Constrained consensus-based optimization

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.

An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation

“…Such methods consider interacting particle systems described by stochastic differential equations (SDEs) that combine a drift towards the estimated minimum and random exploration of the search space [2,10,11,30,31,48,51]. These approaches have been extended also to optimization problems over hypersurfaces [27][28][29], constrained optimization [5,13] and multi-objective optimization [6]. From a mathematical viewpoint, this class of metaheuristic methods is inspired by the corresponding mean-field dynamics based on particle swarming and multi-agent social interactions, which have been widely used to study complex systems in life sciences, social sciences and economics [17,43,44,46,53].…”