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

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.

💡 Research Summary

This paper introduces MADS‑PIP, a novel algorithm that blends a penalty‑interior‑point (PIP) strategy with the Mesh Adaptive Direct Search (MADS) framework to solve nonsmooth black‑box optimization problems containing both inequality and equality constraints. The key idea is to split the inequality constraints into two disjoint subsets. Constraints that are strictly satisfied at the current iterate are placed in the “interior” set (G_{\text{int}}) and are handled by a logarithmic barrier applied to an aggregated interior violation function (c_{\text{int}}(x)). The remaining inequalities belong to the “exterior” set (G_{\text{ext}}) and are treated with a classic quadratic exterior penalty. All equality constraints are incorporated into the same quadratic penalty term.

A merit function is defined as
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