Nonlinear Derivative-free Constrained Optimization with a Penalty-Interior Point Method and Direct Search

TL;DR

This paper introduces a novel derivative-free optimization method combining a penalty-interior point approach with direct search, effectively handling nonlinear constraints without convexity assumptions, and demonstrating robustness through computational experiments.

Contribution

It proposes a new hybrid method integrating penalty-interior point and direct search techniques for nonlinear constrained optimization without convexity.

Findings

Demonstrates robustness and efficiency on analytical problems.

Shows effectiveness in engineering applications.

Outperforms some state-of-the-art solvers.

Abstract

In this work, the joint use of a mixed penalty-interior point method and direct search is proposed, to address {nonlinear} constrained derivative-free optimization problems. A merit function is considered, wherein the set of nonlinear 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 is adapted and incorporated into a direct search method, enabling the effective handling of general nonlinear constraints. Convergence to KKT-stationary points is established under continuous differentiability assumptions, without requiring any kind of convexity. Computational experiments on analytical problems and an engineering application demonstrate the robustness, efficiency, and overall effectiveness of the proposed method, when compared with state-of-the-art solvers.