Nonlinear Derivative-free Constrained Optimization with a Mixed Penalty-Logarithmic Barrier Approach and Direct Search

In this work, we propose the joint use of a mixed penalty-logarithmic barrier approach and generating set search, for addressing nonlinearly constrained derivative-free optimization problems. A merit function is considered, wherein the set of inequality constraints is divided into two groups: one treated with a logarithmic barrier approach, and another, along with the equality constraints, addressed using a penalization term. This strategy, initially proposed in the framework of LOG-DFL 12, is adapted and incorporated into SID-PSM 14,15 algorithm, a generalized pattern search method, allowing to effectively handle general nonlinear constraints. Under reasonable assumptions regarding the smoothness of the functions, convergence is established, without any convexity assumptions. Using CUTEst test problems, numerical experiments demonstrate the robustness, efficiency, and overall effectiveness of the proposed method, when compared with state-of-art solvers and with the original SID-PSM and LOG-DFL implementations.