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Structure of Pareto solutions of generalized polyhedral-valued vector optimization problems in Banach spaces. (English) Zbl 1470.90119

Summary: In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.

MSC:

90C29 Multi-objective and goal programming
90C48 Programming in abstract spaces
90C31 Sensitivity, stability, parametric optimization
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[1] Armand, P., Finding all maximal efficient faces in multiobjective linear programming, Mathematical Programming, 61, 3, 357-375 (1993) · Zbl 0795.90054 · doi:10.1007/BF01582157
[2] Arrow, K. J.; Barankin, E. W.; Blackwell, D., Admissible points of convex sets, Contributions to the Theory of Games. Contributions to the Theory of Games, Annals of Mathematics Studies, no. 28, 87-92 (1953), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 0050.14203
[3] Benson, H. P.; Sun, E., Outcome space partition of the weight set in multiobjective linear programming, Journal of Optimization Theory and Applications, 105, 1, 17-36 (2000) · Zbl 1028.90051 · doi:10.1023/A:1004605810296
[4] Gass, S. I.; Roy, P. G., The compromise hypersphere for multiobjective linear programming, European Journal of Operational Research, 144, 3, 459-479 (2003) · Zbl 1028.90049 · doi:10.1016/S0377-2217(01)00388-5
[5] Hamacher, H. W.; Nickel, S., Multiobjecture planar location problems, European Journal of Operational Research, 94, 66-86 (1996) · Zbl 0929.90047
[6] Luc, D. T., Theory of Vector Optimization. Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319 (1989), Berlin, Germany: Springer, Berlin, Germany
[7] Pérez Gladish, B.; Arenas Parra, M.; Bilbao Terol, A.; Rodríguez Uría, M. V., Management of surgical waiting lists through a possibilistic linear multiobjective programming problem, Applied Mathematics and Computation, 167, 1, 477-495 (2005) · Zbl 1079.90574 · doi:10.1016/j.amc.2004.07.015
[8] Thuan, L. V.; Luc, D. T., On sensitivity in linear multiobjective programming, Journal of Optimization Theory and Applications, 107, 3, 615-626 (2000) · Zbl 1168.90615 · doi:10.1023/A:1026455401079
[9] Zeleny, M., Linear Multiobjective Programming. Linear Multiobjective Programming, Lecture Notes in Economics and Mathematical Systems, 95 (1974), Berlin, Germany: Springer, Berlin, Germany · Zbl 0325.90033
[10] Nickel, S.; Wiecek, M. M., Multiple objective programming with piecewise linear functions, Journal of Multi-Criteria Decision Analysis, 8, 322-332 (1999) · Zbl 0963.90053
[11] Zheng, X.; Yang, X., The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces, Science in China A, 51, 7, 1243-1256 (2008) · Zbl 1176.90562 · doi:10.1007/s11425-008-0021-3
[12] Alexandrov, A. D., Convex Polyhedra. Convex Polyhedra, Springer Monographs in Mathematics (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1067.52011
[13] Zheng, X. Y., Pareto solutions of polyhedral-valued vector optimization problems in Banach spaces, Set-Valued and Variational Analysis, 17, 4, 389-408 (2009) · Zbl 1177.49033 · doi:10.1007/s11228-009-0120-5
[14] Rockafellar, R. T., Convex Analysis. Convex Analysis, Princeton Mathematical Series, No. 28 (1970), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 0193.18401
[15] Fang, Y. P.; Meng, K.; Yang, X. Q., Piecewise linear multicriteria programs: the continuous case and its discontinuous generalization, Operations Research, 60, 2, 398-409 (2012) · Zbl 1274.90345 · doi:10.1287/opre.1110.1014
[16] Gong, X. H., Connectedness of the efficient solution set of a convex vector optimization in normed spaces, Nonlinear Analysis: Theory, Methods & Applications, 23, 9, 1105-1114 (1994) · Zbl 0822.90115 · doi:10.1016/0362-546X(94)90095-7
[17] Luc, D. T., Connectedness of the efficient point sets in quasiconcave vector maximization, Journal of Mathematical Analysis and Applications, 122, 2, 346-354 (1987) · Zbl 0615.90087 · doi:10.1016/0022-247X(87)90264-2
[18] Makarov, E. K.; Rachkovski, N. N., Efficient sets of convex compacta are arcwise connected, Journal of Optimization Theory and Applications, 110, 1, 159-172 (2001) · Zbl 1052.90080 · doi:10.1023/A:1017599614183
[19] Sun, E. J., On the connectedness of the efficient set for strictly quasiconvex vector minimization problems, Journal of Optimization Theory and Applications, 89, 2, 475-481 (1996) · Zbl 0851.90108 · doi:10.1007/BF02192541
[20] Zheng, X. Y., Contractibility and connectedness of efficient point sets, Journal of Optimization Theory and Applications, 104, 3, 717-737 (2000) · Zbl 0974.90023 · doi:10.1023/A:1004649928081
[21] Jahn, J., Vector Optimization: Theory, Applications, and Extensions (2004), Berlin, Germany: Springer, Berlin, Germany · Zbl 1055.90065
[22] Luc, D. T., Contractibility of efficient point sets in normed spaces, Nonlinear Analysis: Theory, Methods & Applications, 15, 6, 527-535 (1990) · Zbl 0718.90077 · doi:10.1016/0362-546X(90)90057-N
[23] Warburton, A. R., Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives, Journal of Optimization Theory and Applications, 40, 4, 537-557 (1983) · Zbl 0496.90073 · doi:10.1007/BF00933970
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