An image set-oriented method for the numerical treatment of bi-level multi-objective optimization problems. (English) Zbl 1477.65092

Junge, Oliver (ed.) et al., Advances in dynamics, optimization and computation. A volume dedicated to Michael Dellnitz on the occasion of his 60th birthday. Cham: Springer. Stud. Syst. Decis. Control 304, 337-354 (2020).
Summary: In this chapter, we consider equality constrained bi-level multi-objective optimization problems, where the lower level problem is convex. Based on a suitable reformulation of the Kuhn-Tucker equations, we present an image set-oriented algorithm of reference point type for the approximation of the solution set, the Pareto set respectively its image, the Pareto front, of such a problem. The algorithm is designed such that the generated representation of the Pareto front is well-distributed with respect to the higher level image space. We first prove convergence for this algorithm and further on indicate its efficiency on two academic test problems.
For the entire collection see [Zbl 1445.37003].


65K05 Numerical mathematical programming methods
90C29 Multi-objective and goal programming
90C30 Nonlinear programming
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory


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[1] Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Berlin (1998) · Zbl 0943.90078
[2] Bogoya, J.M., Vargas, A., Schütze, O.: The averaged Hausdorff distances in multi-objective optimization: a review. Mathematics 7(10), 894 (2019)
[3] Coello Coello, C.A., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-Objective Problems, 2nd edn. Springer, New York (2007). ISBN 978-0-387-33254-3 · Zbl 1142.90029
[4] Colson, B., Marcotte, P., Savard, G.: Bilevel programming: a survey. Q. J. Oper. Res. 3, 87-107 (2005) · Zbl 1134.90482
[5] Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631-657 (1998) · Zbl 0911.90287
[6] Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Hoboken (2001) · Zbl 0970.90091
[7] Deb, K., Sinha, A.: Solving bilevel multi-objective optimization problems using evolutionary algorithms. In: Ehrgott, M., et al. (eds.) Evolutionary Multi-Criterion Optimization (2009)
[8] Dell’Aere, A.: Multi-objective optimization in self-optimizing systems. In: Proceedings of the 32nd Annual Conference of the IEEE Industrial Electronics Society (2006)
[9] Dell’Aere, A.: Numerical methods for the solution of bi-level multi-objective optimization problems. Ph.D. thesis, University of Paderborn (2008)
[10] Dellnitz, M., Schütze, O., Hestermeyer, T.: Covering Pareto sets by multilevel subdivision techniques. J. Optim. Theory Appl. 124(1), 113-155 (2005) · Zbl 1137.90015
[11] Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Berlin (2002) · Zbl 1038.90097
[12] Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333-359 (2003) · Zbl 1140.90493
[13] Ehrgott, M.: Multicriteria Optimization. Lecture Notes in Economics and Mathematical Systems (2000) · Zbl 0956.90039
[14] Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Heidelberg (2008) · Zbl 1145.90001
[15] Eichfelder, G.: Multiobjective bilevel optimization. Math. Program. 123, 419-449 (2010) · Zbl 1198.90347
[16] Figueira, J., Greco, S., Ehrgott, M.: Multiple Criteria Decision Analysis. Lecture Notes in Economics and Mathematical Systems. Springer (2005) · Zbl 1060.90002
[17] Fliege, J.: Gap-free computation of Pareto-points by quadratic scalarizations. Math. Methods Oper. Res. 59, 69-89 (2004) · Zbl 1131.90054
[18] Fliege, J., Fux Svaiter, B.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479-494 (2000) · Zbl 1054.90067
[19] Gass, S., Saaty, T.: The computational algorithm for the parametric objective function. Naval Res. Logist. Q. 2(1), 39-45 (1955)
[20] Gembicki, F.W., Haimes, Y.Y.: Approach to performance and multiobjective sensivity optimization: the goal attainment method. IEEE Trans. Autom. Control 20, 769-771 (1975)
[21] Hernández, C., Naranjani, Y., Sardahi, Y., Liang, W., Schütze, O., Sun, J.-Q.: Simple cell mapping method for multi-objective optimal feedback control design. Int. J. Dyn. Control 1(3), 231-238 (2013)
[22] Hillermeier, C.: Nonlinear Multiobjective Optimization - A Generalized Homotopy Approach. Birkhäuser, Basel (2001) · Zbl 0966.90069
[23] Jahn, J.: Multiobjective search algorithm with subdivision technique. Comput. Optim. Appl. 35(2), 161-175 (2006) · Zbl 1153.90533
[24] Klamroth, K., Tind, J., Wiecek, M.: Unbiased approximation in multicriteria optimization. Math. Methods Oper. Res. 56, 413-437 (2002) · Zbl 1064.90042
[25] Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, Berkeley and Los Angeles, pp. 481-492. University of California Press (1951)
[26] Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) · Zbl 0898.90006
[27] Martín, A., Schütze, O.: Pareto tracer: a predictor corrector method for multi-objective optimization problems. Eng. Optim. 50(3), 516-536 (2018)
[28] Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: Certified parallelotope continuation for one-manifolds. SIAM J. Numer. Anal. 51(6), 3373-3401 (2013) · Zbl 1298.65088
[29] Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach. J. Global Optim., 1-14 (2014) · Zbl 1332.90251
[30] Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Berlin (1999) · Zbl 0949.90082
[31] Naranjani, Y., Hernández, C., Xiong, F.-R., Schütze, O., Sun, J.-Q.: A hybrid algorithm for the simple cell mapping method in multi-objective optimization. In: Emmerich, M., et al. (eds.) EVOLVE-A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV. Advances in Intelligent Systems and Computing, pp. 207-223. Springer, Heidelberg (2013) · Zbl 1322.90088
[32] Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499-524 (1984) · Zbl 0505.90072
[33] Peitz, S., Dellnitz, M.: A survey of recent trends in multiobjective optimal control surrogate models, feedback control and objective reduction. Math. Comput. Appl. 23(2) (2018)
[34] Pereyra, V., Saunders, M., Castillo, J.: Equispaced Pareto front construction for constrained bi-objective optimization. Math. Comput. Model. 57(9-10), 2122-2131 (2013) · Zbl 1286.90138
[35] Qin, Z.-C., Xiong, F.-R., Ding, Q., Hernandez, C., Fernandez, J., Schütze, O., Sun, J.-Q.: Multi-objective optimal design of sliding mode control with parallel simple cell mapping method. J. Vib. Control (2015)
[36] Recchioni, M.C.: A path following method for box-constrained multiobjective optimization with applications to goal programming problems. Math. Methods Oper. Res. 58, 69-85 (2003) · Zbl 1059.90127
[37] Ringkamp, M., Ober-Blöbaum, S., Dellnitz, M., Schütze, O.: Handling high dimensional problems with multi-objective continuation methods via successive approximation of the tangent space. Eng. Optim. 44(6) (2012)
[38] Roy, B.: Problems and methods with multiple objective functions. Math. Program. 1, 239-266 (1971) · Zbl 0254.90061
[39] Schandl, B., Klamroth, K., Wiecek, M.M.: Introducing oblique norms into multiple criteria programming. J. Global Optim. 23, 925-942 (2002) · Zbl 0996.90067
[40] Schütze, O., Dell’Aere, A., Dellnitz, M.: On continuation methods for the numerical treatment of multi-objective optimization problems. In: Branke, J., Deb, K., Miettinen, K., Steuer, R.E. (eds.) Practical Approaches to Multi-Objective Optimization, number 04461 in Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany (2005). http://drops.dagstuhl.de/opus/volltexte/2005/349
[41] Schütze, O., Vasile, M., Junge, O., Dellnitz, M., Izzo, D.: Designing optimal low thrust gravity assist trajectories using space pruning and a multi-objective approach. Eng. Optim. 41(2), 155-181 (2009)
[42] Schütze, O., Witting, K., Ober-Blöbaum, S., Dellnitz, M.: Set oriented methods for the numerical treatment of multiobjective optimization problems. In: Tantar, E., et al. (eds.) EVOLVE - A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation, pp. 187-219. Springer, Heidelberg (2013) · Zbl 1251.90356
[43] Schütze, O., Cuate, O., Martín, A., Peitz, S., Dellnitz, M.: Pareto explorer: a global/local exploration tool for many-objective optimization problems. Eng. Optim. (2019)
[44] Sinha, A., Deb, K.: Bilevel multi-objective optimization and decision making. In: Talbi, E.G. (ed.) Metaheuristics for Bi-Level Optimization. Springer, Heidelberg (2013)
[45] Sun, J.-Q., Jia, T., Xiong, F.-R., Qin, Z.-C., Wu, W., Ding, Q.: Aircraft landing gear control with multi-objective optimization using generalized cell mapping. Trans. Tianjin Univ. 21(2), 140-146 (2015)
[46] Sun, J.-Q., Xiong, F.-R., Schütze, O., Hernández, C.: Cell Mapping Methods - Algorithmic Approaches and Applications. Springer, Heidelberg (2007) · Zbl 1415.37005
[47] Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5, 291-306 (1994) · Zbl 0822.90127
[48] Xiong, F.-R., Qin, Z.-C., Hernández, C., Sardahi, Y., Narajani, Y., Liang, W., Xue, Y., Schütze, O., Sun, J.-Q.: A multi-objective optimal pid control for a nonlinear system with time delay. Theoret. Appl. Mech. Lett 3(6), 140-146 (2013)
[49] Xiong, F. · Zbl 1457.93038
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