Lifting-Free Quadratic Sum-Of-Squares Programming

What the math gives to ML

The paper replaces the usual lifted semidefinite representation of quadratic SOS programs with a norm-regularized conic formulation whose dual is unconstrained and has a Lipschitz-continuous gradient. The transferable asset is not SOS itself, but the combination of cone projection, explicit Fenchel conjugates, and accelerated dual optimization: a large PSD-constrained subproblem can be solved using matrix projections and dual variables associated with equality constraints, without introducing auxiliary Schur-complement variables. This is useful for neural models that learn polynomial certificates, Lyapunov functions, energy functions, or PSD Gram-matrix parameters. The most practical adaptation is a differentiable training-time certificate layer solved by accelerated dual ascent, with the regularization parameter gradually reduced to control approximation bias.