Carlos Hernández
ABSTRACT
One important issue in evolutionary multi-objective optimization (EMO) which still leaves room for improvement is the maintenance of the subset of the obtained candidate solutions that forms the approximation of the Pareto front (in short: selection or archiving). Existing archivers that are entirely based on the distances between the candidate solutions are known to reveal cyclic behavior and do hence not tap their full potential. On the other hand, there exist archivers based on ∊-dominance that guarantee monotonic behaviors as well as certain approximation qualities in the limit. For such methods, however, so far the magnitudes of the final archive size heavily depend on several design parameters and cannot be bounded a priori which is desired by many EMO researchers.
In this paper, we propose a new archiver, ArchiveUpdateBound, for bi-objective problems that is based both on distance and ∊-dominance. ArchiveUpdateBound aims for a uniform spread of the solutions along the Pareto front, and the archive sizes can be bounded above. In particular, the use of ∊-dominance eliminates the occurrence of cyclic behavior. Numerical results using NSGA-II, MOEA/D, and SMS-MOEA on selected benchmark problems indicate that the use of the novel archiver significantly increases the performance of the base MOEAs.
- Nicola Beume, Boris Naujoks, and Michael Emmerich. 2007. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research 181, 3 (2007), 1653--1669.Google Scholar
Cross Ref
- Johan M. Bogoya, Andrés Vargas, Oliver Cuate, and Oliver Schütze. 2018. A (p,q)-Averaged Hausdorff Distance for Arbitrary Measurable Sets. Mathematical and Computational Applications 23, 3 (2018).Google Scholar
- Johan M. Bogoya, Andrés Vargas, and Oliver Schütze. 2019. The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review. Mathematics 7, 10 (2019).Google Scholar
- Dimo Brockhoff, Thanh-Do Tran, and Nikolaus Hansen. 2015. Benchmarking numerical multiobjective optimizers revisited. In Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation. 639--646.Google ScholarDigital Library
- C. A. Coello Coello, G. B. Lamont, and D. A. Van Veldhuizen. 2007. Evolutionary Algorithms for Solving Multi-Objective Problems (second ed.). Springer, New York.Google Scholar
- D. W. Corne and J. D. Knowles. 2003. Some multiobjective optimizers are better than others. In Proceedings of the IEEE Congress on Evolutionary Computation. IEEE Press, 2506--2512.Google Scholar
- K. Deb. 2001. Multi-Objective Optimization using Evolutionary Algorithms. John Wiley & Sons, Chichester, UK. ISBN 0-471-87339-X.Google ScholarDigital Library
- Kalyanmoy Deb and Himanshu Jain. 2014. An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: Solving problems with box constraints. Transactions on Evolutionary Computation 18, 4 (2014), 577--601.Google ScholarCross Ref
- Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and TAMT Meyarivan. 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II. Evolutionary Computation, IEEE Transactions on 6, 2 (2002), 182--197.Google ScholarDigital Library
- Michael Emmerich and AndrDeutz. 2007. Test problems based on Lamé super-spheres. In Evolutionary Multi-Criterion Optimization: 4th International Conference, EMO 2007, Matsushima, Japan, March 5--8, 2007. Proceedings, S. Obayashi, K. Deb, C. Poloni, T. Hiroyasu, and T. Murata (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 922--936.Google ScholarDigital Library
- K. Gerstl, G. Rudolph, O. Schütze, and H. Trautmann. 2011. Finding Evenly Spaced Fronts for Multiobjective Control via Averaging Hausdorff-Measure. In International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2011). IEEE, 1--6.Google Scholar
- T. Hanne. 1999. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research 117 (1999), 553--564. Issue 3.Google ScholarCross Ref
- T. Hanne. 2001. Global multiobjective optimization with evolutionary algorithms: Selection mechanisms and mutation control. In Evolutionary Multi-Criterion Optimization, First International Conference, EMO 2001, Zurich, Switzerland. Springer Berlin, 197--212.Google Scholar
- T. Hanne. 2007. A multiobjective evolutionary algorithm for approximating the efficient set. European Journal of Operational Research 176 (2007), 1723--1734. Issue 3.Google ScholarCross Ref
- X Hanne. 2007. A Primal-Dual Multiobjective Evolutionary Algorithm for Approximating the Efficient Set. In Evolutionary Computation (CEC), 2007 IEEE Congress on. IEEE Press, 3127--3134.Google ScholarCross Ref
- C. Hillermeier. 2001. Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach. Vol. 135. Springer Science & Business Media.Google Scholar
- K. Ikeda, H. Kita, and S. Kobayashi. 2001. Failure of Pareto-based MOEAs: Does non-dominated really mean near to optimal?. In Evolutionary Computation (CEC), 2001 IEEE Congress on IEEE, 957--962.Google Scholar
- Hisao Ishibuchi, Takashi Matsumoto, Naoki Masuyama, and Yusuke Nojima. 2020. Effects of dominance resistant solutions on the performance of evolutionary multi-objective and many-objective algorithms. In GECCO '20: Genetic and Evolutionary Computation Conference, Cancún Mexico, July 8--12, 2020, Carlos Artemio Coello Coello (Ed.). ACM, 507--515. Google ScholarDigital Library
- Hisao Ishibuchi, Lie Meng Pang, and Ke Shang. 2020. A New Framework of Evolutionary Multi-Objective Algorithms with an Unbounded External Archive. (2020).Google Scholar
- Hisao Ishibuchi, Lie Meng Pang, and Ke Shang. 2020. Solution Subset Selection for Final Decision Making in Evolutionary Multi-Objective Optimization. arXiv preprint arXiv.2006.08156 (2020).Google Scholar
- J. D. Knowles and D. W. Corne. 2003. Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Transactions on Evolutionary Computation 7 (2003), 100--116. Issue 2.Google ScholarDigital Library
- J. D. Knowles and D. W. Corne. 2004. Bounded Pareto archiving: theory and practice. In Metaheuristics for Multiobjective Optimisation. Springer, 39--64.Google Scholar
- M. Laumanns, L. Thiele, K. Deb, and E. Zitzler. 2002. Combining convergence and diversity in evolutionary multiobjective optimization. Evolutionary computation 10 (2002), 263--282. Issue 3.Google Scholar
- Marco Laumanns, Eckart Zitzler, and Lothar Thiele. 2001. On the effects of archiving, elitism, and density based selection in evolutionary multi-objective optimization. In International Conference on Evolutionary Multi-Criterion Optimization. Springer, Berlin, Heidelberg, 181--196.Google ScholarCross Ref
- Miqing Li. 2021. Is Our Archiving Reliable? Multiobjective Archiving Methods on "Simple" Artificial Input Sequences. ACM Transactions on Evolutionary Learning and Optimization 1, 3 (2021), 1--19.Google ScholarDigital Library
- P. Loridan. 1984. ∊-Solutions in Vector Minimization Problems. Journal of Optimization, Theory and Application 42 (1984), 265--276.Google ScholarCross Ref
- H. N. Luong and P. A. N. Bosnian. 2012. Elitist Archiving for Multi-Objective Evolutionary Algorithms: To Adapt or Not to Adapt. In Parallel Problem Solving from Nature - PPSN XII. Springer, Berlin, Heidelberg, 72--81.Google Scholar
- Lie Meng Pang, Hisao Ishibuchi, and Ke Shang. 2020. Algorithm Configurations of MOEA/D with an Unbounded External Archive. arXiv preprint arXiv.2007.13352 (2020).Google Scholar
- G. Rudolph. 1998. Finite Markov Chain results in evolutionary computation: A Tour d'Horizon. Fundamenta Informaticae 35 (1998), 67--89.Google ScholarDigital Library
- G. Rudolph. 1998. On a multi-objective evolutionary algorithm and its convergence to the Pareto set. In IEEE International Conference on Evolutionary Computation (ICEC 1998). IEEE Press, 511--516.Google ScholarCross Ref
- G. Rudolph. 2001. Evolutionary Search under Partially Ordered Fitness Sets. In Proceedings of the International NAISO Congress on Information Science Innovations (ISI 2001). ICSC Academic Press, Sliedrecht, The Netherlands, 818--822.Google Scholar
- G. Rudolph and A. Agapie. 2000. Convergence Properties of Some Multi-Objective Evolutionary Algorithms. In Evolutionary Computation (CEC), 2011 IEEE Congress on. IEEE Press.Google Scholar
- G. Rudolph, B. Naujoks, and M. Preuss. 2006. Capabilities of EMOA to Detect and Preserve Equivalent Pareto Subsets. In EMO '03: Proceedings of the Evolutionary Multi-Criterion Optimization Conference. 36--50.Google Scholar
- G. Rudolph, O. Schütze, C. Grimme, C. Domínguez-Medina, and H. Trautmann. 2016. Optimal averaged Hausdorff archives for bi-objective problems: theoretical and numerical results. Computational Optimization and Applications 64, 2 (01 Jun 2016), 589--618.Google Scholar
- G. Rudolph, H. Trautmann, S. Sengupta, and O. Schütze. 2013. Evenly Spaced Pareto Front Approximations for Tricriteria Problems Based on Triangulation. In Evolutionary Muti-Criterion Optimization Conference (EMO 2013). Springer, 443--458.Google Scholar
- S. Schaeffler, R. Schultz, and K. Weinzierl. 2002. Stochastic Method for the Solution of Unconstrained Vector Optimization Problems. Journal of Optimization Theory and Applications 114, 1 (2002), 209--222.Google ScholarDigital Library
- Oliver Schütze, Christian Domínguez-Medina, Nareli Cruz-Cortés, Luis Gerardo de la Fraga, Jian-Qiao Sun, Gregorio Toscano, and Ricardo Landa. 2016. A scalar optimization approach for averaged Hausdorff approximations of the Pareto front. Engineering Optimization 48, 9 (2016), 1593--1617.Google ScholarCross Ref
- O. Schütze, X. Esquivel, A. Lara, and C. A. Coello Coello. 2012. Using the averaged Hausdorff distance as a performance measure in evolutionary multi-objective optimization. IEEE Transactions on Evolutionary Computation 16, 4 (2012), 504--522.Google ScholarDigital Library
- O. Schütze and C. Hernandez. 2021. Archiving Strategies for Evolutionary Multi-objective Optimization Algorithms. Springer.Google Scholar
- O. Schütze, C. Hernandez, E-G. Talbi, J. Q. Sun, Y. Naranjani, and F. R. Xiong. 2019. Archivers for the Representation of the Set of Approximate Solutions for MOPs. Journal of Heuristics 5, 1 (2019), 71--105.Google ScholarDigital Library
- O. Schütze, M. Laumanns, C. A. Coello Coello, M. Dellnitz, and E.-G. Talbi. 2008. Convergence of Stochastic Search Algorithms to Finite Size Pareto Set Approximations. Journal of Global Optimization 41, 4 (2008), 559--577.Google ScholarDigital Library
- O. Schütze, M. Laumanns, E. Tantar, C. A. Coello Coello, and E.-G. Talbi. 2007. Convergence of stochastic search algorithms to gap-free Pareto front approximations. In Genetic and Evolutionary Computation Conference (GECCO-2007). 892--901.Google ScholarDigital Library
- O. Schütze, M. Laumanns, E. Tantar, C. A. Coello Coello, and E.-G. Talbi. 2010. Computing gap free Pareto front approximations with stochastic search algorithms. Evolutionary Computation 18, 1 (2010), 65--96.Google ScholarDigital Library
- Oliver Schütze, Massimiliano Vasile, and Carlos A. Coello Coello. 2011. Computing the Set of Epsilon-Efficient Solutions in Multiobjective Space Mission Design. Journal of Aerospace Computing, Information, and Communication 8, 3 (2011), 53--70.Google ScholarCross Ref
- Rui Wang, Zhongbao Zhou, Hisao Ishibuchi, Tianjun Liao, and Tao Zhang. 2016. Localized weighted sum method for many-objective optimization. IEEE Transactions on Evolutionary Computation 22, 1 (2016), 3--18.Google ScholarCross Ref
- D. J. White. 1986. Epsilon efficiency. Journal of Optimization Theory and Applications 49, 2 (1986), 319--337.Google ScholarDigital Library
- Katrin Witting. 2012. Numerical Algorithms for the Treatment of Parametric Multiobjective Optimization Problems and Applications. Ph.D. Dissertation. PhD thesis, Deptartment of Mathematics, University of Paderborn.Google Scholar
- S. Zapotecas Martinez and C. A. Coello Coello. 2010. An archiving strategy based on the Convex Hull of Individual Minima for MOEAs. In Evolutionary Computation (CEC), 2010 IEEE Congress on. IEEE Press, 1--8.Google Scholar
- Saúl Zapotecas-Martínez, Antonio López-Jaimes, and Abel García-Nájera. 2019. Libea: A Lebesgue indicator-based evolutionary algorithm for multi-objective optimization. Swarm and Evolutionary Computation 44 (2019), 404--419.Google ScholarCross Ref
- Q. Zhang and H. Li. 2007. MOEA/D: A Multi-objective Evolutionary Algorithm Based on Decomposition. IEEE Transactions on Evolutionary Computation 11, 6 (2007), 712--731.Google ScholarDigital Library
- Eckart Zitzler, Kalyanmoy Deb, and Lothar Thiele. 2000. Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary computation 8, 2 (2000), 173--195.Google Scholar
- E. Zitzler, M. Laumanns, and L. Thiele. 2002. SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. In Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001), K.C. Giannakoglou et al. (Eds.). International Center for Numerical Methods in Engineering (CIMNE), 95--100.Google Scholar
Index Terms
- A bounded archive based for bi-objective problems based on distance and e-dominance to avoid cyclic behavior
Recommendations
Bi-goal evolution for many-objective optimization problems
This paper presents a meta-objective optimization approach, called Bi-Goal Evolution (BiGE), to deal with multi-objective optimization problems with many objectives. In multi-objective optimization, it is generally observed that 1) the conflict between ...
Single-objective and multi-objective formulations of solution selection for hypervolume maximization
GECCO '09: Proceedings of the 11th Annual conference on Genetic and evolutionary computationA new trend in evolutionary multi-objective optimization (EMO) is the handling of a multi-objective problem as an optimization problem of an indicator function. A number of approaches have been proposed under the name of indicator-based evolutionary ...
Effects of dominance resistant solutions on the performance of evolutionary multi-objective and many-objective algorithms
GECCO '20: Proceedings of the 2020 Genetic and Evolutionary Computation ConferenceDominance resistant solutions (DRSs) in multi-objective problems have very good values for some objectives and very bad values for other objectives. Whereas DRSs are far away from the Pareto front, they are hardly dominated by other solutions due to ...
Comments