A Newton Method for Hausdorff Approximations of the Pareto Front within Multi-objective Evolutionary Algorithms

TL;DR

This work proposes a set-based Newton method for the Hausdorff approximations of the Pareto front to be used within MOEAs and shows the benefit of the Newton method as a postprocessing step on several benchmark test functions and different base evolutionary algorithms.

Abstract

A common goal in evolutionary multi-objective optimization is to find suitable finite-size approximations of the Pareto front of a given multi-objective optimization problem. While many multi-objective evolutionary algorithms have proven to be very efficient in finding good Pareto front approximations, they may need quite a few resources or may even fail to obtain optimal or nearly approximations. Hereby, optimality is implicitly defined by the chosen performance indicator. In this work, we propose a set-based Newton method for Hausdorff approximations of the Pareto front to be used within multi-objective evolutionary algorithms. To this end, we first generalize the previously proposed Newton step for the performance indicator for the treatment of constrained problems for general reference sets. To approximate the target Pareto front, we propose a particular strategy for generating the reference set that utilizes the data gathered by the evolutionary algorithm during its run. Finally, we show the benefit of the Newton method as a post-processing step on several benchmark test functions and different base evolutionary algorithms.

Figures (13)

• Figure 1: Hypothetical example where the sets $Z$ (blue dots) and $\mathbf{X}$ (the image $F(\mathbf{X})$ in red dots) are matched. The matching is indicated by the lines.
• Figure 2: Application of the $\Delta_p$-Newton method on ZDT1. The starting populations are shown in (a) and (b), the filled set ($Y$) and the unshifted reference set ($T$) are shown in (c), the matching between the initial iterate $\mathbf{X}_0$ and the shifted reference set $Z$ can be seen in (d) and finally in (e) the result of the Newton method can be seen together with its performance in (f).
• Figure 3: Application of the $\Delta_p$-Newton method on DTLZ2. Two starting populations are shown in (a) and (b), the filled set ($Y$) and the unshifted reference set ($T$) are shown in (c), and the matching between the initial iterate $\mathbf{X}_0$ and the shifted reference set $Z$ can be seen in (d) and finally in (e) the result of the Newton method can be seen together with its performance in (f).
• Figure 4: Results on performance comparison of the hybrid method (MOEA + DpN) and MOEA. We show the relative improvement of the hybrid method over the MOEAs in terms of the median $\Delta_2$ values on each problem, i.e., $(\Delta_2$(MOEA) $-$$\Delta_2$(hybrid))$/\Delta_2$(MOEA), where red bars indicate statistically significant improvement according to Mann-Whitney U test; blue ones indicate significant worsening; gray ones show no statistical difference. 30 independent runs are conducted for each method on each problem to obtain the result.
• Figure 5: We show the impact of the number of objectives by aggregating the relative improvement for bi-objective and tri-objective problems, respectively. Since we only consider one four-objective problem (CONV4-2F) in this study, this problem is not shown in this chart.

Theorems & Definitions (1)

• Definition 1