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A derivative-free trust-region approach for Low Order-Value Optimization problems

Anderson E. Schwertner, Francisco N. C. Sobral

TL;DR

This work addresses constrained Low Order-Value Optimization (LOVO), where the objective is $f_{ ext{min}}(x)=\min_{i\in\mathcal I} f_i(x)$ with black-box, differentiable $f_i$ and convex feasible set $\Omega$. The authors develop a derivative-free trust-region algorithm that builds a local model of a chosen $f_i$ and uses two radii to manage model quality and step size, achieving convergence to weakly critical points and providing global convergence and worst-case iteration-complexity analyses. An open-source implementation, LOWDER, uses linear interpolation models and practical enhancements (two-radius framework, sample-set management, and TRS/ALTMOV subroutines) to demonstrate efficiency on standard LOVO test suites. Across MW, HS, and QD benchmarks, LOWDER shows strong performance, especially for small computational budgets and problems with many component functions, underscoring its practical value for black-box constrained optimization in scientific and engineering contexts.

Abstract

The Low Order-Value Optimization (LOVO) problem involves minimizing the minimum among a finite number of function values within a feasible set. LOVO has several practical applications such as robust parameter estimation, protein alignment, portfolio optimization, among others. In this work, we are interested in the constrained nonlinear optimization LOVO problem of minimizing the minimum between a finite number of function values subject to a nonempty closed convex set where each function is a black-box and continuously differentiable, but the derivatives are not available. We develop the first derivative-free trust-region algorithm for constrained LOVO problems with convergence to weakly critical points. Under suitable conditions, we establish the global convergence of the algorithm and also its worst-case iteration complexity analysis. An initial open-source implementation using only linear interpolation models is developed. Extensive numerical experiments and comparison with existing alternatives show the properties and the efficiency of the proposed approach when solving LOVO problems.

A derivative-free trust-region approach for Low Order-Value Optimization problems

TL;DR

This work addresses constrained Low Order-Value Optimization (LOVO), where the objective is with black-box, differentiable and convex feasible set . The authors develop a derivative-free trust-region algorithm that builds a local model of a chosen and uses two radii to manage model quality and step size, achieving convergence to weakly critical points and providing global convergence and worst-case iteration-complexity analyses. An open-source implementation, LOWDER, uses linear interpolation models and practical enhancements (two-radius framework, sample-set management, and TRS/ALTMOV subroutines) to demonstrate efficiency on standard LOVO test suites. Across MW, HS, and QD benchmarks, LOWDER shows strong performance, especially for small computational budgets and problems with many component functions, underscoring its practical value for black-box constrained optimization in scientific and engineering contexts.

Abstract

The Low Order-Value Optimization (LOVO) problem involves minimizing the minimum among a finite number of function values within a feasible set. LOVO has several practical applications such as robust parameter estimation, protein alignment, portfolio optimization, among others. In this work, we are interested in the constrained nonlinear optimization LOVO problem of minimizing the minimum between a finite number of function values subject to a nonempty closed convex set where each function is a black-box and continuously differentiable, but the derivatives are not available. We develop the first derivative-free trust-region algorithm for constrained LOVO problems with convergence to weakly critical points. Under suitable conditions, we establish the global convergence of the algorithm and also its worst-case iteration complexity analysis. An initial open-source implementation using only linear interpolation models is developed. Extensive numerical experiments and comparison with existing alternatives show the properties and the efficiency of the proposed approach when solving LOVO problems.

Paper Structure

This paper contains 14 sections, 17 theorems, 45 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. A derivative-free trust-region LOVO algorithm
  4. Trust-region framework
  5. Algorithm
  6. Convergence analysis
  7. Worst-case complexity
  8. Numerical implementation and experiments
  9. Implementation details
  10. Numerical experiments
  11. MW test set
  12. HS test set
  13. QD test set
  14. Conclusions

Key Result

Lemma 3.1

Suppose that Assumptions hip_f_diferenciavel and hip_grad_limitante hold. Thus, for every iteration $k \in \mathbb{N}$, where $\kappa_{H} := 2\kappa_{g} + L + 1$ and $L := \max_{i \in \mathcal{I}} \{L_{i} \}$.

Figures (4)

  • Figure 1: Contour plot (left) and surface plot (right) for an example of objective function $f_{min}$ defined by the generating function of QD test set.
  • Figure 2: Data profiles for the problems in MW test set with tolerance $\tau \in \left\{10^{-1}, 10^{-3}, 10^{-5}, 10^{-7}\right\}$.
  • Figure 3: Data profiles for the problems in HS test set with tolerance $\tau \in \left\{10^{-1}, 10^{-3}, 10^{-5}, 10^{-7}\right\}$.
  • Figure 4: Data profiles for the problems in QD test set, with $\tau = 10^{-5}$, and $r \in \left\{ 10, 25, 50, 75, 100 \right\}$.

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 27 more