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Analyzing the Landscape of the Indicator-based Subset Selection Problem

Keisuke Korogi, Ryoji Tanabe

TL;DR

This work analyzes the landscape of the indicator-based subset selection problem (ISSP) in evolutionary multi-objective optimization by applying traditional landscape measures and exact local optima networks (LONs) across seven quality indicators and seven Pareto fronts. By fully enumerating ISSP instances at modest scales, it reveals that the landscape structure—global/local optima, neutrality, ruggedness, and global structure measured via FDC—depends critically on the chosen quality indicator and PF shape, with the $\epsilon$-SSP exhibiting particularly rich multimodality and neutrality. The study shows high correlations among certain ISSP instances (e.g., HV and NR2) and highlights that the indicator type drives neutrality and search difficulty, informing algorithm design; notably, using objective-space neighbors (via Wasserstein distance) can improve search guidance. Overall, the findings guide the development of efficient ISSP solvers, suggesting strategies such as leveraging Wasserstein-based neighborhoods and initializing from strongly correlated indicators, while pointing to the need for scalable analyses on larger $n$ and $k$ in future work.

Abstract

The indicator-based subset selection problem (ISSP) involves finding a point subset that minimizes or maximizes a quality indicator. The ISSP is frequently found in evolutionary multi-objective optimization (EMO). An in-depth understanding of the landscape of the ISSP could be helpful in developing efficient subset selection methods and explaining their performance. However, the landscape of the ISSP is poorly understood. To address this issue, this paper analyzes the landscape of the ISSP by using various traditional landscape analysis measures and exact local optima networks (LONs). This paper mainly investigates how the landscape of the ISSP is influenced by the choice of a quality indicator and the shape of the Pareto front. Our findings provide insightful information about the ISSP. For example, high neutrality and many local optima are observed in the results for ISSP instances with the additive $ε$-indicator.

Analyzing the Landscape of the Indicator-based Subset Selection Problem

TL;DR

This work analyzes the landscape of the indicator-based subset selection problem (ISSP) in evolutionary multi-objective optimization by applying traditional landscape measures and exact local optima networks (LONs) across seven quality indicators and seven Pareto fronts. By fully enumerating ISSP instances at modest scales, it reveals that the landscape structure—global/local optima, neutrality, ruggedness, and global structure measured via FDC—depends critically on the chosen quality indicator and PF shape, with the -SSP exhibiting particularly rich multimodality and neutrality. The study shows high correlations among certain ISSP instances (e.g., HV and NR2) and highlights that the indicator type drives neutrality and search difficulty, informing algorithm design; notably, using objective-space neighbors (via Wasserstein distance) can improve search guidance. Overall, the findings guide the development of efficient ISSP solvers, suggesting strategies such as leveraging Wasserstein-based neighborhoods and initializing from strongly correlated indicators, while pointing to the need for scalable analyses on larger and in future work.

Abstract

The indicator-based subset selection problem (ISSP) involves finding a point subset that minimizes or maximizes a quality indicator. The ISSP is frequently found in evolutionary multi-objective optimization (EMO). An in-depth understanding of the landscape of the ISSP could be helpful in developing efficient subset selection methods and explaining their performance. However, the landscape of the ISSP is poorly understood. To address this issue, this paper analyzes the landscape of the ISSP by using various traditional landscape analysis measures and exact local optima networks (LONs). This paper mainly investigates how the landscape of the ISSP is influenced by the choice of a quality indicator and the shape of the Pareto front. Our findings provide insightful information about the ISSP. For example, high neutrality and many local optima are observed in the results for ISSP instances with the additive -indicator.

Paper Structure

This paper contains 18 sections, 3 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-objective optimization
  4. Quality indicators
  5. Indicator-based subset selection problem
  6. Landscape analysis
  7. Experimental setup
  8. Results
  9. Distribution of quality indicator values
  10. Correlation of subset rankings
  11. Number of global and local optima
  12. Ruggedness
  13. Neutrality
  14. Fitness distance correlation (FDC)
  15. Local optima networks (LONs)
  16. ...and 3 more sections

Figures (9)

  • Figure 1: Distribution of quality indicator values $\mathcal{I}(S)$, where those are min-max normalized.
  • Figure 2: Spearman's rank correlation coefficients between the rankings of all subsets by each pair of quality indicators.
  • Figure 3: Number of global and local optima (plateaus).
  • Figure 4: Distributions of two global optimal subsets on the $\epsilon$-SSP with the nonconvex PF.
  • Figure 5: Degree of ruggedness. Each bar indicates the correlation between the quality indicator values of neighbor solutions.
  • ...and 4 more figures