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Fix Sparse.Putinar with Newton polytope (#278)
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* Fix Sparse.Putinar with Newton polytope

* Fixes
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@blegat
blegat authored Mar 15, 2023
1 parent 5823575 commit 530ba09
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Showing 3 changed files with 69 additions and 35 deletions.
31 changes: 26 additions & 5 deletions src/Certificate/Sparsity/monomial.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -152,18 +152,39 @@ _monos(basis::MB.MonomialBasis) = basis.monomials
function _gram_monos(vars, certificate::SumOfSquares.Certificate.MaxDegree{CT, MB.MonomialBasis}) where CT
return _monos(SumOfSquares.Certificate.maxdegree_gram_basis(MB.MonomialBasis, vars, certificate.maxdegree))
end
# poly = s0 + sum si gi
# where `s1` are the multipliers with basis `multiplier_generator_monos`
# we want to get the gram basis of `s0`
function _ideal_monos(poly_monos, multiplier_gram_monos)
monos_set = Set(poly_monos)
for (gram_monos, gen_monos) in multiplier_gram_monos
for a in gram_monos
for b in gram_monos
for m in gen_monos
push!(monos_set, a * b * m)
end
end
end
end
return MP.monovec(collect(monos_set))
end
# The ideal certificate should only ask for `MP.monomial`
struct DummyPolynomial{M}
monomials::M
end
MP.monomials(p::DummyPolynomial) = p.monomials
MP.variables(p::DummyPolynomial) = MP.variables(p.monomials)
function sparsity(poly::MP.AbstractPolynomial, domain::SemialgebraicSets.BasicSemialgebraicSet, sp::Monomial, certificate::SumOfSquares.Certificate.AbstractPreorderCertificate)
gram_monos = _gram_monos(
reduce((v, q) -> unique!(sort!([v..., MP.variables(q)...], rev=true)),
domain.p, init = MP.variables(poly)),
SumOfSquares.Certificate.ideal_certificate(certificate)
)
processed = SumOfSquares.Certificate.preprocessed_domain(certificate, domain, poly)
multiplier_generator_monos = [
(_monos(SumOfSquares.Certificate.multiplier_basis(certificate, index, processed)),
MP.monomials(SumOfSquares.Certificate.generator(certificate, index, processed)))
for index in SumOfSquares.Certificate.preorder_indices(certificate, processed)
]
gram_monos = _monos(SumOfSquares.Certificate.gram_basis(
SumOfSquares.Certificate.ideal_certificate(certificate),
DummyPolynomial(_ideal_monos(MP.monomials(poly), multiplier_generator_monos)),
))
cliques, multiplier_cliques = sparsity(MP.monomials(poly), sp, gram_monos, multiplier_generator_monos)
return MB.MonomialBasis.(cliques), [MB.MonomialBasis.(clique) for clique in multiplier_cliques]
end
6 changes: 3 additions & 3 deletions src/constraint.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -81,15 +81,15 @@ function default_ideal_certificate(domain::FullSpace, basis, cone, maxdegree, ne
end

function default_ideal_certificate(
domain::AbstractAlgebraicSet, sparsity::Certificate.Sparsity.NoPattern, basis::AbstractPolynomialBasis, cone, args...)
::AbstractAlgebraicSet, ::Certificate.Sparsity.NoPattern, basis::AbstractPolynomialBasis, cone, args...)
return Certificate.FixedBasis(cone, basis)
end
function default_ideal_certificate(
domain::AbstractAlgebraicSet, sparsity::Certificate.Sparsity.NoPattern, args...)
domain::AbstractAlgebraicSet, ::Certificate.Sparsity.NoPattern, args...)
return default_ideal_certificate(domain, args...)
end
function default_ideal_certificate(
domain::AbstractAlgebraicSet, sparsity::Certificate.Sparsity.Pattern, args...)
::AbstractAlgebraicSet, sparsity::Certificate.Sparsity.Pattern, args...)
return Certificate.Sparsity.Ideal(sparsity, args...)
end

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67 changes: 40 additions & 27 deletions test/sparsity.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -18,11 +18,6 @@ end

set_monos(bases::Vector{<:MB.MonomialBasis}) = Set([basis.monomials for basis in bases])

function Certificate.Sparsity.sparsity(monos::AbstractVector{<:MP.AbstractMonomial}, domain::SemialgebraicSets.BasicSemialgebraicSet, sp::Sparsity.Monomial, maxdegree, degs)
half_monos = Certificate.maxdegree_gram_basis(MB.MonomialBasis, variables, div(maxdegree, 2))
P = Set(monos)
end

"""
wml19()
Expand Down Expand Up @@ -82,8 +77,11 @@ function wml19()
]))
@test set_monos(Certificate.Sparsity.sparsity(f, SignSymmetry(), certificate)) == expected
end
@testset "Example 5.4" begin
preorder_certificate = Certificate.Putinar(Certificate.MaxDegree(SOSCone(), MB.MonomialBasis, 4), SOSCone(), MB.MonomialBasis, 4)
@testset "Example 5.4 $(typeof(ideal_certificate))" for ideal_certificate in [
Certificate.MaxDegree(SOSCone(), MB.MonomialBasis, 4),
Certificate.Newton(SOSCone(), MB.MonomialBasis, tuple()),
]
preorder_certificate = Certificate.Putinar(ideal_certificate, SOSCone(), MB.MonomialBasis, 4)
@polyvar x[1:2]
f = x[1]^4 + x[2]^4 + x[1] * x[2]
K = @set 1 - 2x[1]^2 - x[2]^2 >= 0
Expand Down Expand Up @@ -168,9 +166,8 @@ function l09()
]))
end
end
function square_domain()
d = 6
preorder_certificate = Certificate.Putinar(Certificate.MaxDegree(SOSCone(), MB.MonomialBasis, 6), SOSCone(), MB.MonomialBasis, 6)
function square_domain(ideal_certificate, d)
preorder_certificate = Certificate.Putinar(ideal_certificate, SOSCone(), MB.MonomialBasis, d)
@polyvar x y
f = x^2*y^4 + x^4*y^2 - 3*x^2*y*2 + 1
K = @set(1 - x^2 >= 0 && 1 - y^2 >= 0)
Expand Down Expand Up @@ -238,22 +235,37 @@ function drop_monomials()
end
@test set_monos(Certificate.Sparsity.sparsity(f, Sparsity.Monomial(ChordalCompletion(), k, use_all_monomials), certificate)) == expected
end
preorder_certificate = Certificate.Putinar(Certificate.MaxDegree(SOSCone(), MB.MonomialBasis, 4), SOSCone(), MB.MonomialBasis, 3)
f = polynomial(x^3)
K = @set x >= 0
@testset "$k $use_all_monomials" for k in 0:3, use_all_monomials in [false, true]
basis, preorder_bases = Certificate.Sparsity.sparsity(f, K, Sparsity.Monomial(ChordalCompletion(), k, use_all_monomials), preorder_certificate)
if k == 1 && !use_all_monomials
@test set_monos(basis) == Set(monovec.([[x^2, x]]))
elseif (k == 2 && !use_all_monomials) || (k == 1 && use_all_monomials)
@test set_monos(basis) == Set(monovec.([[x^2, 1], [x^2, x]]))
else
@test set_monos(basis) == Set(monovec.([[x^2, x, 1]]))
end
if k == 1 && !use_all_monomials
@test set_monos(preorder_bases[1]) == Set(monovec.([[x]]))
else
@test set_monos(preorder_bases[1]) == Set(monovec.([[x, 1]]))
@testset "$(typeof(ideal_certificate))" for ideal_certificate in [
Certificate.MaxDegree(SOSCone(), MB.MonomialBasis, 4),
Certificate.Newton(SOSCone(), MB.MonomialBasis, tuple()),
]
preorder_certificate = Certificate.Putinar(ideal_certificate, SOSCone(), MB.MonomialBasis, 3)
f = polynomial(x^3)
K = @set x >= 0
@testset "$k $use_all_monomials" for k in 0:3, use_all_monomials in [false, true]
basis, preorder_bases = Certificate.Sparsity.sparsity(f, K, Sparsity.Monomial(ChordalCompletion(), k, use_all_monomials), preorder_certificate)
if ideal_certificate isa Certificate.Newton
if use_all_monomials
@test set_monos(basis) == Set(monovec.([[x]]))
@test set_monos(preorder_bases[1]) == Set(monovec.([[x, 1]]))
else
@test isempty(basis)
@test set_monos(preorder_bases[1]) == Set(monovec.([[x]]))
end
else
if k == 1 && !use_all_monomials
@test set_monos(basis) == Set(monovec.([[x^2, x]]))
elseif (k == 2 && !use_all_monomials) || (k == 1 && use_all_monomials)
@test set_monos(basis) == Set(monovec.([[x^2, 1], [x^2, x]]))
else
@test set_monos(basis) == Set(monovec.([[x^2, x, 1]]))
end
if (k == 1 && !use_all_monomials)
@test set_monos(preorder_bases[1]) == Set(monovec.([[x]]))
else
@test set_monos(preorder_bases[1]) == Set(monovec.([[x, 1]]))
end
end
end
end
end
Expand All @@ -262,7 +274,8 @@ end
xor_complement_test()
wml19()
l09()
square_domain()
square_domain(Certificate.MaxDegree(SOSCone(), MB.MonomialBasis, 6), 6)
square_domain(Certificate.Newton(SOSCone(), MB.MonomialBasis, tuple()), 6)
sum_square(8)
@test Certificate.Sparsity.appropriate_type(32) == Int64
sum_square(32)
Expand Down

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