Multi-start Optimization Method via Scalarization based on Target Point-based Tchebycheff Distance for Multi-objective Optimization

TL;DR

This work tackles multi-objective optimization in settings where traditional dominance- and decomposition-based methods struggle with coverage, especially on inverted or nonlinear Pareto fronts and when variable dependencies hinder SBX-based operators. It introduces Target Point-based Tchebycheff Distance (TPTD), a scalarization within a decomposition-like framework, and couples it with Natural Evolution Strategy (NES) to handle variable interactions. A four-step, parallelizable search procedure constructs and relocates target points on a target hyperplane to maximize Pareto front coverage, enabling efficient exploration of 0D to (m−1)D front portions. Experimental results on RP and MED benchmarks show substantial improvements in Hypervolume and execution time, with up to 474x speedups over NSGA-II, NSGA-III, and MOEA/D-DE, highlighting practical potential for large-scale, many-objective problems.

Abstract

Multi-objective optimization is crucial in scientific and industrial applications where solutions must balance trade-offs among conflicting objectives. State-of-the-art methods, such as NSGA-III and MOEA/D, can handle many objectives but struggle with coverage issues, particularly in cases involving inverted triangular Pareto fronts or strong nonlinearity. Moreover, NSGA-III often relies on simulated binary crossover, which deteriorates in problems with variable dependencies. In this study, we propose a novel multi-start optimization method that addresses these challenges. Our approach introduces a newly introduced scalarization technique, the Target Point-based Tchebycheff Distance (TPTD) method, which significantly improves coverage on problems with inverted triangular Pareto fronts. For efficient multi-start optimization, TPTD leverages a target point defined in the objective space, which plays a critical role in shaping the scalarized function. The position of the target point is adaptively determined according to the shape of the Pareto front, ensuring improvement in coverage. Furthermore, the flexibility of this scalarization allows seamless integration with powerful single-objective optimization methods, such as natural evolution strategies, to efficiently handle variable dependencies. Experimental results on benchmark problems, including those with inverted triangular Pareto fronts, demonstrate that our method outperforms NSGA-II, NSGA-III, and MOEA/D-DE in terms of the Hypervolume indicator. Notably, our approach achieves computational efficiency improvements of up to 474 times over these baselines.

Figures (5)

• Figure 1: An example of the target point $\boldsymbol{t}$ and the normalized objective vector of optimal solution $\boldsymbol{f}'\left(\boldsymbol{x}^{*}\right)$ obtained by the Target Point-based Tchebycheff Distance method (TPTD) in a 2-objective problem. $T$ is the target point set. Note that the objective space is normalized.
• Figure 2: Search scenario of the proposed method in a 3-objective problem.
• Figure 3: Pareto front boundary search in a 3-objective problem. The red line is the image of the Pareto front boundary on the target hyperplane, and the red points are the target points generated during the search.
• Figure 4: Example of guide point correspondence and relocation in a 3-objective problem. The red line is the image of the Pareto front boundary on the target hyperplane.
• Figure 5: Execution time of each method on each problem. The red dashed line indicates the maximum execution time. The values of the proposed method (blue) are quite small, which may make them hard to see.