zbMATH — the first resource for mathematics

Structural properties of Pareto fronts: the occurrence of dents in classical and parametric multiobjective optimization problems. (English) Zbl 07271603
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 (ISBN 978-3-030-51263-7/hbk; 978-3-030-51264-4/ebook). Studies in Systems, Decision and Control 304, 315-336 (2020).
Summary: This contribution deals with the occurrence of “dents” in Pareto fronts of continuous and adequately smooth multiobjective optimization problems. After giving a formal definition of this notion, a system of equations is derived that characterizes points on the boundary of the dent. This can be used to obtain information about the structure of the Pareto front without computing the entire Pareto set. Furthermore, the evolution of dents in parametric multiobjective optimization problems is studied using results from bifurcation theory. Theory and computations are illustrated by several examples, whose construction is described as well.
For the entire collection see [Zbl 1445.37003].
65K Numerical methods for mathematical programming, optimization and variational techniques
90 Operations research, mathematical programming
AUTO2000; HomCont; NBI
Full Text: DOI
[1] Bigi, G., Castellani, M.: Uniqueness of KKT multipliers in multiobjective optimization. Appl. Math. Lett. 17(11), 1285-1290 (2004) · Zbl 1087.90067
[2] Coello Coello, C.A., Lamont, G., Veldhuizen, D.V.: Evolutionary Algorithms for Solving Multi-objective Optimization Problems, 2nd edn. Springer, Berlin (2007) · Zbl 1142.90029
[3] Das, I., Dennis, J.: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optim. 14(1), 63-69 (1997)
[4] 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, 631-657 (1998) · Zbl 0911.90287
[5] Deb, K.: Multi-objective Optimization using Evolutionary Algorithms. Wiley-Interscience Series in Systems and Optimization. John Wiley, Chichester (2001)
[6] Dellnitz, M., Schütze, O., Hestermeyer, T.: Covering Pareto sets by multilevel subdivision techniques. J. Optim. Theory Appl. 124(1), 113-136 (2005) · Zbl 1137.90015
[7] Doedel, E.J., Champneys, A.R., Paffenroth, R.C., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B.E., Sandstede, B., Wang, X.J.: AUTO2000: Continuation and bifurcation software for ordinary differential equations (with homcont). Technical report California Institute of Technology, Pasadena, California USA (2000)
[8] Ehrgott, M.: Multicriteria Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 491. Springer, Berlin (2000) · Zbl 0956.90039
[9] Ehrgott, M.: Multicriteria Optimization, second edn. Springer, Berlin (2005) · Zbl 1132.90001
[10] Ehrgott, M., Gandibleaux, X. (eds.): Multiple Criteria Optimization, International Series in Operations Research & Management Science, vol. 52. Kluwer Academic Publishers, Boston (2002)
[11] Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L.: Evolutionary multi-criterion optimization. In: Second International Conference EMO 2003. Springer (2003)
[12] Gass, S., Saaty, T.: The computational algorithm for the parametric objective function. Naval Res. Logist. Q. 2, 39-45 (1955)
[13] Golubitsky, M., Schaeffer, D.G.: Singularities and groups in bifurcation theory. Vol. I, Applied Mathematical Sciences, vol. 51. Springer, New York (1985) · Zbl 0607.35004
[14] Göpfert, A., Nehse, R.: Vektoroptimierung. Teubner Verlagsgesellschaft Leipzig (1990)
[15] Guddat, J., Guerra Vasquez, F., Tammer, K., Wendler, K.: Multiobjective and Stochastic Optimization Based on Parametric Optimization. Akademie-Verlag, Berlin (1985) · Zbl 0583.90055
[16] Hillermeier, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach. Birkhäuser (2001) · Zbl 0966.90069
[17] Kim, B., Gel, E., Fowler, J., Carlyle, W., Wallenius, J.: Evaluation of nondominated solution sets for k-objective optimization problems: an exact method and approximations. Eur. J. Oper. Res. 173(2), 565-582 (2006) · Zbl 1125.90044
[18] Kuhn, H., Tucker, A.: Nonlinear programming. In: Neumann, J. (ed.) Proceedings of 2nd Berkeley Symposium of Mathematical Statistics and Probability, pp. 481-492 (1951)
[19] Martín, A., Schütze, O.: Pareto tracer: a predictor-corrector method for multi-objective optimization problems. Eng. Optim. 50(3), 516-536 (2018). https://doi.org/10.1080/0305215X.2017.1327579
[20] Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Berlin (1999) · Zbl 0949.90082
[21] Moore, G., Spence, A.: The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17(4), 567-576 (1980) · Zbl 0454.65042
[22] Schäffler, S., Schultz, R., Weinzierl, K.: A stochastic method for the solution of unconstrained vector optimization problems. J. Optim. Theory Appl. 114(1), 209-222 (2002) · Zbl 1022.90027
[23] Schütze, O., Cuate, O., Martín, A., Peitz, S., Dellnitz, M.: Pareto explorer: a global/local exploration tool for many-objective optimization problems. Eng. Optim. 52, 1-24 (2019). https://doi.org/10.1080/0305215X.2019.1617286
[24] Schütze, O., Witting, K., Ober-Blöbaum, S., Dellnitz, M.: Set oriented methods for the numerical treatment of multiobjective optimization problems. In: Tantar, E., Tantar, A.A., Bouvry, P., Moral, P.D., Legrand, P., Coello Coello, C.A., Schütze, O. (eds.) EVOLVE - A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation, pp. 187-219. Springer (2013)
[25] Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Application. Wiley Series in Probability & Mathematical Statistics, John Wiley Inc. (1986) · Zbl 0663.90085
[26] Werner, B., Spence, A.: The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21(2), 388-399 (1984) · Zbl 0554.65045
[27] Werner, D.: Funktionalanalysis. Springer, Heidelberg (2005)
[28] Witting, K.: Numerical algorithms for the treatment of parametric multiobjective optimization problems and applications. Dissertation, Universität Paderborn (2012). http://digital.ub.uni-paderborn.de/urn/urn:nbn:de:hbz:466:2-8617
[29] Zadeh, L.: Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control 8, 59-60 (1963)
[30] Zitzler, E.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.