Global Solutions to Nonconvex Problems by Evolution of Hamilton-Jacobi PDEs

TL;DR

This paper introduces HJ-MAD, a Hamilton-Jacobi-based zero-order algorithm that guarantees convergence to global minima for nonconvex, nondifferentiable optimization problems by approximating gradients of the Moreau envelope.

Contribution

It presents a novel method combining Hamilton-Jacobi PDEs and Moreau envelope smoothing to achieve global convergence in nonconvex optimization.

Findings

Demonstrates global convergence in numerical examples

Approximates gradients via Hopf-Lax formula for viscous Hamilton-Jacobi equation

Guarantees convergence assuming continuity of the objective function

Abstract

Computing tasks may often be posed as optimization problems. The objective functions for real-world scenarios are often nonconvex and/or nondifferentiable. State-of-the-art methods for solving these problems typically only guarantee convergence to local minima. This work presents Hamilton-Jacobi-based Moreau Adaptive Descent (HJ-MAD), a zero-order algorithm with guaranteed convergence to global minima, assuming continuity of the objective function. The core idea is to compute gradients of the Moreau envelope of the objective (which is "piece-wise convex") with adaptive smoothing parameters. Gradients of the Moreau envelope \rev{(\ie proximal operators)} are approximated via the Hopf-Lax formula for the viscous Hamilton-Jacobi equation. Our numerical examples illustrate global convergence.