Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization

TL;DR

This paper introduces a convex dual relaxation method, called PDR, for obtaining lower bounds on complex integral variational problems, and demonstrates its effectiveness and convergence through sum-of-squares relaxations in polynomial cases.

Contribution

The paper develops a novel convex dual relaxation framework for integral variational problems, extending previous methods with sum-of-squares relaxations for polynomial cases.

Findings

PDR provides sharp lower bounds for certain classes of problems.

SOS relaxations of PDR converge to the optimal bounds as polynomial degree increases.

The method is computationally feasible for problems with moderate dimensions.

Abstract

We present a method for finding lower bounds on the global infima of integral variational problems, wherein $\int_\Omega f(x,u(x),\nabla u(x)){\rm d}x$ is minimized over functions $u\colon\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$ satisfying given equality or inequality constraints. Each constraint may be imposed over $\Omega$ or its boundary, either pointwise or in an integral sense. These global minimizations are generally non-convex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the original problem. The PDR can be derived by dualizing and relaxing the original problem; its constraints are pointwise equalities or inequalities over finite-dimensional sets, rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of $(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions $m,n$ and number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat. Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincar\'e inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several examples.