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

Andrea Brilli, Ana L. Custódio, Giampaolo Liuzzi, Everton J. Silva

TL;DR

This work tackles derivative-free nonlinear constrained optimization with general inequalities and equalities by proposing LOG-DS, a direct-search method that uses a mixed penalty–logarithmic barrier merit function. The nonlinear constraints are partitioned into interior and exterior groups based on an initial feasible point, enabling a log-barrier treatment for the interior set and an exterior penalty for the rest, with a path-following scheme that drives the barrier parameter $\rho_k$ to zero. Convergence to $KKT$-stationary points is established under standard differentiability assumptions and the extended Mangasarian–Fromovitz constraint qualification, without requiring convexity. The method demonstrates robust and efficient performance on CUTEst problems and a concentrated solar power engineering application, often outperforming state-of-the-art derivative-free solvers and offering practical flexibility for problems with many linear constraints and nonsmooth behavior.

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.

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

TL;DR

This work tackles derivative-free nonlinear constrained optimization with general inequalities and equalities by proposing LOG-DS, a direct-search method that uses a mixed penalty–logarithmic barrier merit function. The nonlinear constraints are partitioned into interior and exterior groups based on an initial feasible point, enabling a log-barrier treatment for the interior set and an exterior penalty for the rest, with a path-following scheme that drives the barrier parameter to zero. Convergence to -stationary points is established under standard differentiability assumptions and the extended Mangasarian–Fromovitz constraint qualification, without requiring convexity. The method demonstrates robust and efficient performance on CUTEst problems and a concentrated solar power engineering application, often outperforming state-of-the-art derivative-free solvers and offering practical flexibility for problems with many linear constraints and nonsmooth behavior.

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.
Paper Structure (28 sections, 10 theorems, 114 equations, 7 figures, 1 table)

This paper contains 28 sections, 10 theorems, 114 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Notation
  4. A direct search algorithm for nonlinear constrained optimization
  5. LOG-DS algorithm
  6. Poll directions and linear inequalities
  7. Preliminary results
  8. Convergence analysis
  9. Implementation details of LOG-DS
  10. LOG-DS vs SID-PSM
  11. Implementation choices
  12. Numerical experiments
  13. Performance and data profiles
  14. Results and analysis
  15. Strategies for addressing linear constraints
  16. ...and 13 more sections

Key Result

Proposition 2.4

Let $\{{\bf{x}}_k\}_{k\in\mathbb{N}}$ be a sequence of points in $\mathcal{X}$, converging to ${\bf{x}}^*\in \mathcal{X}$. Then, there exists an $\varepsilon^*>0$ (depending only on ${\bf{x}}^*$) such that for any $\varepsilon\in (0,\varepsilon^*]$ there exists $k_\varepsilon\in\mathbb{N}$ such that for all $k\geq k_\varepsilon$.

Figures (7)

  • Figure 1: Performance (on top) and data (on bottom) profiles comparing LOG-DS using two different approaches to address linear inequality constraints.
  • Figure 2: Performance (on top) and data (on bottom) profiles comparing LOG-DS and SID-PSM.
  • Figure 3: Performance (on top) and data (on bottom) comparing LOG-DS, NOMAD, and X-LOG-DFL on the complete problem collection.
  • Figure 4: Performance (on top) and data (on bottom) profiles comparing LOG-DS, NOMAD, and LOG-DFL on the subset of problems with only inequality constraints.
  • Figure 5: Performance (on top) and data (on bottom) profiles comparing LOG-DS and NOMAD on the complete problem collection, for different levels of violation of the nonlinear constraints.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2: Active constraints and tangent related sets
  • Definition 2.3: $\varepsilon$-Active constraints and tangent related sets
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • proof
  • Definition 3.1
  • ...and 13 more