Efficient search strategies for constrained multiobjective blackbox optimization

TL;DR

This work tackles constrained multiobjective blackbox optimization without derivatives by enhancing the DMulti-MADS direct-search framework with two native search strategies: a quadratic surrogate search and a Nelder-Mead–based sampling strategy, both integrated via single-objective reformulations. The authors establish theoretical underpinnings using Clarke nonsmooth analysis and present a comprehensive algorithmic framework that handles inequality constraints with a progressive barrier and uses a mesh-based search/poll scheme. Through extensive experiments on synthetic benchmarks and three engineering problems, they demonstrate consistent performance gains over the baseline, with NM-based strategies often delivering robust improvements and quadratic strategies offering broader front coverage on constrained problems. The results provide practical guidance for selecting default strategies in NOMAD and point to future work in parallelism and smarter poll designs that leverage the proposed reformulations.

Abstract

Multiobjective blackbox optimization deals with problems where the objective and constraint functions are the outputs of a numerical simulation. In this context, no derivatives are available, nor can they be approximated by finite differences, which precludes the use of classical gradient-based techniques. The DMulti-MADS algorithm implements a state-of-the-art direct search procedure for multiobjective blackbox optimization based on the mesh adaptive direct search (MADS) algorithm. Since its conception, many search strategies have been proposed to improve the practical efficiency of the single-objective MADS algorithm. Inspired by this previous research, this work proposes the integration of two search heuristics into the DMulti-MADS algorithm. The first uses quadratic models, built from previously evaluated points, which act as surrogates for the true objectives and constraints, to suggest new promising candidates. The second exploits the sampling strategy of the Nelder-Mead algorithm to explore the decision space for new non-dominated points. Computational experiments on analytical problems and three engineering applications show that the use of such search steps considerably improves the performance of the DMulti-MADS algorithm.

Figures (15)

• Figure 1: Levels sets in the objective space of the $R^\text{Do}_Y$ formulation for a biobjective minimization problem.
• Figure 2: Illustration for a biobjective minimization problem: $\mathbf{x}^1 \prec \mathbf{x}^2$ but $\psi_Y^\text{Do}(\mathbf{x}^1) = \psi_Y^\text{Do}(\mathbf{x}^2)$.
• Figure 3: Zones of interest (dotted) in the objective space for the MultiMADS search strategy for different configuration of incumbent points $\mathbf{x}^k$ in an iterate list $L^k$ for a biobjective minimization problem.
• Figure 4: Zones of interest (dotted) in the objective space for the dominance move search strategy for different configuration of incumbent points $\mathbf{x}^k$ in an iterate list $L^k$ for a biobjective minimization problem.
• Figure 5: Data profiles obtained from 100 multiobjective bound-constrained analytical problems taken from CuMaVaVi2010 for DMulti-MADS basic, NM-DoM, NM-Multi, Quad-DMS, Quad-DoM, and Quad-Multi variants with tolerance $\varepsilon_\tau \in \{10^{-2}, 5 \times 10^{-2}, 10^{-1}\}$.

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1: Clarke tangent vector
• Definition 3.2: Hypertangent vector
• Theorem 3.1
• Definition 4.1: Dominance for constrained optimization
• Definition 5.1: adapted from AuSaZg2008aAuSaZg2010a
• Lemma 5.1
• proof