[Paper Review] Multimodal Multi-objective Optimization: Comparative Study of the State-of-the-Art

Wenhua Li, Tao Zhang+2 more|arXiv (Cornell University)|Jul 11, 2022

Advanced Multi-Objective Optimization Algorithms4 citations

TL;DR

This paper conducts a comprehensive comparative study of 12 state-of-the-art multimodal multi-objective evolutionary algorithms (MMEAs) on standard test suites, evaluating their performance in maintaining diversity across both decision and objective spaces. The study reveals that no single algorithm dominates across all problem types, but techniques emphasizing local Pareto set identification show superior effectiveness in solving multimodal multi-objective problems (MMOPs).

ABSTRACT

Multimodal multi-objective problems (MMOPs) commonly arise in real-world problems where distant solutions in decision space correspond to very similar objective values. To obtain all solutions for MMOPs, many multimodal multi-objective evolutionary algorithms (MMEAs) have been proposed. For now, few studies have encompassed most of the recently proposed representative MMEAs and made a comparative comparison. In this study, we first review the related works during the last two decades. Then, we choose 12 state-of-the-art algorithms that utilize different diversity-maintaining techniques and compared their performance on existing test suites. Experimental results indicate the strengths and weaknesses of different techniques on different types of MMOPs, thus providing guidance on how to select/design MMEAs in specific scenarios.

Motivation & Objective

• To evaluate the performance of 12 state-of-the-art MMEAs with diverse diversity-maintaining techniques on existing MMOP test suites.
• To identify the strengths and weaknesses of different diversity-maintaining strategies in handling various types of MMOPs.
• To provide practical guidance for selecting or designing MMEAs based on problem characteristics and desired solution distribution.
• To highlight critical gaps in current MMOP research, including lack of multi-modality detection, discrete optimization benchmarks, and comprehensive performance metrics.
• To advocate for embedding multimodal techniques into existing MOEAs to improve global exploration and robustness in real-world applications.

Proposed method

• The study selects 12 representative MMEAs that employ distinct diversity-maintaining mechanisms, including crowding distance in decision space, niching, and specialized environmental selection strategies.
• A standardized experimental setup is applied across multiple benchmark test suites, including MMF, IDMP, and other widely used MMOP problems.
• Performance is evaluated using metrics that assess convergence to the true Pareto front and distribution diversity in both decision and objective spaces.
• The analysis compares solution sets across different algorithmic approaches, focusing on their ability to locate and preserve multiple Pareto sets, including local and global ones.
• The study evaluates the impact of algorithmic design on convergence, diversity, and robustness, particularly in high-dimensional and complex MMOPs.
• The authors analyze the limitations of existing metrics and benchmarks, emphasizing the need for comprehensive, multi-space evaluation frameworks.

Experimental results

Research questions

• RQ1 Which MMEA performs best across a wide range of MMOP test problems, and under what conditions does it excel?
• RQ2 How do different diversity-maintaining techniques (e.g., niching, crowding distance in decision space) affect the ability to locate and preserve multiple Pareto sets?
• RQ3 To what extent do current performance metrics adequately capture both convergence and diversity in both decision and objective spaces?
• RQ4 Why do existing MMEAs struggle with high-dimensional MMOPs, and what are the key limitations in current benchmark suites?
• RQ5 What are the critical gaps in MMOP research, such as lack of multi-modality detection, discrete optimization benchmarks, and comprehensive evaluation metrics?

Key findings

• No single MMEA outperforms all others across all test suites, indicating that algorithm choice is highly problem-dependent.
• Algorithms that explicitly target the identification of local Pareto sets demonstrate superior performance in most MMOPs, especially those with multiple disconnected Pareto sets.
• Traditional MOEAs with convergence-first strategies fail to maintain sufficient diversity in the decision space, leading to premature convergence and loss of multiple optimal solutions.
• Existing performance metrics often neglect the decision space diversity, and no comprehensive metric currently balances both objective and decision space quality with adjustable weights.
• The IDMP test suite poses significant challenges, as previous MMEAs fail to locate all Pareto sets, highlighting the need for more robust and scalable algorithms.
• There is a critical lack of discrete optimization benchmarks and multi-modality detection tools, which hinders the practical adoption of MMEAs in real-world problems.