Constrained Consensus-Based Optimization

SIAM Journal on Optimization, vol. 33, pp. 211–236

Abstract

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 of 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 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.

Authors 3

1. Giacomo Borghi Aachen Institute for Geometry and Applied Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Applied Mathematics, RWTH Aachen University, Aachen, North Rhine-Westphalia 52062, Germany

2. Michaël Herty Aachen Institute for Geometry and Applied Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Applied Mathematics, RWTH Aachen University, Aachen, North Rhine-Westphalia 52062, Germany

3. Lorenzo Pareschi

   University of Ferrara

   Affiliation as printed

   Department of Mathematics and Computer Science, University of Ferrara, Ferrara 44121, Italy