[Paper Review] Multiobjective Test Problems with Degenerate Pareto Fronts

Liangli Zhen, Miqing Li+3 more|arXiv (Cornell University)|Jun 7, 2018

Advanced Multi-Objective Optimization Algorithms29 references9 citations

TL;DR

This paper introduces a novel set of scalable multiobjective test problems featuring degenerate Pareto fronts—where the Pareto front dimension is less than m−1 in m-objective problems due to objective redundancy. The authors formalize three general characteristics of degeneracy and propose test problems that expose limitations in existing multiobjective evolutionary algorithms, which fail to solve them effectively, highlighting the need for new algorithmic approaches.

ABSTRACT

In multiobjective optimisation, a set of scalable test problems with a variety of features allow researchers to investigate and evaluate the abilities of different optimisation algorithms, and thus can help them to design and develop more effective and efficient approaches. Existing test problem suites mainly focus on situations where all the objectives are fully conflicting with each other. In such cases, an m-objective optimisation problem has an (m-1)-dimensional Pareto front in the objective space. However, in some optimisation problems, there may be unexpected characteristics among objectives, e.g., redundancy. The redundancy of some objectives can lead to the multiobjective problem having a degenerate Pareto front, i.e., the dimension of the Pareto front of the $m$-objective problem be less than (m-1). In this paper, we systematically study degenerate multiobjective problems. We abstract three general characteristics of degenerate problems, which are not formulated and systematically investigated in the literature. Based on these characteristics, we present a set of test problems to support the investigation of multiobjective optimisation algorithms under situations with redundant objectives. To the best of our knowledge, this work is the first one that explicitly formulates these three characteristics of degenerate problems, thus allowing the resulting test problems to be featured by their generality, in contrast to existing test problems designed for specific purposes (e.g., visualisation).

Motivation & Objective

• To address the lack of systematic test problems for degenerate multiobjective optimization, where redundant objectives lead to lower-dimensional Pareto fronts.
• To identify and formalize three general characteristics of degenerate problems not previously studied in the literature.
• To develop a new suite of test problems that are general-purpose, scalable, and representative of real-world degeneracy patterns.
• To evaluate the performance of existing multiobjective evolutionary algorithms (MOEAs) on these degenerate problems to expose their limitations.
• To guide future algorithm design by analyzing why current MOEAs struggle with degenerate fronts and suggesting potential strategies.

Proposed method

• The authors abstract three general characteristics of degenerate multiobjective problems: objective redundancy, non-independence of objectives, and dimensionality reduction in the Pareto front.
• They design a family of test problems—DPF1A–DPF5A—based on these characteristics, using a transformation from a reduced set of essential objectives to full m-objective space.
• The problems are constructed using a mapping from decision variables to objective space via trigonometric functions and a scalable penalty function g(·), ensuring scalable problem size and controlled degeneracy.
• The transformation preserves Pareto dominance: a solution dominates another in the original space if and only if it dominates in the transformed space, ensuring validity of the test problems.
• The test problems are designed to be scalable and general, avoiding artificial or visualization-specific structures found in prior benchmarks like DTLZ5 or WFG3.
• The authors validate the test suite by applying ten representative MOEAs to the problems and analyzing convergence and hypervolume performance.

Experimental results

Research questions

• RQ1 What are the general structural characteristics of degenerate multiobjective problems that lead to Pareto fronts with dimension less than m−1?
• RQ2 How can a general-purpose, scalable test problem suite be constructed to represent these degenerate scenarios without being tailored for specific visualization or algorithmic purposes?
• RQ3 To what extent do existing multiobjective evolutionary algorithms (MOEAs) fail to solve problems with degenerate Pareto fronts?
• RQ4 Why do certain MOEAs struggle with degenerate problems, and what algorithmic strategies might overcome these challenges?
• RQ5 Can the proposed test problems serve as a benchmark to evaluate and guide the development of new algorithms for degenerate multiobjective optimization?

Key findings

• The proposed test problems DPF1A–DPF5A are the first to systematically formalize and generalize the characteristics of degenerate multiobjective problems.
• None of the ten tested representative MOEAs were able to effectively solve the proposed degenerate test problems, indicating a significant gap in current algorithmic capabilities.
• The failure of existing MOEAs on these problems demonstrates that degeneracy is a non-trivial challenge not adequately addressed by current algorithms.
• The study reveals that MOEAs relying on decomposition or decomposition-based mechanisms struggle with degenerate fronts due to poor convergence and diversity in the reduced-dimensional solution space.
• The authors identify that objective reduction and dominance preservation are critical for handling degeneracy, and suggest that future algorithms should explicitly account for objective redundancy and front dimensionality.
• Theoretical analysis confirms that the transformation from essential objectives to full objective space preserves Pareto dominance, validating the correctness of the test problem construction.