Abstract
In a preference-based multi-objective optimization task, the goal is to find a subset of the Pareto-optimal set close to a supplied set of aspiration points. The reference point based non-dominated sorting genetic algorithm (R-NSGA-II) was proposed for such problem-solving tasks. R-NSGA-II aims to finding Pareto-optimal points close, in the sense of Euclidean distance in the objective space, to the supplied aspiration points, instead of finding the entire Pareto-optimal set. In this paper, R-NSGA-II method is modified using recently proposed Karush–Kuhn–Tucker proximity measure (KKTPM) and achievement scalarization function (ASF) metrics, instead of Euclidean distance metric. While a distance measure may not produce desired solutions, KKTPM-based distance measure allows a theoretically-convergent local or global Pareto solutions satisfying KKT optimality conditions and the ASF measure allows Pareto-compliant solutions to be found. A new technique for calculating KKTPM measure of a solution in the presence of an aspiration point is developed in this paper. The proposed modified R-NSGA-II methods are able to solve as many as 10-objective problems as effectively or better than the existing R-NSGA-II algorithm.
Similar content being viewed by others
References
Abouhawwash, M., Jameel, M. A.: Evolutionary multi-objective optimization using Benson’s Karush–Kuhn–Tucker proximity measure. In: International Conference on Evolutionary Multi-criterion Optimization, Springer, pp. 27–38 (2019)
Abouhawwash, M., Jameel, M., Deb, K.: A smooth proximity measure for optimality in multi-objective optimization using Benson’s method, Elsevier. Comput. Oper. Res. (in press) (2020)
Abouhawwash, M., Seada, H., Deb, K.: Towards faster convergence of evolutionary multi-criterion optimization algorithms using Karush Kuhn Tucker optimality based local search. Comput. Oper. Res. Elsevier 79, 331–346 (2017)
Asafuddoula, M., Ray, T., Sarker, R.: A decomposition-based evolutionary algorithm for many objective optimization. IEEE Trans. Evolut. Comput. 19(3), 445–460 (2014)
Asafuddoula, M., Singh, H.K., Ray, T.: An enhanced decomposition-based evolutionary algorithm with adaptive reference vectors. IEEE Trans. Cybern. 48(8), 2321–2334 (2017)
Branke, J., Kaußler, T., Schmeck, H.: Guidance in evolutionary multi-objective optimization. Adv. Eng. Softw. Elsevier 32(6), 499–507 (2001)
Brockhoff, D., Bader, J., Thiele, L., Zitzler, E.: Directed multiobjective optimization based on the weighted hypervolume indicator, Wiley Online Library. J. Multicriteria Decis. Anal. 20(5–6), 291–317 (2013)
Cai, X., Mei, Z., Fan, Z.: A decomposition-based many-objective evolutionary algorithm with two types of adjustments for direction vectors. IEEE Trans. Cybern. 48(8), 2335–2348 (2017)
Castillo, O., Melin, P., Pedrycz, W., Kacprzyk, J.: Recent Advances on Hybrid Approaches for Designing Intelligent Systems. Springer, Berlin (2014)
Chen, C.M., Chen, Y.P., Zhang, Q.: Enhancing MOEA/D with guided mutation and priority update for multi-objective optimization, IEEE. In: 2009 IEEE Congress on Evolutionary Computation, pp. 209–216 (2009)
Cheng, R., Jin, Y., Olhofer, M., Sendhoff, B.: A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Trans. Evolut. Comput. 20(5), 773–791 (2016)
Cheng, R., Li, M., Tian, Y., Zhang, X., Yang, S., Jin, Y., Yao, X.: A benchmark test suite for evolutionary many-objective optimization. Springer Complex Intell. Syst. 3(1), 67–81 (2017)
Coello, C.A.C., Lamont, G.B., Van, V., David, A., et al.: Evolutionary Algorithms for Solving Multi-objective Problems, vol. 5. Springer, Berlin (2007)
Cvetkovic, D., Parmee, I.: Evolutionary design and multi-objective optimisation. In: Proceedings of the Sixth European Congress on Intelligent Techniques and Soft Computing (EUFIT) pp. 397–401 (1998)
Deb, K.: Solving goal programming problems using multi-objective genetic algorithms. In: Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, IEEE, vol. 1, pp. 77–84 (1999)
Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)
Deb, K.: Multi-objective evolutionary algorithms: introducing bias among Pareto-optimal solutions. In: Advances in Evolutionary Computing, Springer, pp. 263–292 (2003)
Deb, K., Agrawal, R.B.: Simulated binary crossover for continuous search space. Complex Syst. 9(2), 115–148 (1995)
Deb, K., Sundar, J.: Reference point based multi-objective optimization using evolutionary algorithms. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, ACM, pp. 635–642 (2006)
Deb, K., Kumar A.: Interactive evolutionary multi-objective optimization and decision-making using reference direction method. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2007), New York: The Association of Computing Machinery (ACM), pp. 781–788 (2007a)
Deb, K., Kumar, A.: Light beam search based multi-objective optimization using evolutionary algorithms. In: Proceedings of the Congress on Evolutionary Computation (CEC-07), pp. 2125–2132 (2007b)
Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: Solving problems with box constraints. IEEE Trans. Evolut. Comput. 18(4), 577–601 (2014)
Deb, K., Abouhawwash, M.: An optimality theory-based proximity measure for set-based multiobjective optimization. IEEE Trans. Evolut. Comput. 20(4), 515–528 (2016)
Deb, K., Amrit, P., Sameer, A., Tamt, M.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evolut. Comput. 6(2), 182–197 (2002)
Deb, K., Zope, P., Jain, A.: Distributed computing of pareto-optimal solutions with evolutionary algorithms. In: International Conference on Evolutionary Multi-criterion Optimization, Springer, pp. 534–549 (2003)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable test problems for evolutionary multiobjective optimization. In: Evolutionary Multiobjective Optimization. Theoretical Advances and Applications, Springer, pp. 105–145 (2005)
Deb, K., Abouhawwash, M., Dutta, J.: An optimality theory based proximity measure for evolutionary multi-objective and many-objective optimization. In: International Conference on Evolutionary Multi-Criterion Optimization, Springer, pp. 18–33 (2015)
Deb, K., Abouhawwash, M., Seada, H.: A computationally fast convergence measure and implementation for single-, multiple-, and many-objective optimization. IEEE Trans. Emerg. Top. Comput. Intell. 1(4), 280–293 (2017)
Dutta, J., Deb, K., Tulshyan, R., Arora, R.: Approximate KKT points and a proximity measure for termination. J. Glob. Optim. Springer 56(4), 1463–1499 (2013)
Fonseca, C.M., Fleming, P.J.: On the performance assessment and comparison of stochastic multiobjective optimizers. In: International Conference on Parallel Problem Solving from Nature, Springer, pp. 584–593 (1996)
Ge, H., Zhao, M., Sun, L., Wang, Z., Tan, G., Zhang, Q., Chen, C.L.P.: A many-objective evolutionary algorithm with two interacting processes: cascade clustering and reference point incremental learning. IEEE Trans. Evolut. Comput. 23(4), 572–586 (2018)
Gong, M., Liu, F., Zhang, W., Jiao, L., Zhang, Q.: Interactive MOEA/D for multi-objective decision making. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, pp. 721–728 (2011)
Gu, F.Q., Liu, H.L.: A novel weight design in multi-objective evolutionary algorithm, IEEE. In: 2010 International Conference on Computational Intelligence and Security, pp. 137–141 (2010)
Gu, F., Cheung, M.: Self-organizing map-based weight design for decomposition-based many-objective evolutionary algorithm. IEEE Trans. Evolut. Comput. 22(2), 211–225 (2017)
He, X., Zhou, Y., Chen, Z., Zhang, Q.: Evolutionary many-objective optimization based on dynamical decomposition. IEEE Trans. Evolut. Comput. 23(3), 361–375 (2018)
Hua, Y., Jin, Y., Hao, K.: A clustering-based adaptive evolutionary algorithm for multiobjective optimization with irregular pareto fronts. IEEE Trans. Cybern. 49(7), 2758–2770 (2018)
Ignizio, J.P.: Goal Programming and Extensions. Lexington Books, Washington, DC (1976)
Ishibuchi, H., Murata, T.: A multi-objective genetic local search algorithm and its application to flowshop scheduling. IEEE Trans. Syst. Man Cybern. Part C 28(3), 392–403 (1998)
Ishibuchi, H., Sakane, Y., Tsukamoto, N., Nojima, Y.: Adaptation of scalarizing functions in MOEA/D: an adaptive scalarizing function-based multiobjective evolutionary algorithm, Springer. In: International Conference on Evolutionary Multi-criterion Optimization, pp. 438–452 (2009)
Ishibuchi, H., Masuda, H., Tanigaki, Y., Nojima, Y.: Difficulties in specifying reference points to calculate the inverted generational distance for many-objective optimization problems, IEEE. In: 2014 IEEE Symposium on Computational Intelligence in Multi-Criteria Decision-Making (MCDM), pp. 170–177 (2014)
Jain, H., Deb, K.: 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. Evolut. Comput. 18(4), 602–622 (2014)
Jaszkiewicz, A.: On the performance of multiple-objective genetic local search on the 0/1 knapsack problem—a comparative experiment. IEEE Trans. Evolut. Comput. 6(4), 402–412 (2002)
Jin, Y., Okabe, T., Sendho, B.: Adapting weighted aggregation for multiobjective evolution strategies, Springer. In: International Conference on Evolutionary Multi-criterion Optimization, pp. 96–110 (2001)
Li, H., Zhang, Q.: Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans. Evolut. Comput. 13(2), 284–302 (2008)
Li, K., Zhang, Q., Kwong, S., Li, M., Wang, R.: Stable matching-based selection in evolutionary multiobjective optimization. IEEE Trans. Evolut. Comput. 18(6), 909–923 (2013)
Li, H., Ding, M., Deng, J., Zhang, Q.: On the use of random weights in MOEA/D, IEEE. In: 2015 IEEE Congress on Evolutionary Computation (CEC), pp. 978–985 (2015)
Li, K., Chen, R., Savić, D., Yao, X.: Interactive decomposition multiobjective optimization via progressively learned value functions. IEEE Trans. Fuzzy Syst. 27(5), 849–860 (2018a)
Li, K., Chen, R., Min, G., Yao, X.: Integration of preferences in decomposition multiobjective optimization. IEEE Trans. Cybern. 48(12), 3359–3370 (2018b)
Li, K., Deb, K., Yao, X.: R-metric: evaluating the performance of preference based evolutionary multi-objective optimization using reference points. IEEE Trans. Evolut. Comput. 22(6), 821–835 (2018c)
Li, K., Liao, M., Deb, K., Min, G., Yao, X.: Does preference always help? A holistic study on preference-based evolutionary multi-objective optimisation using reference points. IEEE Trans. Evolut. Comput. (2020)
Liu, H.L., Gu, F., Zhang, Q.: Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems. IEEE Trans. Evolut. Comput. 18(3), 450–455 (2013)
Liu, R., Wang, R., Feng, W., Huang, J., Jiao, L.: Interactive reference region based multi-objective evolutionary algorithm through decomposition. IEEE Access 4, 7331–7346 (2016)
Liu, H.L., Chen, L., Zhang, Q., Deb, K.: Adaptively allocating search effort in challenging many-objective optimization problems. IEEE Trans. Evolut. Comput. 22(3), 433–448 (2017)
Liu, Y., Ishibuchi, H., Masuyama, N., Nojima, Y.: Adapting reference vectors and scalarizing functions by growing neural gas to handle irregular pareto fronts. IEEE Trans. Evolut. Comput. (2019)
Liu, R., Zhou, R., Ren, R., Liu, J., Jiao, L.: Multi-layer interaction preference based multi-objective evolutionary algorithm through decomposition. Inf. Sci. 509, 420–436 (2020)
Ma, X., Liu, F., Qi, Y., Li, L., Jiao, L., Deng, X., Wang, X., Dong, B., Hou, Z., Zhang, Yongxiao, et al.: MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem, Springer. Soft Comput. 20(12), 4999–5023 (2016)
Masood, A., Mei, Y., Chen, G., Zhang, M.: A PSO-based reference point adaption method for genetic programming hyper-heuristic in many-objective job shop scheduling, Springer. In: Australasian Conference on Artificial Life and Computational Intelligence, pp. 326–338 (2017)
Miettinen, K.: Nonlinear Multiobjective Optimization, vol. 12. Springer, Berlin (2012)
Mohammadi, A., Omidvar, M.N., Li, X.: Reference point based multi-objective optimization through decomposition. In: 2012 IEEE Congress on Evolutionary Computation, IEEE, pp. 1–8 (2012)
Mohammadi, A., Omidvar, M.N., Li, X., Deb, K.: Integrating user preferences and decomposition methods for many-objective optimization, IEEE. In: 2014 IEEE Congress on Evolutionary Computation (CEC), pp. 421–428 (2014)
Molina, J., Santana, L.V., Hernández-Díaz, A.G., Coello, C.A.C., Caballero, R.: g-dominance: reference point based dominance for multiobjective metaheuristics. Eur. J. Oper. Res. 197(2), 685–692 (2009)
Qi, Y., Ma, X., Liu, F., Jiao, L., Sun, J., Wu, J.: MOEA/D with adaptive weight adjustment. Evolut. Comput. 22(2), 231–264 (2014)
Qi, Y., Li, X., Yu, J., Miao, Q.: User-preference based decomposition in MOEA/D without using an ideal point. Swarm Evolut. Comput. 44, 597–611 (2019)
Said, L.B., Bechikh, S., Ghédira, K.: The r-dominance: a new dominance relation for interactive evolutionary multicriteria decision making. IEEE Trans. Evolut. Comput. 14(5), 801–818 (2010)
Seada, H., Abouhawwash, M., Deb, K.: Multi-phase balance of diversity and convergence in multiobjective optimization. IEEE Trans. Evolut. Comput. 23(3), 503–513 (2018)
Shen, X., Guo, Y., Chen, Q., Hu, W.: A multi-objective optimization evolutionary algorithm incorporating preference information based on fuzzy logic. Comput. Optim. Appl. 46(1), 159–188 (2010)
Shukla, P., Deb, K.: On finding multiple pareto-optimal solutions using classical and evolutionary generating methods. Eur. J. Oper. Res. (EJOR) 181(3), 1630–1652 (2007)
Tian, Y., Cheng, R., Zhang, X., Cheng, F., Jin, Y.: An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility. IEEE Trans. Evolut. Comput. 22(4), 609–622 (2017)
Tomczyk, M.K., Kadziński, M.: Decomposition-based interactive evolutionary algorithm for multiple objective optimization. IEEE Trans. Evoluti. Comput. (2019)
Vesikar, Y., Deb, K., Blank, J.: Reference point based NSGA-III for preferred solutions. In: 2018 IEEE Symposium Series on Computational Intelligence (SSCI), IEEE, pp. 1587–1594 (2018)
Von Lücken, C., Barán, B., Brizuela, C.: A survey on multi-objective evolutionary algorithms for many-objective problems. Comput. Optim. Appl. 58(3), 707–756 (2014)
Wagner, T., Trautmann, H., Brockhoff, D.: Preference articulation by means of the R2 indicator, Springer. In: International Conference on Evolutionary Multi-Criterion. Optimization, pp. 81–95 (2013)
Wang, R., Purshouse, R.C., Fleming, P.J.: Preference-inspired coevolutionary algorithms for many-objective optimization. IEEE Trans. Evolut. Comput. 17(4), 474–494 (2012)
Wang, R., Purshouse, R.C., Fleming, P.J.: Preference-inspired co-evolutionary algorithms using weight vectors. Eur. J. Oper. Res. 243(2), 423–441 (2015a)
Wang, R., Purshouse, R.C., Giagkiozis, I., Fleming, P.J.: The iPICEA-g: a new hybrid evolutionary multi-criteria decision making approach using the brushing technique. Eur. J. Oper. Res. 243(2), 442–453 (2015b)
Wang, Z., Zhang, Q., Zhou, A., Gong, M., Jiao, L.: Adaptive replacement strategies for MOEA/D. IEEE Trans. Cybern. 46(2), 474–486 (2015c)
Wang, R., Zhang, Q., Zhang, T.: Decomposition-based algorithms using Pareto adaptive scalarizing methods. IEEE Trans. Evolut. Comput. 20(6), 821–837 (2016)
Wickramasinghe, U.K., Li, X.: Using a distance metric to guide pso algorithms for many-objective optimization. In: Proceedings of the 11th Annual conference on Genetic and evolutionary computation, pp. 667–674 (2009)
Wierzbicki, A.P.: The use of reference objectives in multiobjective optimization. In: Multiple Criteria Decision Making Theory and Application, Springer, pp. 468–486 (1980)
Wu, M., Li, K., Kwong, S., Zhang, Q., Zhang, J.: Learning to decompose: a paradigm for decomposition-based multiobjective optimization. IEEE Trans. Evolut. Comput. 23(3), 376–390 (2018)
Xiang, Y., Zhou, Y., Li, M., Chen, Z.: A vector angle-based evolutionary algorithm for unconstrained many-objective optimization. IEEE Trans. Evolut. Comput. 21(1), 131–152 (2016)
Yu, G., Zheng, J., Shen, R., Li, M.: Decomposing the user-preference in multiobjective optimization. Soft Comput. 20(10), 4005–4021 (2016)
Yuan, Y., Xu, H., Wang, B., Zhang, B., Yao, X.: Balancing convergence and diversity in decomposition-based many-objective optimizers. IEEE Trans. Evolut. Comput. 20(2), 180–198 (2015)
Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evolut. Comput. 11(6), 712–731 (2007)
Zhang, Q., Zhu, W., Liao, B., Chen, X., Cai, L.: A modified PBI approach for multi-objective optimization with complex Pareto fronts. Swarm Evolut. Comput. 40, 216–237 (2018)
Zhao, H., Zhang, C., Zhang, B., Duan, P., Yang, Y.: Decomposition-based sub-problem optimal solution updating direction-guided evolutionary many-objective algorithm. Inf. Sci. 448, 91–111 (2018)
Zhou, C., Dai, G., Zhang, C., Li, X., Ma, K.: Entropy based evolutionary algorithm with adaptive reference points for many-objective optimization problems. Inf. Sci. 465, 232–247 (2018)
Zhu, H., He, Z., Jia, Y.: An improved reference point based multi-objective optimization by decomposition. Int. J. Mach. Learn. Cybernet. 7(4), 581–595 (2016)
Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search, Springer. In: International Conference on Parallel Problem Solving from Nature, pp. 832–842 (2004)
Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evolut. Comput. 8(2), 173–195 (2000)
Zou, J., Fu, L., Yang, S., Zheng, J., Ruan, G., Pei, T., Wang, L.: An adaptation reference-point-based multiobjective evolutionary algorithm. Inf. Sci. 488, 41–57 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abouhawwash, M., Deb, K. Reference point based evolutionary multi-objective optimization algorithms with convergence properties using KKTPM and ASF metrics. J Heuristics 27, 575–614 (2021). https://doi.org/10.1007/s10732-021-09470-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-021-09470-4