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

Charles Audet, Andrea Brilli, Youssef Diouane, Sébastien Le Digabel, Everton J. Silva, Christophe Tribes

TL;DR

This paper tackles constrained nonsmooth blackbox optimization where derivatives are unavailable, introducing MADS-PIP, a Penalty-Interior Point framework that integrates a logarithmic barrier for an interior inequality subset with an exterior penalty for the remaining inequalities and all equalities. The method transforms the constrained problem into a sequence of unconstrained subproblems via a merit function z(; rho) that combines f with barrier and penalty terms, driving the barrier-penalty parameter  rho to zero while solving with MADS. Under Lipschitz conditions and appropriate constraint qualifications (FCQ for feasibility and SCQ for stationarity), end-path limit points are feasible and satisfy Clarke stationarity for inequality constraints; the analysis handles nonsmoothness and the mixed barrier/penalty structure. Empirically, MADS-PIP is competitive with the progressive barrier approach on inequality-only problems and shows clear advantages when equality constraints are present, validated on CUTEst and challenging real-world blackbox problems, and implemented in NOMAD 4. The work provides rigorous convergence guarantees for a hybrid penalty-interior-point strategy within the MADS framework and demonstrates practical impact for derivative-free constrained optimization.

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.

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

TL;DR

This paper tackles constrained nonsmooth blackbox optimization where derivatives are unavailable, introducing MADS-PIP, a Penalty-Interior Point framework that integrates a logarithmic barrier for an interior inequality subset with an exterior penalty for the remaining inequalities and all equalities. The method transforms the constrained problem into a sequence of unconstrained subproblems via a merit function z(; rho) that combines f with barrier and penalty terms, driving the barrier-penalty parameter  rho to zero while solving with MADS. Under Lipschitz conditions and appropriate constraint qualifications (FCQ for feasibility and SCQ for stationarity), end-path limit points are feasible and satisfy Clarke stationarity for inequality constraints; the analysis handles nonsmoothness and the mixed barrier/penalty structure. Empirically, MADS-PIP is competitive with the progressive barrier approach on inequality-only problems and shows clear advantages when equality constraints are present, validated on CUTEst and challenging real-world blackbox problems, and implemented in NOMAD 4. The work provides rigorous convergence guarantees for a hybrid penalty-interior-point strategy within the MADS framework and demonstrates practical impact for derivative-free constrained optimization.

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.
Paper Structure (18 sections, 11 theorems, 50 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 18 sections, 11 theorems, 50 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Constraint handling in DFO
  3. Contributions and structure of the paper
  4. A penalty-interior point method for MADS
  5. Partitioning the constraints
  6. The algorithm: MADS-PIP
  7. Convergence analysis
  8. Preliminary results
  9. Feasibility
  10. Stationarity
  11. Practical implementation
  12. Coding algorithm
  13. Switches and numerical choices
  14. Computational experiments
  15. Settings
  16. ...and 3 more sections

Key Result

Proposition 3.2

Let ass:bounded_level_sets be satisfied. If the sequence of penalty-barrier parameters $\{\rho_k\}$ produced by LOG-MADS-1 satisfies $\rho_k = \rho_0$ for all $k \in \mathbb{N}$, then $\liminf_{k\to\infty} \Delta_k = 0.$

Figures (4)

  • Figure 1: Data profiles on a selection of analytical problems from the literature.
  • Figure 2: Data profiles obtained on $30$ inequality constrained problem instances of ${\sf solar 6}$.
  • Figure 3: Data profiles on $25$ CUTEst analytical problems having equality constraints. The left plot displays the portion of feasible instances obtained. Middle and right plots show the portion of $\tau$-solved instances for $\tau=10^{-1}$ (middle) and $\tau=10^{-3}$ (right).
  • Figure 4: Data profiles on 100 problem instances of Aircraft Range MDO. The left plot displays the portion of feasible instances obtained. Middle and right plots show the portion of $\tau$-solved instances for $\tau=10^{-1}$ (middle) and $\tau=10^{-2}$ (right).

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Proposition 3.6
  • proof
  • ...and 17 more