GroupsWithPropertyT/paper_data/check_positivity.jl

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using AbstractAlgebra
using Groups
using GroupRings
using PropertyT
using IntervalArithmetic
using SCS
using JLD
include("sqadjop.jl")
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 (numerical) parameters: `-k` and `-lambda`"""
if !(iseven(length(ARGS)))
throw(invalid_use_message)
end
K = LAMBDA = nothing
for i in 1:2:length(ARGS)
if ARGS[i] == "-k"
K = parse(Float64, ARGS[i+1])
elseif ARGS[i] == "-lambda"
LAMBDA = parse(Float64, ARGS[i+1])
end
end
if K == nothing || LAMBDA == nothing
throw(invalid_use_message)
end
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info("Running checks for 36(Adj₅ + $K·Op₅) - $LAMBDA·Δ₅")
G = AutGroup(FreeGroup(5), special=true)
S = generating_set(G)
const prefix = "oSAutF5_r2"
isdir(prefix) || mkpath(prefix)
const DELTA_FILE = joinpath(prefix,"delta.jld")
const SQADJOP_FILE = joinpath(prefix, "SqAdjOp_coeffs.jld")
const ORBITDATA_FILE = joinpath(prefix, "OrbitData.jld")
const fullpath = joinpath(prefix, string(LAMBDA))
isdir(fullpath) || mkpath(fullpath)
const SOLUTION_FILE = joinpath(fullpath, "solution.jld")
info("Looking for delta.jld, SqAdjOp_coeffs.jld and OrbitData.jld in $prefix")
if isfile(DELTA_FILE) && isfile(SQADJOP_FILE) && isfile(ORBITDATA_FILE)
# cached
Δ = PropertyT.loadGRElem(DELTA_FILE, G)
RG = parent(Δ)
orbit_data = load(ORBITDATA_FILE, "OrbitData")
sq_c, adj_c, op_c = load(SQADJOP_FILE, "Sq", "Adj", "Op")
sq = GroupRingElem(sq_c, RG)
adj = GroupRingElem(adj_c, RG)
op = GroupRingElem(op_c, RG);
else
info("Computing Laplacian")
Δ = PropertyT.Laplacian(S, 2)
PropertyT.saveGRElem(DELTA_FILE, Δ)
RG = parent(Δ)
info("Computing Sq, Adj, Op")
@time sq, adj, op = Sq(RG), Adj(RG), Op(RG)
save(SQADJOP_FILE, "Sq", sq.coeffs, "Adj", adj.coeffs, "Op", op.coeffs)
info("Compute OrbitData")
if !isfile(ORBITDATA_FILE)
orbit_data = PropertyT.OrbitData(RG, sett.autS)
save(ORBITDATA_FILE, "OrbitData", orbit_data)
else
orbit_data = load(ORBITDATA_FILE, "OrbitData")
end
end;
orbit_data = PropertyT.decimate(orbit_data);
elt = adj + K*op;
info("Looking for solution.jld in $fullpath")
if !isfile(SOLUTION_FILE)
info("solution.jld not found, attempting to recreate one.")
λ = Ps = ws = nothing
SDP_problem, varλ, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=LAMBDA)
begin
scs_solver = SCS.SCSSolver(linear_solver=SCS.Direct,
eps=1e-12,
max_iters=200_000,
alpha=1.5,
acceleration_lookback=1)
JuMP.setsolver(SDP_problem, scs_solver)
end
i = 0
# for i in 1:6
status= :Unknown
while status !=:Optimal
i += 1
status, (λ, Ps, ws) = PropertyT.solve(SDP_problem, varλ, varP, ws);
precision = abs(λ - LAMBDA)
println("i = $i, \t precision = $precision")
end
info("Reconstructing P...")
@time P = PropertyT.reconstruct(Ps, orbit_data);
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info("Computing Q = √P")
@time const Q = real(sqrtm(P));
save(SOLUTION_FILE, "λ", λ, "Q", Q)
end
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info("Checking the sum of squares solution for 36(Adj₅ + $K·Op₅) - $LAMBDA·Δ₅")
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Q, λ = load(SOLUTION_FILE, "Q", "λ")
function SOS_residual(eoi::GroupRingElem, Q::Matrix)
RG = parent(eoi)
@time sos = PropertyT.compute_SOS(RG, Q);
return eoi - sos
end
info("In floating point arithmetic:")
EOI = elt - λ*Δ
b = SOS_residual(EOI, Q)
@show norm(b, 1);
info("In interval arithmetic:")
EOI_int = elt - @interval(λ)*Δ;
Q_int = PropertyT.augIdproj(Q);
@assert all([zero(eltype(Q)) in sum(view(Q_int, :, i)) for i in 1:size(Q_int, 2)])
b_int = SOS_residual(EOI_int, Q_int)
@show norm(b_int, 1);
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λ_cert = @interval(λ) - 2^2*norm(b_int,1)
info("λ is certified to be > ", λ_cert.lo)
info("i.e Adj₅ + $K·Op₅ - $((λ_cert/36).lo)·Δ₅ ∈ Σ²₂ ISAut(F₅)")