Simplifying deflation for non-convex optimization with applications in Bayesian inference and topology optimization

TL;DR

This paper introduces a simple, general deflation constraint method to efficiently find multiple solutions in non-convex optimization problems, with applications in Bayesian inference and topology optimization.

Contribution

It proposes a novel, versatile deflation constraint that integrates with existing solvers, enhancing the exploration of multiple solutions in non-convex problems.

Findings

Effective in exploring multiple local optima

Applicable to Bayesian inference and topology optimization

Improves solution diversity without re-initialization

Abstract

Non-convex optimization problems have multiple local optimal solutions. Non-convex optimization problems are commonly found in numerous applications. One of the methods recently proposed to efficiently explore multiple local optimal solutions without random re-initialization relies on the concept of deflation. In this paper, different ways to use deflation in non-convex optimization and nonlinear system solving are discussed. A simple, general and novel deflation constraint is proposed to enable the use of deflation together with existing nonlinear programming solvers or nonlinear system solvers. The connection between the proposed deflation constraint and a minimum distance constraint is presented. Additionally, a number of variations of deflation constraints and their limitations are discussed. Finally, a number of applications of the proposed methodology in the fields of approximate Bayesian inference and topology optimization are presented.