Constrained consensus-based optimization

TL;DR

This paper introduces a novel particle-based, gradient-free optimization method for high-dimensional constrained nonlinear problems, combining consensus-based optimization with penalization and mean-field analysis to ensure convergence and effectiveness.

Contribution

It develops a new constrained CBO method with an iterative penalty update strategy, extending CBO to constrained problems and providing convergence analysis via mean-field limits.

Findings

Method converges to the constrained minimum.

Effective in high-dimensional problems.

Numerical experiments confirm theoretical results.

Abstract

In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function and extends to the constrained settings the class of CBO methods. Exact penalization is employed and, since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation. Using a mean-field description of the the many particle limit of the arising CBO dynamics, we are able to show convergence of the proposed method to the minimum for general nonlinear constrained problems. Properties of the new algorithm are analyzed. Several numerical examples, also in high dimensions, illustrate the theoretical findings and the good performance of the new numerical method.