A penalty-interior point method combined with MADS for equality and inequality constrained optimization

TL;DR

This paper presents MADS-PIP, a novel optimization framework combining penalty-interior point methods with MADS to efficiently solve nonsmooth constrained blackbox problems, demonstrating superior performance over existing strategies.

Contribution

It introduces a new integrated penalty-interior point approach within MADS for nonsmooth constrained optimization, with proven convergence and improved computational results.

Findings

MADS-PIP outperforms traditional MADS with progressive barrier.

The method effectively handles both equality and inequality constraints.

Convergence to feasible and stationary points is theoretically established.

Abstract

This work introduces MADS-PIP, an efficient framework that integrates a penalty-interior point strategy into the mesh adaptive direct search (MADS) algorithm for solving nonsmooth blackbox optimization problems with general inequality and equality constraints. Inequality constraints are partitioned into two subsets: one treated via a logarithmic barrier applied to an aggregated interior constraint violation, and the other handled through an exterior quadratic penalty. All equality constraints are treated by the exterior penalty. A merit function defines a sequence of unconstrained subproblems, which are solved approximately using MADS, while a carefully designed update rule drives the penalty-barrier parameter to zero. In the nonsmooth setting, we establish convergence results ensuring feasibility for general constraints as well as Clarke stationarity for inequality-constrained problems. Computational experiments on both analytical test sets and challenging blackbox problems demonstrate that the proposed MADS-PIP algorithm is competitive with, and often outperforms, MADS with the progressive barrier strategy, particularly in the presence of equality constraints.