On Steering Dominated Points in Hypervolume Indicator Gradient Ascent for Bi-Objective Optimization

Multi-objective optimization problems are commonly encountered in real world applications. In some applications, where the gradient information of the objective functions is available, it is natural to consider a gradient-based multi-objective optimization algorithm for relatively high convergence speed and stability. In this chapter, we consider a recently proposed gradient-based approach, called the hyper-volume indicator gradient ascent method. It is designed to maximize the hypervolume indicator in the steepest direction by calculating its gradient fieldwith respect to decision vectors. The hypervolume indicator gradient derivation will be covered in this chapter. Despite the elegance of this approach, a critical issue arises when applying the gradient computation for some of the decision vectors: the gradient at a dominated point is either zero or undefined, which restricts the usage of this approach. To remedy this, five methods are proposed to provide a search direction for dominated points (at which the hypervolume indicator gradient fails to do so). These five methods are devised for the bi-objective optimization case and are illustrated in detail. In addition, a thorough empirical study is carried out to investigate the convergence behavior of these five methods. The combination of the hypervolume indicator gradient and the proposed five methods constitute a novel gradient-based, bi-objective optimization algorithm, which is validated and benchmarked. The benchmark results show interesting performance comparisons among the five proposed methods.