An Instance Space Analysis of Constrained Multi-Objective Optimization Problems

TL;DR

This work addresses CMOPs by applying Instance Space Analysis (ISA) to relate algorithm performance to instance characteristics, extending landscape analysis with a novel multi-objective-violation landscape. It introduces 12 new features and six landscape-related metrics, evaluating 15 CMOEAs across six benchmark suites (383 bi-objective instances) using Hypervolume ($HV$) as the performance measure. The analysis identifies two key instance properties—isolation of the non-dominant set and evolvability correlations between constraints and objectives—as the dominant factors shaping algorithm success, while highlighting limited diversity in current benchmarks. The study demonstrates ISA’s utility for diagnosing algorithm strengths/weaknesses and for guiding algorithm selection, though it notes the need for more diverse, real-world benchmarks and new CMOP-specific features for broader challenge.

Abstract

Multi-objective optimization problems with constraints (CMOPs) are generally considered more challenging than those without constraints. This in part can be attributed to the creation of infeasible regions generated by the constraint functions, and/or the interaction between constraints and objectives. In this paper, we explore the relationship between constrained multi-objective evolutionary algorithms (CMOEAs) performance and CMOP instances characteristics using Instance Space Analysis (ISA). To do this, we extend recent work focused on the use of Landscape Analysis features to characterise CMOP. Specifically, we scrutinise the multi-objective landscape and introduce new features to describe the multi-objective-violation landscape, formed by the interaction between constraint violation and multi-objective fitness. Detailed evaluation of problem-algorithm footprints spanning six CMOP benchmark suites and fifteen CMOEAs, illustrates that ISA can effectively capture the strength and weakness of the CMOEAs. We conclude that two key characteristics, the isolation of non-dominate set and the correlation between constraints and objectives evolvability, have the greatest impact on algorithm performance. However, the current benchmarks problems do not provide enough diversity to fully reveal the efficacy of CMOEAs evaluated.

Figures (8)

• Figure 1: Summary of the Instance Space Analysis framework SmithMiles2021.
• Figure 2: Effect of constraints on the $PF$: \ref{['fig:CPF1']} The $UPF$ is no longer feasible, and the true $PF$ lies completely on bounds of the feasible region; \ref{['fig:CPF2']} part of the $UPF$ is no longer feasible, and parts of the true $PF$ lies on bounds of the feasible region; \ref{['fig:CPF3']} the true $PF$ is only a proportion of the $UPF$; and \ref{['fig:CPF4']} an infeasible region blocks the way toward the $PF$.
• Figure 3: Distribution of the instances in 2D space by using the projection matrix in Equation \ref{['eq:IS']}. Instances are color-coded based on the source.
• Figure 4: Number of algorithms performed well for each instance, where good performance means a normalized $HV$ within 1% of the best algorithm. The color scale corresponds to the total number of algorithms. A color closer to dark blue means fewer number of algorithms performed well.
• Figure 5: Seven algorithms footprints in the projected instance space, where good performance means a normalized $HV$ within 1% of the best algorithm.