284 lines
8.5 KiB
Julia
284 lines
8.5 KiB
Julia
using Printf
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###############################################################################
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#
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# Settings and filenames
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#
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###############################################################################
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abstract type Settings end
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struct Naive{El} <: Settings
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name::String
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G::Group
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S::Vector{El}
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radius::Int
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upper_bound::Float64
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solver::JuMP.OptimizerFactory
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warmstart::Bool
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end
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struct Symmetrized{El} <: Settings
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name::String
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G::Group
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S::Vector{El}
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autS::Group
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radius::Int
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upper_bound::Float64
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solver::JuMP.OptimizerFactory
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warmstart::Bool
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end
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function Settings(name::String,
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G::Group, S::Vector{<:GroupElem}, solver::JuMP.OptimizerFactory;
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radius::Integer=2, upper_bound::Float64=1.0, warmstart=true)
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return Naive(name, G, S, radius, upper_bound, solver, warmstart)
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end
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function Settings(name::String,
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G::Group, S::Vector{<:GroupElem}, autS::Group, solver::JuMP.OptimizerFactory;
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radius::Integer=2, upper_bound::Float64=1.0, warmstart=true)
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return Symmetrized(name, G, S, autS, radius, upper_bound, solver, warmstart)
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end
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prefix(s::Naive) = s.name
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prefix(s::Symmetrized) = "o"*s.name
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suffix(s::Settings) = "$(s.upper_bound)"
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prepath(s::Settings) = prefix(s)
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fullpath(s::Settings) = joinpath(prefix(s), suffix(s))
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filename(sett::Settings, s::Symbol) = filename(sett, Val{s})
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filename(sett::Settings, ::Type{Val{:fulllog}}) =
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joinpath(fullpath(sett), "full_$(string(now())).log")
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filename(sett::Settings, ::Type{Val{:solverlog}}) =
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joinpath(fullpath(sett), "solver_$(string(now())).log")
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filename(sett::Settings, ::Type{Val{:Δ}}) =
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joinpath(prepath(sett), "delta.jld")
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filename(sett::Settings, ::Type{Val{:OrbitData}}) =
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joinpath(prepath(sett), "OrbitData.jld")
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filename(sett::Settings, ::Type{Val{:warmstart}}) =
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joinpath(fullpath(sett), "warmstart.jld")
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filename(sett::Settings, ::Type{Val{:solution}}) =
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joinpath(fullpath(sett), "solution.jld")
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###############################################################################
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#
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# λandP
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#
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###############################################################################
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function warmstart(sett::Settings)
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if sett.warmstart && isfile(filename(sett, :warmstart))
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ws = load(filename(sett, :warmstart), "warmstart")
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else
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ws = nothing
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end
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return ws
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end
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function computeλandP(sett::Naive, Δ::GroupRingElem;
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solverlog=tempname()*".log")
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@info("Creating SDP problem...")
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SDP_problem = SOS_problem(Δ^2, Δ, upper_bound=sett.upper_bound)
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@info(Base.repr(SDP_problem))
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ws = warmstart(sett)
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@time status, ws = PropertyT.solve(solverlog, SDP_problem, sett.solver, ws)
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@info("Solver's status: $status")
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P = value.(SDP_problem[:P])
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λ = value(SDP_problem[:λ])
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save(filename(sett, :warmstart),
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"warmstart", (ws.primal, ws.dual, ws.slack), "P", P, "λ", λ)
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return λ, P
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end
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function computeλandP(sett::Symmetrized, Δ::GroupRingElem;
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solverlog=tempname()*".log")
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if !isfile(filename(sett, :OrbitData))
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isdefined(parent(Δ), :basis) || throw("You need to define basis of Group Ring to compute orbit decomposition!")
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orbit_data = OrbitData(parent(Δ), sett.autS)
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save(filename(sett, :OrbitData), "OrbitData", orbit_data)
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end
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orbit_data = load(filename(sett, :OrbitData), "OrbitData")
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orbit_data = decimate(orbit_data)
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@info("Creating SDP problem...")
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SDP_problem, varP = SOS_problem(Δ^2, Δ, orbit_data, upper_bound=sett.upper_bound)
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@info(Base.repr(SDP_problem))
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ws = warmstart(sett)
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@time status, ws = PropertyT.solve(solverlog, SDP_problem, sett.solver, ws)
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@info("Solver's status: $status")
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λ = value(SDP_problem[:λ])
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Ps = [value.(P) for P in varP]
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save(filename(sett, :warmstart),
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"warmstart", (ws.primal, ws.dual, ws.slack), "Ps", Ps, "λ", λ)
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@info("Reconstructing P...")
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@time P = reconstruct(Ps, orbit_data)
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return λ, P
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end
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###############################################################################
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#
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# Checking solution
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#
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###############################################################################
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function distance_to_positive_cone(Δ::GroupRingElem, λ, Q; R::Int=2)
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separator = "-"^76
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info_strs = [separator,
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"Checking in floating-point arithmetic...",
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"λ = $λ"]
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@info(join(info_strs, "\n"))
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eoi = Δ^2-λ*Δ
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@info("Computing sum of squares decomposition...")
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@time residual = eoi - compute_SOS(parent(eoi), augIdproj(Q))
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L1_norm = norm(residual,1)
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distance = λ - 2.0^(2ceil(log2(R)))*L1_norm
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info_strs = ["Numerical metrics:",
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"ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ≈ $(@sprintf("%.10f", aug(residual)))",
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"‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ≈ $(@sprintf("%.10f", L1_norm))",
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"Floating point distance (to positive cone) ≈",
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"$(@sprintf("%.10f", distance))"]
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@info(join(info_strs, "\n"))
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if distance ≤ 0
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return distance
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end
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λ = @interval(λ)
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info_strs = [separator,
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"Checking in interval arithmetic...",
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"λ ∈ $λ"]
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@info(join(info_strs, "\n"))
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eoi = Δ^2 - λ*Δ
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@info("Projecting columns of Q to the augmentation ideal...")
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@time Q, check = augIdproj(Interval, Q)
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info_strs = ["Checking that sum of every column contains 0.0...",
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(check ? "DONE!" : "FAILED!")]
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@info(join(info_strs, "\n"))
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check || @warn("The following numbers are meaningless!")
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@info("Computing sum of squares decomposition...")
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@time residual = eoi - compute_SOS(parent(eoi), Q)
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L1_norm = norm(residual,1)
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distance = λ - 2.0^(2ceil(log2(R)))*L1_norm
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info_strs = ["Numerical metrics:",
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"ɛ(Δ² - λΔ - ∑ξᵢ*ξᵢ) ∈ $(aug(residual))",
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"‖Δ² - λΔ - ∑ξᵢ*ξᵢ‖₁ ∈ $(L1_norm)",
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"Interval distance (to positive cone) ∈",
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"$(distance)",
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separator]
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@info(join(info_strs, "\n"))
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return distance.lo
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end
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###############################################################################
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#
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# Interpreting the numerical results
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#
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###############################################################################
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Kazhdan(λ::Number, N::Integer) = sqrt(2*λ/N)
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function check_property_T(sett::Settings)
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print_summary(sett)
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certified_sgap = spectral_gap(sett)
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return interpret_results(sett, certified_sgap)
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end
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function print_summary(sett::Settings)
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separator = "="^76
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info_strs = [separator,
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"Running tests for $(sett.name):",
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"Upper bound for λ: $(sett.upper_bound), on radius $(sett.radius).",
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"Warmstart: $(sett.warmstart)",
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"Results will be stored in ./$(PropertyT.prepath(sett))",
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"Solver: $(typeof(sett.solver()))",
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"Solvers options: "]
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append!(info_strs, [rpad(" $k", 30)* "→ \t$v" for (k,v) in sett.solver().options])
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push!(info_strs, separator)
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@info(join(info_strs, "\n"))
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end
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function interpret_results(sett::Settings, sgap::Number)
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if sgap > 0
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Kazhdan_κ = Kazhdan(sgap, length(sett.S))
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if Kazhdan_κ > 0
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@info("κ($(sett.name), S) ≥ $Kazhdan_κ: Group HAS property (T)!")
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return true
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end
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end
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info_strs = ["The certified lower bound on the spectral gap is negative:",
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"λ($(sett.name), S) ≥ 0.0 > $sgap",
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"This tells us nothing about property (T)"]
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@info(join(info_strs, "\n"))
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return false
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end
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function spectral_gap(sett::Settings)
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fp = PropertyT.fullpath(sett)
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isdir(fp) || mkpath(fp)
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if isfile(filename(sett,:Δ))
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# cached
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@info("Loading precomputed Δ...")
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Δ = loadGRElem(filename(sett,:Δ), sett.G)
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else
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# compute
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Δ = Laplacian(sett.S, sett.radius)
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saveGRElem(filename(sett, :Δ), Δ)
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end
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if !sett.warmstart && isfile(filename(sett, :solution))
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λ, P = load(filename(sett, :solution), "λ", "P")
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else
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λ, P = computeλandP(sett, Δ,
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solverlog=filename(sett, :solverlog))
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save(filename(sett, :solution), "λ", λ, "P", P)
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if λ < 0
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@warn("Solver did not produce a valid solution!")
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end
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end
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info_strs = ["λ = $λ",
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"sum(P) = $(sum(P))",
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"maximum(P) = $(maximum(P))",
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"minimum(P) = $(minimum(P))"]
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@info(join(info_strs, "\n"))
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isapprox(eigvals(P), abs.(eigvals(P))) ||
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@warn("The solution matrix doesn't seem to be positive definite!")
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@time Q = real(sqrt( (P.+ P')./2 ))
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certified_sgap = distance_to_positive_cone(Δ, λ, Q, R=sett.radius)
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return certified_sgap
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end
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