  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.
