A Smoothing Consensus-Based Optimization Algorithm for Nonsmooth Nonconvex Optimization

TL;DR

This paper introduces a novel smoothing consensus-based optimization algorithm capable of effectively solving nonsmooth, nonconvex, and non-Lipschitz continuous optimization problems, with proven convergence and promising numerical results.

Contribution

It develops a new CBO algorithm for nonsmooth, nonconvex problems that does not depend on mean-field limits, with theoretical convergence guarantees.

Findings

Algorithm achieves global consensus.

Converges to a global minimum.

Demonstrates strong performance on numerical examples.

Abstract

Lately, a novel swarm intelligence model, namely the consensus-based optimization (CBO) algorithm, was introduced to deal with the global optimization problems. Limited by the conditions of Ito's formula, the convergence analysis of the previous CBO finite particle system mainly focuses on the problem with smooth objective function. With the help of smoothing method, this paper achieves a breakthrough by proposing an effective CBO algorithm for solving the global solution of a nonconvex, nonsmooth, and possible non-Lipschitz continuous minimization problem with theoretical analysis, which dose not rely on the mean-field limit. We indicate that the proposed algorithm exhibits a global consensus and converges to a common state with any initial data. Then, we give a more detailed error estimation on the objective function values along the state of the proposed algorithm towards the global minimum. Finally, some numerical examples are presented to illustrate the appreciable performance of the proposed method on solving the nonsmooth, nonconvex minimization problems.