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add 1812.03456 positivity tests in SL(n,Z)

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kalmarek 2019-06-30 13:20:39 +02:00
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test/1812.03456.jl
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@ -91,3 +91,135 @@
end
end
@testset "1812.03456 examples" begin
with_SCS = with_optimizer(SCS.Optimizer,
linear_solver=SCS.Direct,
eps=2e-10,
max_iters=20000,
alpha=1.5,
acceleration_lookback=10,
warm_start=true)
function SOS_residual(x::GroupRingElem, Q::Matrix)
RG = parent(x)
@time sos = PropertyT.compute_SOS(RG, Q);
return x - sos
end
function check_positivity(elt, Δ, orbit_data, upper_bound, warm=nothing; with_solver=with_SCS)
SDP_problem, varP = PropertyT.SOS_problem(elt, Δ, orbit_data; upper_bound=upper_bound)
status, warm = PropertyT.solve(SDP_problem, with_solver, warm);
@info "Optimization status:" status
λ = value(SDP_problem[:λ])
Ps = [value.(P) for P in varP]
Qs = real.(sqrt.(Ps));
Q = PropertyT.reconstruct(Qs, orbit_data);
b = SOS_residual(elt - λ*Δ, Q)
return b, λ, warm
end
@testset "SL(3,Z)" begin
N = 3
halfradius = 2
M = MatrixSpace(Nemo.ZZ, N,N)
S = PropertyT.generating_set(M)
Δ = PropertyT.Laplacian(S, halfradius)
RG = parent(Δ)
orbit_data = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
orbit_data = PropertyT.decimate(orbit_data);
@testset "Sq₃ is SOS" begin
elt = PropertyT.Sq(RG)
UB = 0.05 # 0.105?
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
@test 2^2*norm(residual, 1) < λ/100
end
@testset "Adj₃ is SOS" begin
elt = PropertyT.Adj(RG)
UB = 0.1 # 0.157?
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) < λ
@test 2^2*norm(residual, 1) < λ/100
end
@testset "Op₃ is empty, so can not be certified" begin
elt = PropertyT.Op(RG)
UB = Inf
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) > λ
end
end
@testset "SL(4,Z)" begin
N = 4
halfradius = 2
M = MatrixSpace(Nemo.ZZ, N,N)
S = PropertyT.generating_set(M)
Δ = PropertyT.Laplacian(S, halfradius)
RG = parent(Δ)
orbit_data = PropertyT.OrbitData(RG, WreathProduct(PermGroup(2), PermGroup(N)))
orbit_data = PropertyT.decimate(orbit_data);
@testset "Sq₄ is SOS" begin
elt = PropertyT.Sq(RG)
UB = 0.2 # 0.3172
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
@test 2^2*norm(residual, 1) < λ/100
end
@testset "Adj₄ is SOS" begin
elt = PropertyT.Adj(RG)
UB = 0.3 # 0.5459?
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
@test 2^2*norm(residual, 1) < λ/100
end
@testset "we can't cerify that Op₄ SOS" begin
elt = PropertyT.Op(RG)
UB = 2.0
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) > λ # i.e. we can certify positivity
end
@testset "Adj₄ + Op₄ is SOS" begin
elt = PropertyT.Adj(RG) + PropertyT.Op(RG)
UB = 0.6 # 0.82005
residual, λ, _ = check_positivity(elt, Δ, orbit_data, UB)
@info "obtained λ and residual" λ norm(residual, 1)
@test 2^2*norm(residual, 1) < λ # i.e. we can certify positivity
@test 2^2*norm(residual, 1) < λ/100
end
end
end