Speeding up Local Search for the Indicator-based Subset Selection Problem by a Candidate List Strategy

TL;DR

This work tackles the computational bottleneck of local search for the indicator-based subset selection problem (ISSP) in evolutionary multi-objective optimization. It introduces a candidate list strategy that restricts swaps to a per-point list of $l$ candidates, reducing per-iteration evaluations from $k(n-k)$ to $kl$, and presents two list types: nearest-neighbor and random-neighbor, plus a two-phase sequential scheme to handle discontinuities. Empirical results across seven quality indicators, including HV and IGD, show substantial speedups—especially with the two-list approach—while maintaining high-quality subsets; the strategy scales well with the number of objectives $d$ and subset size $k$, and proves particularly effective on continuous PFs, with careful handling needed for discontinuous PFs via the random list. The work also analyzes the behavior of swaps and provides guidance on list sizes and configurations, offering a practical, generalizable acceleration technique for ISSP within indicator-based environmental selection.

Abstract

In evolutionary multi-objective optimization, the indicator-based subset selection problem involves finding a subset of points that maximizes a given quality indicator. Local search is an effective approach for obtaining a high-quality subset in this problem. However, local search requires high computational cost, especially as the size of the point set and the number of objectives increase. To address this issue, this paper proposes a candidate list strategy for local search in the indicator-based subset selection problem. In the proposed strategy, each point in a given point set has a candidate list. During search, each point is only eligible to swap with unselected points in its associated candidate list. This restriction drastically reduces the number of swaps at each iteration of local search. We consider two types of candidate lists: nearest neighbor and random neighbor lists. This paper investigates the effectiveness of the proposed candidate list strategy on various Pareto fronts. The results show that the proposed strategy with the nearest neighbor list can significantly speed up local search on continuous Pareto fronts without significantly compromising the subset quality. The results also show that the sequential use of the two lists can address the discontinuity of Pareto fronts.

Figures (5)

• Figure 1: Distribution of dist$^{\mathrm{s}}$ and dist$^{\mathrm{us}}$ values in a representative run of LS, where $\square$ indicates a successful swap, and $\bullet$ indicates an unsuccessful swap. The results are shown for the ISSPs using the seven quality indicators.
• Figure 2: Examples of distributions of points on continuous and discontinuous PFs.
• Figure 3: Results of the four LS methods on the ISSP with the linear PF using HV, IGD, R2, and $\epsilon$. The average number of subset evaluations by $\mathcal{I}$ and the wall-clock time of the run are shown in figures (a)--(d) and (e)--(h), respectively.
• Figure 4: Scalability of the four LS methods to the number of objectives $d$ on the ISSP with the linear PF using HV and $\epsilon$.
• Figure 5: