Abstract
In this paper we address the problem of computing suitable representations of the set of approximate solutions of a given multi-objective optimization problem via stochastic search algorithms. For this, we will propose different archiving strategies for the selection of the candidate solutions maintained by the generation process of the stochastic search process, and investigate them further on analytically and empirically. For all archivers we will provide upper bounds on the approximation quality as well as on the cardinality of the limit solution set. We conclude this work by a comparative study on some test problems in order to visualize the effect of all novel archiving strategies.
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Al Moubayed, N., Petrovski, A., McCall, J.: D2MOPSO: MOPSO based on decomposition and dominance with archiving using crowding distance in objective and solution spaces. Evol. Comput. 22(1), 47–77 (2014)
Blanquero, R., Carrizosa, E.: A d.c. biobjective location model. J. Glob. Optim. 23(2), 569–580 (2002)
Bolintineanu, S.: Vector variational principles; \(\epsilon \)-efficiency and scalar stationarity. J. Convex Anal. 8, 71–85 (2001)
Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, Chichester (2001). ISBN 0-471-87339-X
Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)
Eiben, A.E., Rudolph, G.: Theory of evolutionary algorithms: a bird’s eye view. Theor. Comput. Sci. 1, 3–9 (1999)
Engau, A., Wiecek, M.M.: Generating epsilon-efficient solutions in multiobjective programming. Eur. J. Oper. Res. 177(3), 1566–1579 (2007)
Forster, O.: Analysis, vol. 3. Vieweg, Braunschweig (1984)
Hanne, T.: On the convergence of multiobjective evolutionary algorithms. Eur. J. Oper. Res. 117(3), 553–564 (1999)
Hanne, T.: Global multi-objective optimization using evolutionary algorithms. J. Heuristics 6(3), 347–360 (2000)
He, M., Xiong, F., Sun, J.: Multi-objective optimization of elastic beams for noise reduction. ASME. J. Vib. Acoust. 139(5), 051014 (2017). https://doi.org/10.1115/1.4036680
Hernández, C., Sun, J.Q., Schütze, O.: Computing the set of approximate solutions of a multi-objective optimization problem by means of cell mapping techniques. In: Emmerich, M., et al. (eds.) EVOLVE—A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation IV, pp. 171–188. Springer, Berlin (2013)
Hillermeier, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach, vol. 135. Springer, Berlin (2001)
Horoba, C., Neumann, F.: Benefits and drawbacks for the use of epsilon-dominance in evolutionary multi-objective optimization. In: Genetic and Evolutionary Computation Conference (GECCO-2008), pp. 641–648 (2008)
Knowles, J.D., Corne, D.W.: Approximating the non-dominated front using the Pareto archived evolution strategy. Evol. Comput. 8(2), 149–172 (2000)
Knowles, J.D., Corne, D.W.: Metaheuristics for multiobjective optimisation. Lecture Notes in Economics and Mathematical Systems, vol. 535, chapter Bounded Pareto Archiving: Theory and Practice, pp. 39–64. Springer (2004)
Lagunas-Jiménez, J.R., Moo-Yam, V., Ortíz-Moctezuma, B.: Two-degrees-of-freedom robust PID controllers tuning via a multiobjective genetic algorithm. Comput. Sist. 18(2), 259 (2014)
Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)
Laumanns, M., Zenklusen, R.: Stochastic convergence of random search methods to fixed size pareto front approximations. Eur. J. Oper. Res. 213(2), 414–421 (2011)
López-Ibáñez, M., Knowles, J.D., Laumanns, M.: On sequential online archiving of objective vectors. In: EMO, pp. 46–60. Springer (2011)
Loridan, P.: \(\epsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 42, 265–276 (1984)
Machin Navas, M., Nebro Urbaneja, A.J.: Multiobjective adaptive metaheuristics. Comput. Sist. 17(1), 53–62 (2013)
Novoa-Hernández, P.: Evolutionary multi-objective optimization for scheduling professor evaluations in Cuban higher education. Comput. Sist. 19(2), 321–335 (2015)
Okabe, T., Jin, Y., Olhofer, M., Sendhoff, B.: Parallel problem solving from nature—PPSN VIII. In: 8th International Conference, Birmingham, UK, 18–22 Sept 2004. Proceedings, chapter On Test Functions for Evolutionary Multi-objective Optimization, pp. 792–802 (2004)
Pareto, V.: Manual of Political Economy. The MacMillan Press, New York (1971). (original edition in French in 1927)
Pérez, N., Cuate, O., Schütze, O., Alvarado, A.: Including users preferences in the decision making for discrete many objective optimization problems. Comput. Sist. 20(4), 589–607 (2016)
Rudolph, G.: Finite Markov chain results in evolutionary computation: a tour d’horizon. Fundam. Inform. 35, 67–89 (1998)
Rudolph, G., Agapie, A.: Convergence properties of some multi-objective evolutionary algorithms. In: Proceedings of the 2000 Conference on Evolutionary Computation, vol. 2, pp. 1010–1016, Piscataway, New Jersey. IEEE Press (2000)
Rudolph, G., Naujoks, B., Preuss, M.: Capabilities of EMOA to detect and preserve equivalent pareto subsets. In: EMO ’03: Proceedings of the Evolutionary Multi-criterion Optimization Conference, pp. 36–50 (2006)
Ruhe, G., Fruhwirt, B.: \(\epsilon \)-optimality for bicriteria programs and its application to minimum cost flows. Computing 44, 21–34 (1990)
Ruiz, E.R., Segura, C.: Memetic algorithm with hungarian matching based crossover and diversity preservation. Comput. Sist. 22(2), 347–361 (2018)
Saha, A., Ray, T., Smith, W.: Towards practical evolutionary robust multi-objective optimization. In: 2011 IEEE Congress on Evolutionary Computation (CEC), pp. 2123–2130 (2011)
Schaeffler, S., Schultz, R., Weinzierl, K.: Stochastic method for the solution of unconstrained vector optimization problems. J. Optim. Theory Appl. 114(1), 209–222 (2002)
Schütze, O., Coello Coello, C.A., Talbi, E.-G.: Approximating the \(\epsilon \)-efficient set of an MOP with stochastic search algorithms. In: Gelbukh, A., Morales, A.F.K. (eds.) Mexican International Conference on Artificial Intelligence (MICAI-2007), pp. 128–138. Springer, Berlin (2007a)
Schütze, O., Laumanns, M., Tantar, E., Coello, C.A.C., Talbi, E.-G.: Convergence of stochastic search algorithms to gap-free Pareto front approximations. In: Genetic and Evolutionary Computation Conference (GECCO-2007), pp. 892–901 (2007b)
Schütze, O., Coello, C.A.C., Tantar, E., Talbi, E.-G.: Computing finite size representations of the set of approximate solutions of an MOP with stochastic search algorithms. In: GECCO ’08: Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, pp. 713–720, New York, NY, USA. ACM (2008a)
Schütze, O., Coello, C.A.C., Mostaghim, S., Talbi, E.-G., Dellnitz, M.: Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems. Eng. Optim. 40(5), 383–402 (2008b)
Schütze, O., Laumanns, M., Coello, C.A.C., Dellnitz, M., Talbi, E.-G.: Convergence of stochastic search algorithms to finite size Pareto set approximations. J. Glob. Optim. 41(4), 559–577 (2008c)
Schütze, O., Vasile, M., Coello, C.A.C.: Approximate solutions in space mission design. In: PPSN ’08: Proceedings of the 10th International Conference on Parallel Problem Solving From Nature, pp. 805–814 (2008d)
Schütze, O., Esquivel, X., Lara, A., Coello, C.A.C.: Using the averaged Hausdorff distance as a performance measure in evolutionary multi-objective optimization. IEEE Trans. Evol. Comput. 16(4), 504–522 (2012)
Schütze, O., Laumanns, M., Tantar, E., Coello, C.A.C., Talbi, E.-G.: Computing gap free Pareto front approximations with stochastic search algorithms. Evol. Comput. 18(1), 65–96 (2010)
Sun, J.Q.: Vibration and sound radiation of non-uniform beams. J. Sound Vib. 185(5), 827–843 (1995)
Tanaka, T.: A new approach to approximation of solutions in vector optimization problems. In: Fushini, M., Tone, K., (eds.) Proceedings of APORS 1994, pp. 497–504 (1995)
Tantar, A.A., Tantar, E., Schütze, O.: Asymmetric quadratic landscape approximation model. In: GECCO ’14: Proceedings of the 16th Annual Conference on Genetic and Evolutionary Computation, pp. 493–500 (2014)
Vargas, A., Bogoya, J.: A generalization of the averaged Hausdorff distance. Comput. Sist. 22(2), 331–345 (2018)
White, D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49(2), 319–337 (1986)
White, D.J.: Epsilon-dominating solutions in mean–variance portfolio analysis. Eur. J. Oper. Res. 105(3), 457–466 (1998)
Yuen, J., Gao, S., Wagner, M., Neumann, F.: An adaptive data structure for evolutionary multi-objective algorithms with unbounded archives. In: IEEE Congress on Evolutionary Computation (CEC), 2012 , pp. 1–8 (2012)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)
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Schütze, O., Hernández, C., Talbi, EG. et al. Archivers for the representation of the set of approximate solutions for MOPs. J Heuristics 25, 71–105 (2019). https://doi.org/10.1007/s10732-018-9383-z
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DOI: https://doi.org/10.1007/s10732-018-9383-z