Lasry-Lions Envelopes and Nonconvex Optimization: A Homotopy Approach

TL;DR

This paper introduces a homotopy method using Lasry-Lions envelopes to efficiently approximate and solve large-scale nonconvex, nonsmooth optimization problems, demonstrating advantages in signal decoding and spectral unmixing.

Contribution

It develops a novel homotopy approach leveraging Lasry-Lions envelopes for nonconvex optimization, extending the applicability of envelope methods beyond convex problems.

Findings

The method converges under certain conditions.

It outperforms classical approaches in specific signal processing tasks.

Experimental results show improved efficiency in decoding and spectral unmixing.

Abstract

In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their respective Moreau envelopes. In this work, we study the use of Lasry-Lions double envelopes to approximate nonsmooth terms that are also not convex. These envelopes are an extension of the Moreau ones but exhibit an additional smoothness property that makes them amenable to fast optimization algorithms. Lasry-Lions envelopes can also be seen as an "intermediate" between a given function and its convex envelope, and we make use of this property to develop a method that builds a sequence of approximate subproblems that are easier to solve than the original problem. We discuss convergence properties of this method when used to address composite minimization problems; additionally, based on a number of experiments, we discuss settings where it may be more useful than classical alternatives in two domains: signal decoding and spectral unmixing.