A set of new multi- and many-objective test problems for continuous optimization and a comprehensive experimental evaluation

Yuren Zhou (周育人)

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Xiaoyu He

Yi Xiang

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Shaowei Cai (蔡少伟)

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Artif. Intell., pp. 105-129, 2019.

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inverted generational distanceevolutionary many objective optimizationdistance functioninverted PBIreal worldMore(20+)

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We provide a penalty method to augment the position function, which further increases the difficulties of the problems

Abstract:

Multi- and many-objective optimization problems have wide applications in the real world, and they have received growing attention from the evolutionary computation community. To promote the algorithm development in this area, numerous studies have been devoted to designing multi- and many-objective test problems. Most of these studies fo...More

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Introduction

There are a variety of problems involving more than one conflicting objectives.
These problems, referred to as the multiobjective optimization problems (MOPs), can be stated as follows:.
✩ This paper is supported in part by the Natural Science Foundation of China (Grant nos.
61773410, 61673403, and 61906069), and in part by the Fundamental Research Funds for the Central Universities under Grant x2rjD2190840.
For other types of problems that involve discrete or combinatorial variables, please refer to [3,4,5,6,7]

Highlights

In the real world, there are a variety of problems involving more than one conflicting objectives
This paper presents a novel framework to design multiobjective test problems by considering the geometrical properties of both Pareto sets (PSs) and Pareto fronts (PFs)
The first m decision variables are transformed into the hyperspherical coordinates to form the PF while the rest are designed to optimize a certain distance function
A position function is introduced to impose non-linear dependencies on the first m decision variables such that they form a unit hypersphere centered at the origin and enveloped in the first orthant
We provide a penalty method to augment the position function, which further increases the difficulties of the problems
F1 to F12 cover a variety of difficulties in both PF irregularity and PS complexity

Methods

Number of objectives (m).
The authors consider four values for the number of objectives, namely m ∈ {2, 3, 5, 10}.
The authors choose m = 2 and m = 3 because MOPs with 2 or 3 objectives have been studied in depth over the past few decades.
The authors choose m = 5 and m = 10 for constructing MaOPs, noting that 10 objectives are sufficient to cause considerable difficulties for MOEAs regarding the impact of the dominance resistance

Conclusion

This paper presents a novel framework to design multiobjective test problems by considering the geometrical properties of both PS and PF.
In this framework, a decision vector is divided into two parts.
F13 to F24 further increase difficulties by using the penalty-based position functions
They differ from existing MOPs because the position variables are correlated

Summary

Introduction:
There are a variety of problems involving more than one conflicting objectives.
These problems, referred to as the multiobjective optimization problems (MOPs), can be stated as follows:.
✩ This paper is supported in part by the Natural Science Foundation of China (Grant nos.
61773410, 61673403, and 61906069), and in part by the Fundamental Research Funds for the Central Universities under Grant x2rjD2190840.
For other types of problems that involve discrete or combinatorial variables, please refer to [3,4,5,6,7]
Methods:
Number of objectives (m).
The authors consider four values for the number of objectives, namely m ∈ {2, 3, 5, 10}.
The authors choose m = 2 and m = 3 because MOPs with 2 or 3 objectives have been studied in depth over the past few decades.
The authors choose m = 5 and m = 10 for constructing MaOPs, noting that 10 objectives are sufficient to cause considerable difficulties for MOEAs regarding the impact of the dominance resistance
Conclusion:
This paper presents a novel framework to design multiobjective test problems by considering the geometrical properties of both PS and PF.
In this framework, a decision vector is divided into two parts.
F13 to F24 further increase difficulties by using the penalty-based position functions
They differ from existing MOPs because the position variables are correlated

Tables

Table1: Summary of the proposed test problems
Table2: Detailed definitions of F1 to F24
Table3: The number of divisions, and that of the generated reference points for calculating IGD+
Table4: The number of weight vectors for different values of m
Table5: Medians of IGD+ results on F13-F16 and four representative DTLZ/WFG test problems (n = m + 10)
Table6: Medians of IGD+ results on F13-F16 and four representative DTLZ/WFG test problems (n = m + 30)
Table7: The proportion of converged solutions for both NSGA-III and MOEA/D on the 10-objective F13-F16 with n = m + 10
Table8: Medians of IGD+ results on F1-F4 (without penalty) and F13-F16 (with penalty), where n = 10 + m. In this table, the best results are shown in bold
Table9: The proportion of converged solutions for θ -DEA on {F1, F13} and {F2, F14} with n = m + 10
Table10: Medians of HV results on F9-F12 (without penalty) and F21-F24 (with penalty), where n = 10 + m. In this table, the best results are shown in bold
Table11: Medians of IGD+ results on F1-F12, where n = 30 + m. For each type of PF, the best and the second-best results are shown in a dark-gray and a light-gray background, respectively
Table12: Medians of IGD+ results on F13-F24, where n = 30 + m. For each type of PF, the best and the second-best results are shown in a dark-gray and a light-gray background, respectively
Table13: Medians of IGD+ results on the 2- and 3-objective F5-F8 with C =10, 30 and 60
Table14: Medians of IGD+ results on the 5- and 10-objective F5-F8 with C =10, 30 and 60

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A set of new multi- and many-objective test problems for continuous optimization and a comprehensive experimental evaluationFull Text

Introduction:

Methods:

Conclusion:

A set of new multi- and many-objective test problems for continuous optimization and a comprehensive experimental evaluation