×

A Minty variational principle for set optimization. (English) Zbl 1301.49022

Summary: Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty-type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient”. A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for the proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert’s theorem to improper functions.

MSC:

49J40 Variational inequalities
49J53 Set-valued and variational analysis
49J52 Nonsmooth analysis

Software:

BENSOLVE
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alonso, M.; Rodríguez-Marín, L., Set-relations and optimality conditions in set-valued maps, Nonlinear Anal., 63, 8, 1167-1179 (2005) · Zbl 1091.90090
[2] Aubin, J. P.; Frankowska, H., Set-Valued Analysis, Systems Control Found. Appl., vol. 2 (1990), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA
[3] Benoist, J.; Borwein, J. M.; Popovici, N., A characterization of quasiconvex vector-valued functions, Proc. Amer. Math. Soc., 131, 4, 1109-1114 (2003) · Zbl 1024.26020
[4] Benoist, J.; Popovici, N., Characterizations of convex and quasiconvex set-valued maps, Math. Methods Oper. Res., 57, 3, 427-435 (2003) · Zbl 1047.54012
[5] Birkhoff, G., Lattice Theory (1940), AMS Colloquium Publications: AMS Colloquium Publications Providence, RI · Zbl 0126.03801
[6] Borwein, J. M., Multivalued convexity and optimization: a unified approach to inequality and equality constraints, Math. Program., 13, 2, 183-199 (1977) · Zbl 0375.90062
[7] Brink, C., Power structures, Algebra Universalis, 30, 2, 177-216 (1993) · Zbl 0787.08001
[8] Cambini, A.; Martein, L., Generalized Convexity and Optimization: Theory and Applications (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1157.49001
[9] Corley, H. W., Existence and Lagrangian duality for maximizations of set-valued functions, J. Optim. Theory Appl., 54, 3, 489-501 (1987) · Zbl 0595.90085
[10] Corley, H. W., Optimality conditions for maximizations of set-valued functions, J. Optim. Theory Appl., 58, 1, 1-10 (1988) · Zbl 0956.90509
[11] Crespi, G. P.; Ginchev, I.; Rocca, M., Minty variational inequalities, increase-along-rays property and optimization, J. Optim. Theory Appl., 123, 3, 479-496 (2004) · Zbl 1059.49010
[12] Crespi, G. P.; Ginchev, I.; Rocca, M., First-order optimality conditions in set-valued optimization, Math. Methods Oper. Res., 63, 1, 87-106 (2006) · Zbl 1103.90089
[13] Crespi, G. P.; Ginchev, I.; Rocca, M., Minty variational principle for set-valued variational inequalities, Pac. J. Optim., 6, 1, 39-56 (2010) · Zbl 1190.49023
[14] Crespi, G. P.; Schrage, C., Set optimization meets variational inequalities, in: Springer Proceedings ‘Set Optimization, State of Art’, forthcoming · Zbl 1337.49023
[15] Crouzeix, J. P., Continuity and differentiability of quasiconvex functions, (Hadjisavvas, N.; Komlósi, S.; Schaible, S., Handbook of Generalized Convexity and Generalized Monotonicity (2005), Springer-Verlag: Springer-Verlag Berlin), 121-149 · Zbl 1077.49015
[16] Dedekind, R., Supplement XI von Dirichlets Vorlesungen über Zahlentheorie, (Fricke, R.; Noether, E.; Ore, Ö., Richard Dedekind Gesammelte Mathematische Werke (1932), Druck und Verlag von Friedr. Vieweg & Sohn Akt.-Ges.: Druck und Verlag von Friedr. Vieweg & Sohn Akt.-Ges. Braunschweig) (1894), (1863)
[17] Dedekind, R., Stetigkeit und irrationale Zahlen, (Fricke, R.; Noether, E.; Ore, Ö., Richard Dedekind Gesammelte Mathematische Werke (1932), Druck und Verlag von F. Vieweg & Sohn Akt.-Ges.: Druck und Verlag von F. Vieweg & Sohn Akt.-Ges. Braunschweig) (1927), (1872) · JFM 53.0182.06
[18] Demyanov, V. F.; Lemaréchal, C.; Zowe, J., Approximation to a set-valued mapping, I: a proposal, Appl. Math. Optim., 14, 1, 203-214 (1986) · Zbl 0619.49005
[19] Diewert, W. E., Alternative characterizations of six kinds of quasiconvexity in the nondifferentiable case with applications to nonsmooth programming, (Schaible, S.; Ziemba, W. T., Generalized Concavity in Optimization and Economics (1981), Academic Press: Academic Press New York), 51-93 · Zbl 0539.90088
[20] Fuchs, L., Teilweise geordnete algebraische Strukturen (1966), Vandenhoeck u. Ruprecht: Vandenhoeck u. Ruprecht Göttingen · Zbl 0154.00708
[21] Galatos, N., Residuated Lattices: An Algebraic Glimpse at Substructural Logics (2007), Elsevier Science Ltd · Zbl 1171.03001
[22] Getan, J.; Martinez-Legaz, J. E.; Singer, I., \((\ast, s)\)-dualities, J. Math. Sci., 115, 4, 2506-2541 (2003) · Zbl 1136.49315
[23] Ginchev, I.; Ivanov, V. I., Higher-order pseudoconvex functions, (Konnov, I. V.; Rubinov, A. M., Generalized Convexity and Related Topics. Generalized Convexity and Related Topics, Lecture Notes in Econom. and Math. Systems, vol. 583 (2007), Springer-Verlag: Springer-Verlag Berlin), 247-264 · Zbl 1123.26013
[24] Göpfert, A.; Riahi, H.; Tammer, C.; Zălinescu, C., Variational Methods in Partially Ordered Spaces, CMS Books Math., vol. 17 (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1140.90007
[25] Hamel, A. H., Variational principles on metric and uniform spaces (2005), Halle, Habilitationsschrift
[26] Hamel, A. H., A duality theory for set-valued functions I: Fenchel conjugation theory, Set-Valued Var. Anal., 17, 2, 153-182 (2009) · Zbl 1168.49031
[27] Hamel, A. H.; Schrage, C., Notes about extended real-and set-valued functions, J. Convex Anal., 19, 2 (2012) · Zbl 1244.49065
[28] Hamel, A. H.; Schrage, C., Directional derivatives and subdifferentials of set-valued convex functions (2012)
[29] Hernández, E.; Rodríguez-Marín, L., Lagrangian duality in set-valued optimization, J. Optim. Theory Appl., 134, 1, 119-134 (2007) · Zbl 1129.49029
[30] Heyde, F.; Löhne, A., Solution concepts in vector optimization: a fresh look at an old story, Optimization, 60, 12, 1421-1440 (2011) · Zbl 1258.90064
[31] Heyde, F.; Schrage, C., Continuity of set-valued maps and a fundamental duality formula for set-valued optimization, J. Math. Anal. Appl., 397, 2, 772-784 (2013) · Zbl 1261.46070
[32] Ivanov, V. I., First order characterizations of pseudoconvex functions, Serdica Math. J., 27, 6, 203-218 (2011) · Zbl 0982.26009
[33] Jahn, J., Vector Optimization: Theory, Applications, and Extensions (2010), Springer-Verlag: Springer-Verlag Berlin-Heidelberg
[34] Jahn, J.; Rauh, R., Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 25, 1-9 (1997) · Zbl 0889.90123
[35] Jahn, J.; Truong, X. D.H., New order relations in set optimization, J. Optim. Theory Appl., 148, 2, 209-236 (2011) · Zbl 1226.90092
[36] Karamardian, S., Duality in mathematical programming, J. Math. Anal. Appl., 20, 344-358 (1967) · Zbl 0157.49603
[37] Khan, A. A.; Raciti, F., A multiplier rule in set-valued optimization, Bull. Aust. Math. Soc., 68, 1, 93-100 (2003) · Zbl 1168.90574
[38] Kuroiwa, D., Convexity for set-valued maps, Appl. Math. Lett., 9, 2, 97-101 (1996) · Zbl 0856.26015
[39] Kuroiwa, D., Some criteria in set-valued optimization, Investigations on Nonlinear Analysis and Convex Analysis. Investigations on Nonlinear Analysis and Convex Analysis, Kyoto. Investigations on Nonlinear Analysis and Convex Analysis. Investigations on Nonlinear Analysis and Convex Analysis, Kyoto, RIMS Kokyuroku, 985, 71-176 (1997), (in Japanese) · Zbl 0925.90335
[40] Kuroiwa, D., The natural criteria in set-valued optimization, RIMS Kokyuroku, 1031, 85-90 (1998) · Zbl 0938.90504
[41] Kuroiwa, D., On natural criteria in set-valued optimization, RIMS Kokyuroku, 1048, 86-92 (1998) · Zbl 0954.90046
[42] Kuroiwa, D.; Tanaka, T.; Truong, X. D.H., On cone of convexity of set-valued maps, Nonlinear Anal., 30, 3, 1487-1496 (1997) · Zbl 0895.26010
[43] Löhne, A., Vector Optimization with Infimum and Supremum (2011), Springer-Verlag: Springer-Verlag Berlin · Zbl 1230.90002
[44] Luc, Dinh The, Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems, vol. 318 (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0688.90051
[45] Minty, G. J., On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73, 314-321 (1967) · Zbl 0157.19103
[46] Mordukhovich, B. S., Variational Analysis and Generalized Differentiation. I, Grundlehren Math. Wiss., vol. 330 (2006), Springer-Verlag: Springer-Verlag Berlin
[47] Penot, J. P., The directional subdifferential of the difference of two convex functions, J. Global Optim., 49, 3, 505-519 (2011) · Zbl 1223.26030
[48] Ponstein, J., Seven kinds of convexity, SIAM Rev., 9, 1, 115-119 (1967) · Zbl 0164.06501
[49] Rockafellar, R. T.; Wets, R. J.B., Variational Analysis, Grundlehren Math. Wiss., vol. 317 (1998), Springer-Verlag: Springer-Verlag Berlin-Heidelberg · Zbl 0888.49001
[50] Rodríguez-Marín, L.; Sama, M., Epidifferentiability and hypodifferentiability of pseudoconvex maps in set-optimization problems, Nonlinear Anal., 71, 1, 321-331 (2009) · Zbl 1163.49018
[51] Schrage, C., Scalar representation and conjugation of set-valued functions, Optimization (2012), published online
[52] Yang, X. Q., Directional derivatives for set-valued mappings and applications, Math. Methods Oper. Res., 48, 273-285 (1998) · Zbl 0929.90077
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.