Optimizing quasi-dissipative evolution equations with the moment-SOS hierarchy

TL;DR

This paper establishes the equivalence between certain nonlinear evolution equations and their measure-based reformulations, enabling globally convergent numerical optimization via the moment-SOS hierarchy, especially for reaction-diffusion equations.

Contribution

It proves the absence of relaxation gaps for quasi-dissipative equations and introduces a novel approach for global optimization using the moment-SOS hierarchy.

Findings

No relaxation gap between equations and measure reformulation.

Global convergence guarantees for the moment-SOS hierarchy.

Applicable to reaction-diffusion equations with polynomial nonlinearity.

Abstract

We prove that there is no relaxation gap between a quasi-dissipative nonlinear evolution equation in a Hilbert space and its linear Liouville equation reformulation on probability measures. In other words, strong and generalized solutions of such equations are unique in the class of measure-valued solutions. As a major consequence, non-convex numerical optimization over these non-linear partial differential equations can be carried out with the infinite-dimensional moment-SOS hierarchy with global convergence guarantees. This covers in particular all reaction-diffusion equations with polynomial nonlinearity.