[Bridges] Fix computing Q=U'U in QuadtoSOCBridge when Q has 0 eigenvalue

[odow]

Closes #1971

There aren't many good options here.

My trivial test case is (x + y)^2 / 2.

The permissive options all fail:

julia> using LDLFactorizations, LinearAlgebra, QDLDL, SparseArrays

julia> A = [1.0 1.0; 1.0 1.0]
2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0

julia> Q = SparseArrays.sparse(A);
julia> LinearAlgebra.cholesky(Q)
ERROR: PosDefException: matrix is not positive definite; Factorization failed.
Stacktrace:

julia> LinearAlgebra.ldlt(Q)
ERROR: ZeroPivotException: factorization encountered one or more zero pivots. Consider switching to a pivoted LU factorization.
Stacktrace:

julia> QDLDL.qdldl(Q)
ERROR: Zero entry in D (matrix is not quasidefinite)
Stacktrace:

LDLFactorizations.jl works:

julia> LDLFactorizations.ldl(Q)
LDLFactorizations.LDLFactorization{Float64, Int64, Int64, Int64}(true, false, false, 2, [2, -1], [1, 0], [2, 2], [1, 2], [1, 2], [1, 2, 2], Int64[], Int64[], [2], [1.0], [1.0, 0.0], [0.0, 0.0], [1, 1], 0.0, 0.0, 0.0, 2)

but it requires us to add a GPL dependency.

This PR implements an option I don't think we've previously considered: do something that isn't upper triangular...

Am I missing something obviously wrong?

[mlubin]

I don't think the eivenvalue decomposition is wrong, just potentially expensive and, as a result, surprising to users.

[odow]

The alternative here is that we're going to error. I don't know if people solve problems with super large Q matrices? And need to use this bridge, and worry about performance?

[odow]

The suggestion from #1971 is to add LDLFactorizations as a package extension.

[blegat]

Maybe give the minimal eigenvalue

[blegat]

Using eigenvalues as a fallback sounds like a good idea, maybe we should allow a small tolerance. Historically, we have allowed 1e-10 to be considered as zero:

MathOptInterface.jl/src/Bridges/Constraint/bridges/SquareBridge.jl

Lines 110 to 123 in 05cb226

	if MOI.Utilities.isapprox_zero(diff, 1e-10)
	return nothing
	end
	if MOI.Utilities.isapprox_zero(diff, 1e-8)
	i, j = ij
	@warn(
	"The entries ($i, $j) and ($j, $i) of the matrix are " *
	"almost identical, but a constraint has been added " *
	"to ensure their equality because the largest " *
	"difference between the coefficients is smaller than " *
	"1e-8 but larger than 1e-10. This usually means that " *
	"there is a modeling error in your formulation.",
	)
	end

We can use a very tight tolerance and only loosen it upon request. Maybe starting with 0 is good but it should be easy to find example of reasonable PSD matrices having negative eigenvalues like -1e-15