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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Artificial Intelligence, pp. 105-129, 2019.

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Multi/many-objective optimizationTest problemsPerformance evaluationConvergence and diversity

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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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Reference

K. Miettinen, Nonlinear Multiobjective Optimization, International Series in Operations Research & Management Science, vol. 12, Springer US, Boston, MA, 1998.
B. Li, J. Li, K. Tang, X. Yao, Many-objective evolutionary algorithms: a survey, ACM Comput. Surv. 48 (2015) 1–35, https://doi.org/10.1145/2792984.
J. Knowles, D. Corne, Instance generators and test suites for the multiobjective quadratic assignment problem, in: Evolutionary Multi-Criterion Optimization, in: Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2003, pp. 295–310.
D.W. Corne, J.D. Knowles, Techniques for highly multiobjective optimisation: some nondominated points are better than others, in: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation - GECCO ’07, ACM Press, London, England, 2007, p. 773.
Y. Zhang, M. Harman, S.A. Mansouri, The multi-objective next release problem, in: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation - GECCO ’07, ACM Press, London, England, 2007, p. 1129.
H. Ishibuchi, H. Masuda, Y. Tanigaki, Y. Nojima, Review of Coevolutionary Developments of Evolutionary Multi-Objective and Many-Objective Algorithms and Test Problems, IEEE, 2014, pp. 178–184.
H. Ishibuchi, N. Akedo, Y. Nojima, Behavior of multiobjective evolutionary algorithms on many-objective Knapsack problems, IEEE Trans. Evol. Comput. 19 (2015) 264–283, https://doi.org/10.1109/TEVC.2014.2315442.
K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, 1st ed., Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Chichester, New York, 2001.
J.D. Knowles, D.W. Corne, Approximating the nondominated front using the Pareto archived evolution strategy, Evol. Comput. 8 (2000) 149–172, https://
D.W. Corne, N.R. Jerram, J.D. Knowles, M.J. Oates PESA-II, Region-based selection in evolutionary multiobjective optimization, in: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, GECCO’01, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2001, pp. 283–290.
E. Zitzler, M. Laumanns, L. Thiele, SPEA2: Improving the Strength Pareto Evolutionary Algorithm, Technical Report TIK-Report 103, ETH, Zurich, 2001, https://doi.org/10.3929/ethz-a-004284029.
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2002) 182–197, https://doi.org/10.1109/4235.996017, 17703.
Qingfu Zhang, Hui Li, MOEA/D: a multiobjective evolutionary algorithm based on decomposition, IEEE Trans. Evol. Comput. 11 (2007) 712–731, https://
Hui Li, Qingfu Zhang, Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II, IEEE Trans. Evol. Comput. 13 (2009) 284–302, https://doi.org/10.1109/TEVC.2008.925798.
Y. Qi, X. Ma, F. Liu, L. Jiao, J. Sun, J. Wu, MOEA/D with adaptive weight adjustment, Evol. Comput. 22 (2014) 231–264, https://doi.org/10.1162/EVCO_a_
K. Deb, H. Jain, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints, IEEE Trans. Evol. Comput. 18 (2014) 577–601, https://doi.org/10.1109/TEVC.2013.2281535.
R. Cheng, Y. Jin, M. Olhofer, B. Sendhoff, A reference vector guided evolutionary algorithm for many-objective optimization, IEEE Trans. Evol. Comput. 20 (2016) 773–791, https://doi.org/10.1109/TEVC.2016.2519378.
Y. Xiang, Y. Zhou, M. Li, Z. Chen, A vector angle-based evolutionary algorithm for unconstrained many-objective optimization, IEEE Trans. Evol. Comput. 21 (2017) 131–152, https://doi.org/10.1109/TEVC.2016.2587808.
Z. Chen, Y. Zhou, Y. Xiang, A many-objective evolutionary algorithm based on a projection-assisted intra-family election, Appl. Soft Comput. 61 (2017) 394–411, https://doi.org/10.1016/j.asoc.2017.07.052.
E. Zitzler, S. Künzli, Indicator-Based Selection in Multiobjective Search, Parallel Problem Solving From Nature - PPSN VIII, vol. 3242, Springer Berlin Heidelberg, Berlin, Heidelberg, 2004, pp. 832–842.
N. Beume, B. Naujoks, M. Emmerich SMS-EMOA, Multiobjective selection based on dominated hypervolume, Eur. J. Oper. Res. 181 (2007) 1653–1669, https://doi.org/10.1016/j.ejor.2006.08.008.
J. Bader, E. Zitzler, HypE: an algorithm for fast hypervolume-based many-objective optimization, Evol. Comput. 19 (2011) 45–76, https://doi.org/10.
R. Hernández Gómez, C.A. Coello Coello, Improved metaheuristic based on the R2 indicator for many-objective optimization, in: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, GECCO ’15, ACM, Madrid, Spain, 2015, pp. 679–686.
T. Weise, R. Chiong, K. Tang, Evolutionary optimization: Pitfalls and Booby traps, J. Comput. Sci. Technol. 27 (2012) 907–936, https://doi.org/10.1007/
E. Zitzler, K. Deb, L. Thiele, Comparison of multiobjective evolutionary algorithms: empirical results, Evol. Comput. 8 (2000) 173–195, https://doi.org/10.1162/106365600568202.
K. Deb, L. Thiele, M. Laumanns, E. Zitzler, Scalable Multi-Objective Optimization Test Problems, vol. 1, IEEE, 2002, pp. 825–830.
S. Huband, P. Hingston, L. Barone, L. While, A review of multiobjective test problems and a scalable test problem toolkit, IEEE Trans. Evol. Comput. 10 (2006) 477–506, https://doi.org/10.1109/TEVC.2005.861417.
K. Deb, Multi-objective genetic algorithms: problem difficulties and construction of test problems, Evol. Comput. 7 (1999) 205–230, https://doi.org/10.
H. Ishibuchi, Y. Setoguchi, H. Masuda, Y. Nojima, Performance of decomposition-based many-objective algorithms strongly depends on Pareto front shapes, IEEE Trans. Evol. Comput. 21 (2017) 169–190, https://doi.org/10.1109/TEVC.2016.2587749.
H.L. Liu, L. Chen, Q. Zhang, K. Deb, Adaptively allocating search effort in challenging many-objective optimization problems, IEEE Trans. Evol. Comput. 22 (3) (2018) 433–448, https://doi.org/10.1109/TEVC.2017.2725902.
R. Cheng, M. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, X. Yao, A benchmark test suite for evolutionary many-objective optimization, Complex Intell. Syst. 3 (2017) 67–81, https://doi.org/10.1007/s40747-017-0039-7.
H. Jain, K. Deb, An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: handling constraints and extending to an adaptive approach, IEEE Trans. Evol. Comput. 18 (2014) 602–622, https://doi.org/10.1109/TEVC.2013.2281534.
Z. Wang, Y.S. Ong, H. Ishibuchi, On scalable multiobjective test problems with hardly-dominated boundaries, IEEE Trans. Evol. Comput. 23 (2) (2019) 217–231, https://doi.org/10.1109/TEVC.2018.2844286.
K. Deb, L. Thiele, M. Laumanns, E. Zitzler, Scalable test problems for evolutionary multiobjective optimization, in: A. Abraham, L. Jain, R. Goldberg (Eds.), Evolutionary Multiobjective Optimization, Springer-Verlag, London, 2005, pp. 105–145.
S. Zapotecas-Martínez, C.A.C. Coello, H.E. Aguirre, K. Tanaka, A review of features and limitations of existing scalable multi-objective test suites, IEEE Trans. Evol. Comput. 23 (1) (2019) 130–142, https://doi.org/10.1109/TEVC.2018.2836912.
Qingfu Zhang, Aimin Zhou, RM-MEDA Yaochu Jin, A regularity model-based multiobjective estimation of distribution algorithm, IEEE Trans. Evol. Comput. 12 (2008) 41–63, https://doi.org/10.1109/TEVC.2007.894202.
R. Cheng, Y. Jin, K. Narukawa, B. Sendhoff, A multiobjective evolutionary algorithm using Gaussian process-based inverse modeling, IEEE Trans. Evol. Comput. 19 (2015) 838–856, https://doi.org/10.1109/TEVC.2015.2395073.
R. Cheng, Y. Jin, M. Olhofer, B. sendhoff, Test problems for large-scale multiobjective and many-objective optimization, IEEE Trans. Cybern. 47 (2017) 4108–4121, https://doi.org/10.1109/TCYB.2016.2600577.
M. Emmerich, N. Beume, B. Naujoks, An EMO Algorithm Using the Hypervolume Measure as Selection Criterion, Evolutionary Multi-Criterion Optimization, vol. 3410, Springer Berlin Heidelberg, Berlin, Heidelberg, 2005, pp. 62–76.
E. Zitzler, M. Laumanns, L. Thiele, SPEA2: Improving the Strength Pareto Evolutionary Algorithm, TIK-Report 103, ETH, Zurich, Switzerland, 2001.
Y. Yuan, H. Xu, B. Wang, X. Yao, A new dominance relation-based evolutionary algorithm for many-objective optimization, IEEE Trans. Evol. Comput. 20 (2016) 16–37, https://doi.org/10.1109/TEVC.2015.2420112.
Y. Zhou, Y. Xiang, Z. Chen, J. He, J. Wang, A scalar projection and angle-based evolutionary algorithm for many-objective optimization problems, IEEE Trans. Cybern. (2018) 1–12, https://doi.org/10.1109/TCYB.2018.2819360.
H. Sato, Inverted PBI in MOEA/D and Its Impact on the Search Performance on Multi and Many-Objective Optimization, ACM Press, 2014, pp. 645–652.
L. Pan, C. He, Y. Tian, Y. Su, X. Zhang, A region division based diversity maintaining approach for many-objective optimization, Integr. Comput.-Aided Eng. 24 (2017) 279–296, https://doi.org/10.3233/ICA-170542.
C.A. Coello Coello, G.B. Lamont, D.A. Van Veldhuizen, Evolutionary Algorithms for Solving Multi-Objective Problems, 2nd ed., Genetic and Evolutionary Computation Series, Springer, New York, NY, 2007, OCLC: 255368493.
P. Bosman, D. Thierens, The balance between proximity and diversity in multiobjective evolutionary algorithms, IEEE Trans. Evol. Comput. 7 (2003) 174–188, https://doi.org/10.1109/TEVC.2003.810761.
Q. Zhang, A. Zhou, S. Zhao, P.N. Suganthan, W. Liu, S. Tiwari, Multiobjective Optimization Test Instances for the CEC 2009 Special Session and Competition, Technical Report CES-487, 2008, The School of Computer Science and Electronic Engieering University of Essex, Colchester, C04, 3SQ, UK; School of Electrical and Electronic Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore; Department of Mechanical Engineering Clemson University, Clemson, SC 29634, US.
S. Yang, M. Li, X. Liu, J. Zheng, A grid-based evolutionary algorithm for many-objective optimization, IEEE Trans. Evol. Comput. 17 (2013) 721–736, https://doi.org/10.1109/TEVC.2012.2227145.
M. Li, S. Yang, X. Liu, Shift-based density estimation for Pareto-based algorithms in many-objective optimization, IEEE Trans. Evol. Comput. 18 (2014) 348–365, https://doi.org/10.1109/TEVC.2013.2262178.
H. Ishibuchi, H. Masuda, Y. Tanigaki, Y. Nojima, Difficulties in specifying reference points to calculate the inverted generational distance for many-objective optimization problems, in: 2014 IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making (MCDM), 2014, pp. 170–177.
H. Ishibuchi, H. Masuda, Y. Nojima, A study on performance evaluation ability of a modified inverted generational distance indicator, in: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, GECCO ’15, ACM, New York, NY, USA, 2015, pp. 695–702.
I. Das, J.E. Dennis, Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems, SIAM J. Optim. 8 (1998) 631–657, https://doi.org/10.1137/S1052623496307510.
H. Ishibuchi, H. Masuda, Y. Nojima, Pareto fronts of many-objective degenerate test problems, IEEE Trans. Evol. Comput. 20 (2016) 807–813, https://doi.org/10.1109/TEVC.2015.2505784.
E. Zitzler, L. Thiele, Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach, IEEE Trans. Evol. Comput. 3 (1999) 257–271.
L. While, L. Bradstreet, L. Barone, A fast way of calculating exact hypervolumes, IEEE Trans. Evol. Comput. 16 (2012) 86–95, https://doi.org/10.1109/ TEVC.2010.2077298.

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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