The Set-Based Hypervolume Newton Method for Bi-Objective Optimization

In this paper, we propagate the use of a set-based Newton method that enables computing a finite size approximation of the Pareto front (PF) of a given twice continuously differentiable bi-objective optimization problem (BOP). To this end, we first derive analytically the Hessian matrix of the hypervolume indicator, a widely used performance indicator for PF approximation sets. Based on this, we propose the hypervolume Newton method (HNM) for hypervolume maximization of a given set of candidate solutions. We first address unconstrained BOPs and focus further on first attempts for the treatment of inequality constrained problems. The resulting method may even converge quadratically to the optimal solution, however, this property is-as for all Newton methods-of local nature. We hence propose as a next step a hybrid of HNM and an evolutionary strategy in order to obtain a fast and reliable algorithm for the treatment of such problems. The strengths of both HNM and hybrid are tested on several benchmark problems and comparisons of the hybrid to state-of-the-art evolutionary algorithms for hypervolume maximization are presented.