CONSTRAINED CONSENSUS-BASED OPTIMIZATION

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.

Automated summary

In this work, a numerical method for high-dimensional constrained nonlinear optimization problems using particle-based gradient-free techniques is developed. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced. The method reformulates the constrained problem into an unconstrained one and extends to constrained settings. An iterative strategy is proposed to update the penalty parameter based on the constrained violation. Convergence of the proposed method is shown using a mean-field description of the CBO dynamics, and the properties of the algorithm are analyzed. Numerical examples demonstrate the theoretical findings and the good performance of the new method, even in high dimensions.