Polyconvexity with Moments and Sums of Squares
Abstract
A function of a matrix is polyconvex when it can be expressed as a convex function of the matrix minors. Polyconvexity is a regularity condition ensuring existence of minimizers in nonlinear elasticity and, more broadly, in vectorial problems of the calculus of variations, when minimizing integral gradient functionals. The polyconvex envelope of a function is the largest polyconvex lower bound. Yet deciding whether a given energy is polyconvex, or computing the polyconvex envelope, are generally difficult problems. This paper focuses on polynomial matrix functions. We propose (i) tractable convex-optimization based sufficient conditions to certify polyconvexity via sum-of-squares (SOS) technology, and (ii) a principled numerical method to compute the polyconvex envelope pointwise, based on the moment-SOS hierarchy from polynomial optimization.
Keywords
Cite
@article{arxiv.2604.11124,
title = {Polyconvexity with Moments and Sums of Squares},
author = {Giovanni Fantuzzi and Didier Henrion and Martin Kru{ž}ík and Ajay Murali and Stephan Weis},
journal= {arXiv preprint arXiv:2604.11124},
year = {2026}
}