Polyconvexity with Moments and Sums of Squares

TL;DR

This paper introduces convex-optimization methods using sum-of-squares technology to certify polyconvexity and compute polyconvex envelopes for polynomial matrix functions.

Contribution

It provides tractable sufficient conditions for polyconvexity and a numerical approach to approximate the polyconvex envelope using moment-SOS hierarchy.

Findings

Convex-optimization based criteria effectively certify polyconvexity.

Numerical methods accurately compute polyconvex envelopes.

Approach applicable to polynomial matrix functions in elasticity problems.

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.