Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit (1612.07603v3)

Abstract: Multi-objective evolutionary algorithms (MOEAs) have progressed significantly in recent decades, but most of them are designed to solve unconstrained multi-objective optimization problems. In fact, many real-world multi-objective problems contain a number of constraints. To promote research on constrained multi-objective optimization, we first propose a problem classification scheme with three primary types of difficulty, which reflect various types of challenges presented by real-world optimization problems, in order to characterize the constraint functions in constrained multi-objective optimization problems (CMOPs). These are feasibility-hardness, convergence-hardness and diversity-hardness. We then develop a general toolkit to construct difficulty-adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPs when the number of objectives is greater than three) with three types of parameterized constraint functions developed to capture the three proposed types of difficulty. Based on this toolkit, we suggest nine difficulty-adjustable and scalable CMOPs and nine CMaOPs. The experimental results reveal that mechanisms in MOEA/D-CDP may be more effective in solving convergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be more effective in solving DAS-CMOPs with simultaneous diversity-, feasibility- and convergence-hardness. Mechanisms in C-NSGA-III may be more effective in solving feasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective in solving CMaOPs with convergence-hardness. In addition, none of them can solve these problems efficiently, which stimulates us to continue to develop new CMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

Insights into Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit

The paper "Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit" by Zhun Fan et al. offers a novel approach to the development of test problems for constrained multi-objective optimization, each with customized levels of difficulty. This essay seeks to provide a comprehensive overview of the methodologies and findings presented in the paper.

Overview of the Problem Toolkit

Multi-objective evolutionary algorithms (MOEAs) have historically been predisposed to address unconstrained scenarios, despite real-world problems often presenting multiple constraints alongside several objective functions. The authors introduce Difficulty Adjustable and Scalable Constrained Multi-objective Test Problems (DAS-CMOPs), stretching multi-objective frameworks to aid the research community in evaluating and enhancing constrained multi-objective optimization algorithms (CMOEAs).

The toolkit effectively categorizes constraints into three types: feasibility-hardness, convergence-hardness, and diversity-hardness. Each constraint type captures specific challenges typical to real-world constraints:

• Feasibility-hardness poses difficulty in finding feasible solutions due to the small feasible region.
• Convergence-hardness creates obstacles on the path to the Pareto front (PF), complicating convergence.
• Diversity-hardness hampers solution distribution across the PF, forming discrete segments.

Construction of DAS-CMOPs

The paper presents a framework for constructing DAS-CMOPs using three types of constraints correlated to the primary difficulty types:

1. Type-I Constraint Functions: These cause diversity-hardness by dividing the Pareto front into disconnected segments, thus demanding strategic solution distribution.
2. Type-II Constraint Functions: Enforcing feasibility-hardness, they control the proportion of feasible regions in the search space.
3. Type-III Constraint Functions: Dedicated to producing convergence-hardness, these restrict objective boundaries, making it difficult for algorithms to progress toward the PF.

Each difficulty type's level is parameterized, offering a tripartite segregation akin to the primary colors in digital color spaces, enabling precise difficulty calibration.

Scalability and Test Problem Examples

The authors detail the paper's scalability in both objectives and constraints, capable of addressing constrained many-objective optimization problems (CMaOPs) when objectives exceed three. The DAS-CMOPs constructed in the paper demonstrate this scalability, offering nine multifaceted test problems with varied constraint setups.

Additionally, the DAS-CMOPs were validated with existing CMOEAs, including MOEA/D with constraint dominance principle (MOEA/D-CDP) and NSGA-II-CDP. Notably, the paper found that MOEA/D-CDP handles convergence-hard DAS-CMOPs more effectively, whereas NSGA-II-CDP excels in addressing DAS-CMOPs with simultaneous diversity and feasibility constraints.

Implications and Future Directions

The methodologies propounded by Zhun Fan et al. pave a clear path for assessing and refining CMOEAs and CMaOEAs, encouraging newer designs for improved constraint handling. The customizable difficulty levels facilitate testing across various constraint scenarios, reinforcing the importance of diversity and convergence balance in optimization solutions.

The toolkit's future potential includes exploration into problems featuring varying variable counts and types, particularly mixed continuous and discrete variables. Furthermore, evolving the toolkit to address large-scale optimization problems with expansive variable dimensions would provide deeper insights into constraint management efficiency within CMOEAs.

In conclusion, the toolkit introduced provides a reliable, scalable, and adjustable framework for algorithm testing within constrained multi-objective optimization problems, promoting progressive enhancement and understanding of CMOEAs. The robust nature of DAS-CMOPs ensures their efficacy as benchmark tools in the advancement of real-world optimization challenges.