian")
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Δ = PropertyT.Laplacian(S, 2)
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PropertyT.saveGRElem(DELTA_FILE, Δ)
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RG = parent(Δ)
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info("Computing Sq, Adj, Op")
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@time sq, adj, op = Sq(RG), Adj(RG), Op(RG)
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save(SQADJOP_FILE, "Sq", sq.coeffs, "Adj", adj.coeffs, "Op", op.coeffs)
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info("Compute OrbitData")
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if !isfile(ORBITDATA_FILE)
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orbit_data = PropertyT.OrbitData(RG, sett.autS)
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save(ORBITDATA_FILE, "OrbitData", orbit_data)
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else
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orbit_data = load(ORBITDATA_FILE, "OrbitData")
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end
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end;
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orbit_data = PropertyT.decimate(orbit_data);
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elt = adj + K*op;
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info("Looking for solution.jld in $fullpath")
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if !isfile(SOLUTION_FILE)
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info("solution.jld not found, attempting to recreate one.")
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λ = Ps = ws = nothing
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SDP_problem, varλ, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=LAMBDA)
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begin
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scs_solver = SCS.SCSSolver(linear_solver=SCS.Direct,
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eps=1e-12,
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max_iters=200_000,
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alpha=1.5,
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acceleration_lookback=1)
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JuMP.setsolver(SDP_problem, scs_solver)
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end
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i = 0
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# for i in 1:6
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status= :Unknown
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while status !=:Optimal
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i += 1
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status, (λ, Ps, ws) = PropertyT.solve(SDP_problem, varλ, varP, ws);
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precision = abs(λ - LAMBDA)
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println("i = $i, \t precision = $precision")
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end
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info("Reconstructing P...")
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@time P = PropertyT.reconstruct(Ps, orbit_data);
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save(SOLUTION_FILE, "λ", λ, "P", P)
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end
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info("Checking the sum of squares solution for Adj₅ + $K·Op₅ - $LAMBDA·Δ₅")
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P, λ = load(SOLUTION_FILE, "P", "λ")
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info("Computing Q = √P")
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@time const Q = real(sqrtm(P));
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function SOS_residual(eoi::GroupRingElem, Q::Matrix)
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RG = parent(eoi)
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@time sos = PropertyT.compute_SOS(RG, Q);
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return eoi - sos
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end
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info("In floating point arithmetic:")
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EOI = elt - λ*Δ
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b = SOS_residual(EOI, Q)
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@show norm(b, 1);
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info("In interval arithmetic:")
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EOI_int = elt - @interval(λ)*Δ;
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Q_int = PropertyT.augIdproj(Q);
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@assert all([zero(eltype(Q)) in sum(view(Q_int, :, i)) for i in 1:size(Q_int, 2)])
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b_int = SOS_residual(EOI_int, Q_int)
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@show norm(b_int, 1);
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info("λ is certified to be > ", (@interval(λ) - 2^2*norm(b_int,1))).lo
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51
paper_data/sqadjop.jl
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51
paper_data/sqadjop.jl
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@ -0,0 +1,51 @@
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indexing(n) = [(i,j) for i in 1:n for j in 1:n if i≠j]
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function generating_set(G::AutGroup{N}, n=N) where N
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rmuls = [Groups.rmul_autsymbol(i,j) for (i,j) in indexing(n)]
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lmuls = [Groups.lmul_autsymbol(i,j) for (i,j) in indexing(n)]
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gen_set = G.([rmuls; lmuls])
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return [gen_set; inv.(gen_set)]
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end
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function Sq(RG::GroupRing{AutGroup{N}}) where N
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S₂ = generating_set(RG.group, 2)
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ℤ = Int64
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Δ₂ = length(S₂)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
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elt = sum(σ(Δ₂)^2 for σ in Alt_N)
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# return RG(elt.coeffs÷factorial(N-2))
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return elt
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end
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function Adj(RG::GroupRing{AutGroup{N}}) where N
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S₂ = small_generating_set(RG, 2)
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ℤ = Int64
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Δ₂ = length(T2)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
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adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
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adj(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 1]
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@time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in adj(σ)) for σ in Alt_N);
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# return RG(elt.coeffs÷factorial(N-2)^2)
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return elt
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end
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function Op(RG::GroupRing{AutGroup{N}}) where N
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S₂ = small_generating_set(RG, 2)
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ℤ = Int64
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Δ₂ = length(T2)*one(RG, ℤ) - RG(S₂, ℤ);
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Alt_N = [g for g in elements(PermutationGroup(N)) if parity(g) == 0]
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adj(σ::perm, τ::perm, i=1, j=2) = Set([σ[i], σ[j]]) ∩ Set([τ[i], τ[j]])
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op(σ::perm) = [τ for τ in Alt_N if length(adj(σ, τ)) == 0]
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@time elt = sum(σ(Δ₂)*sum(τ(Δ₂) for τ in op(σ)) for σ in Alt_N);
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# return RG(elt.coeffs÷factorial(N-2)^2)
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return elt
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end
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