using JuMP using SCS export Settings, OrbitData immutable Settings name::String N::Int G::Group S::Vector AutS::Group radius::Int solver::SCSSolver upper_bound::Float64 tol::Float64 end immutable OrbitData name::String Us::Vector Ps::Vector{Array{JuMP.Variable,2}} cnstr::Vector laplacian::Vector laplacianSq::Vector dims::Vector{Int} end function OrbitData(name::String) splap = load(joinpath(name, "delta.jld"), "Δ"); pm = load(joinpath(name, "pm.jld"), "pm"); cnstr = PropertyT.constraints_from_pm(pm); splap² = similar(splap) splap² = GroupRings.mul!(splap², splap, splap, pm); Uπs = load(joinpath(name, "U_pis.jld"), "Uπs"); #dimensions of the corresponding πs: dims = load(joinpath(name, "U_pis.jld"), "dims") m, P = init_model(Uπs); orbits = load(joinpath(name, "orbits.jld"), "orbits"); n = size(Uπs[1],1) orb_spcnstrm = [orbit_constraint(cnstr[collect(orb)], n) for orb in orbits] orb_splap = orbit_spvector(splap, orbits) orb_splap² = orbit_spvector(splap², orbits) orbData = OrbitData(name, Uπs, P, orb_spcnstrm, orb_splap, orb_splap², dims); # orbData = OrbitData(name, Uπs, P, orb_spcnstrm, splap, splap², dims); return m, orbData end include("OrbitDecomposition.jl") dens(M::SparseMatrixCSC) = length(M.nzval)/length(M) dens(M::AbstractArray) = sum(abs.(M) .!= 0)/length(M) function sparsify{T}(U::AbstractArray{T}, check=true) W = deepcopy(U) W[abs.(W) .< eps(T)] .= zero(T) if check && rank(W) != rank(U) info("Sparsification would decrease the rank!") W = U else info("Sparsified:", rpad(dens(U), 10), "\t", rpad(dens(W),10)) end W = sparse(W) dropzeros!(W) return W end function init_orbit_data(logger, sett::Settings; radius=2) ex(fname) = isfile(joinpath(sett.name, fname)) files_exists = ex.(["delta.jld", "pm.jld", "U_pis.jld", "orbits.jld"]) if !all(files_exists) compute_orbit_data(logger, sett.name, sett.G, sett.S, sett.AutS, radius=radius) end return 0 end function transform(U::AbstractArray, V::AbstractArray; sparse=true) if sparse return sparsify(U'*V*U) else return U'*V*U end end A(data::OrbitData, π, t) = data.dims[π]*transform(data.Us[π], data.cnstr[t]) function constrLHS(m::JuMP.Model, data::OrbitData, t) l = endof(data.Us) lhs = @expression(m, sum(vecdot(A(data, π, t), data.Ps[π]) for π in 1:l)) return lhs end function addconstraints!(m::JuMP.Model, data::OrbitData, l::Int=length(data.laplacian); var::Symbol = :λ) λ = m[var] # orbits = load(joinpath(data.name, "orbits.jld"), "orbits"); # locate(t, orb=orbits) = findfirst(x->t in x, orb) for t in 1:l # lhs = constrLHS(m, data, locate(t)) lhs = constrLHS(m, data, t) d, d² = data.laplacian[t], data.laplacianSq[t] if lhs == zero(lhs) if d == 0 && d² == 0 info("Detected empty constraint") continue else warn("Adding unsatisfiable constraint!") end end JuMP.@constraint(m, lhs == d² - λ*d) end end function init_model(Uπs) m = JuMP.Model(); l = size(Uπs,1) P = Vector{Array{JuMP.Variable,2}}(l) for k in 1:l s = size(Uπs[k],2) P[k] = JuMP.@variable(m, [i=1:s, j=1:s]) JuMP.@SDconstraint(m, P[k] >= 0.0) end JuMP.@variable(m, λ >= 0.0) JuMP.@objective(m, Max, λ) return m, P end function create_SDP_problem(name::String; upper_bound=Inf) info(PropertyT.logger, "Loading orbit data....") t = @timed SDP_problem, orb_data = OrbitData(name); info(PropertyT.logger, PropertyT.timed_msg(t)) if upper_bound < Inf λ = JuMP.getvariable(SDP_problem, :λ) JuMP.@constraint(SDP_problem, λ <= upper_bound) end info(PropertyT.logger, "Adding constraints... ") t = @timed addconstraints!(SDP_problem, orb_data) info(PropertyT.logger, PropertyT.timed_msg(t)) return SDP_problem, orb_data end function λandP(m::JuMP.Model, data::OrbitData) varλ = m[:λ] varP = data.Ps λ, Ps = PropertyT.λandP(data.name, m, varλ, varP) return λ, Ps end function λandP(m::JuMP.Model, data::OrbitData, sett::Settings) info(PropertyT.logger, "Solving SDP problem...") λ, Ps = λandP(m, data) info(PropertyT.logger, "Reconstructing P...") mreps = matrix_reps(sett.G, sett.S, sett.AutS, sett.radius) recP = reconstruct_sol(mreps, data.Us, Ps, data.dims) fname = PropertyT.λSDPfilenames(data.name)[2] save(fname, "origP", Ps, "P", recP) return λ, recP end function check_property_T(sett::Settings) init_orbit_data(logger, sett, radius=sett.radius) Δ = PropertyT.ΔandSDPconstraints(sett.name, sett.G)[1] fnames = PropertyT.λSDPfilenames(sett.name) if all(isfile.(fnames)) λ, P = PropertyT.λandP(sett.name) else info(logger, "Creating SDP problem...") SDP_problem, orb_data = create_SDP_problem(sett.name, upper_bound=sett.upper_bound) JuMP.setsolver(SDP_problem, sett.solver) λ, P = λandP(SDP_problem, orb_data, sett) end info(logger, "λ = $λ") info(logger, "sum(P) = $(sum(P))") info(logger, "maximum(P) = $(maximum(P))") info(logger, "minimum(P) = $(minimum(P))") if λ > 0 isapprox(eigvals(P), abs.(eigvals(P)), atol=sett.tol) || warn("The solution matrix doesn't seem to be positive definite!") # @assert P == Symmetric(P) Q = real(sqrtm(Symmetric(P))) sgap = PropertyT.check_distance_to_positive_cone(Δ, λ, Q, 2*sett.radius, tol=sett.tol, rational=false) if isa(sgap, Interval) sgap = sgap.lo end if sgap > 0 info(logger, "λ ≥ $(Float64(trunc(sgap,12)))") Kazhdan_κ = PropertyT.Kazhdan_from_sgap(sgap, length(sett.S)) Kazhdan_κ = Float64(trunc(Kazhdan_κ, 12)) info(logger, "κ($(sett.name), S) ≥ $Kazhdan_κ: Group HAS property (T)!") return true else sgap = Float64(trunc(sgap, 12)) info(logger, "λ($(sett.name), S) ≥ $sgap: Group may NOT HAVE property (T)!") return false end end info(logger, "κ($(sett.name), S) ≥ $λ < 0: Tells us nothing about property (T)") return false end