Weak Pareto Boundary: The Achilles' Heel of Evolutionary Multi-Objective Optimization

TL;DR

This work introduces the weak Pareto boundary (WPB) as a critical obstacle for evolutionary multi-objective optimization, defining WPB and its ν-based categories to quantify dominance resistance. It derives a hypervolume-based DDR analysis showing how the proximity of a dominance-resistant solution to WPB governs the difficulty of eliminating it, with the asymptotic scaling depending on WPB shape and category. To enable rigorous evaluation, the authors propose two WPB-focused test problem generators that produce continuous or discontinuous PFs with controllable WPB attributes, validated via extensive experiments. Across multiple MOEAs and instance families, the results show that no single algorithm robustly handles all WPB configurations, underscoring the need for WPB-aware strategies and benchmark suites with diverse WPB characteristics.

Abstract

The weak Pareto boundary ($WPB$) refers to a boundary in the objective space of a multi-objective optimization problem, characterized by weak Pareto optimality rather than Pareto optimality. The $WPB$ brings severe challenges to multi-objective evolutionary algorithms (MOEAs), as it may mislead the algorithms into finding dominance-resistant solutions (DRSs), i.e., solutions that excel on some objectives but severely underperform on the others, thereby missing Pareto-optimal solutions. Although the severe impact of the $WPB$ on MOEAs has been recognized, a systematic and detailed analysis remains lacking. To fill this gap, this paper studies the attributes of the $WPB$. In particular, the category of a $WPB$, as an attribute derived from its weakly Pareto-optimal property, is theoretically analyzed. The analysis reveals that the dominance resistance degrees of DRSs induced by different categories of $WPB$s exhibit distinct asymptotic growth rates as the DRSs in the objective space approach the $WPB$s, where a steeper asymptotic growth rate indicates a greater hindrance to MOEAs. Beyond that, experimental studies are conducted on various new test problems to investigate the impact of $WPB$'s attributes. The experimental results demonstrate consistency with our theoretical findings. Experiments on other attributes show that the performance of an MOEA is highly sensitive to some attributes. Overall, no existing MOEAs can comprehensively address challenges brought by these attributes.

Figures (20)

• Figure 1: Examples of the $WPB$ in the 2-objective case.
• Figure 2: Examples of the $WPB_{\nu,i}$ in the 3-objective case.
• Figure 3: Illustration of a DRS and its enclosed region in the 2-objective case.
• Figure 4: (a) Illustration of the $PF$ and all $WPB$s in the case with 3 objectives and $p=1$. (b) Visualization of moving $\mathbf{r}$ toward the $WPB_{1,i}$ and the $WPB_{2,i}$. For better clarity, $\Delta$ is set to a relatively large value of 0.5 in (b) and (c).
• Figure 5: Curves of $\mathcal{H}(PF,\mathbf{r})$ with respect to $\Delta$ for the case where $p=1$ and $r_j=1.3$ for every $j \in \overline{I}_{2,i}$.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Proposition 1