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

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 Tcheby-cheff 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. This is because, if the target points are distributed uniformly, it is expected that single-objective function optimization using TPTD could construct an approximate solution set with good coverage. The positions of the target points are adaptively determined according to the shape of the Pareto front, ensuring improvement in coverage. The proposed method first searches for target points corresponding to objective vectors on the Pareto front boundary using a binary search method and then relocates the target points corresponding to objective vectors within the boundary according to the shape of the boundary. This operation is computationally efficient because the positions of the non-boundary target points are determined in a single relocation. 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.