Hypervolume maximization via set based Newton’s method

Abstract

The hypervolume indicator is one of the most widely used tool to measure the performance in evolutionary multi-objective optimization. While derivative free methods such as specialized evolutionary algorithms received considerable attention in the past, the investigation of derivative based methods is still scarce. In this work, we aim to make a contribution to fill this gap. Based on the hypervolume gradient that has recently been proposed for general unconstrained multi-objective optimization problems, we first investigate the behavior of the related hypervolume flow. Under this flow, populations evolve toward a final state (population) whose hypervolume indicator is locally maximal. Some insights obtained on selected test functions explain to a certain extend observations made in previous studies and give some possible insights into the application of mathematical programming techniques to this problem. Further, we apply a population-based version of the Newton Raphson method for the maximization of the hypervolume. Fast set-based convergence can be observed towards optimal populations, however, the results indicate that the success depends crucially on the choice of the initial population.