It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.
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Trying to answer a question of one of the anonymous referees, we realized that the first equivalence in Proposition 1 is known for a long time.
Bao TQ, Tammer C (2012) Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications. Nonlinear Anal 75(3):1089–1103
Bao TQ, Mordukhovich BS (2007a) Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints. Appl Math 52(6):453–472
Bao TQ, Mordukhovich BS (2007b) Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern 36(3):531–562
Bao TQ, Mordukhovich BS (2010) Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math Program 122(2, Ser. A):301–347
Borwein JM (1982) Continuity and differentiability properties of convex operators. Proc Lond Math Soc (3) 44(3):420–444
Durea M, Strugariu R, Tammer C (2013) Scalarization in geometric and functional vector optimization revisited. J Optim Theory Appl 159(3):635–655
Durea M, Tammer C (2009) Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58(4):449–467
Dutta J, Tammer C (2006) Lagrangian conditions for vector optimization in Banach spaces. Math Methods Oper Res 64(3):521–540
Gerth C, Weidner P (1990) Nonconvex separation theorems and some applications in vector optimization. J Optim Theory Appl 67(2):297–320
Göpfert A, Riahi H, Tammer C, Zălinescu C (2003) Variational methods in partially ordered spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 17. Springer, New York
Ha TXD (2010) Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Nonlinear analysis and variational problems, volume 35 of Springer Optim Appl, pp 305–324. Springer, New York
Ha TXD (2012) Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Nonlinear Anal 75(3):1305–1323
Hiriart-Urruty J-B (1979) Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math Oper Res 4(1):79–97
Johannes J (2004) Vector optimization: theory, applications, and extensions. Springer, Berlin
Jameson G (1970) Ordered linear spaces, Lecture Notes in Mathematics, vol 141. Springer, Berlin-New York
Khan AA, Tammer C, Zălinescu C (2015) Set-valued optimization. Vector optimization. Springer, Heidelberg An introduction with applications
Luc DT, Tan NX, Tinh PN (1998) Convex vector functions and their subdifferential. Acta Math Vietnam 23(1):107–127
Luc DT (1989) Theory of vector optimization, volume 319 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin
Minh NB, Tan NX (2002a) On the \(C\)-approximations of multivalued mappings. Vietnam J Math 30(4):343–363
Minh NB, Tan NX (2002b) On the continuity of vector convex multivalued functions. Acta Math Vietnam 27(1):13–25
Mordukhovich BS (2006a) Variational analysis and generalized differentiation. I, volume 330 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Basic theory. Springer, Berlin
Mordukhovich BS (2006b) Variational analysis and generalized differentiation. II, volume 331 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Application. Springer, Berlin
Papageorgiou NS (1983) Nonsmooth analysis on partially ordered vector spaces. I. Convex case. Pac J Math 107(2):403–458
Phelps RR (1989) Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol 1364. Springer, Berlin
Reiland TW (1992) Nonsmooth analysis and optimization on partially ordered vector spaces. Int J Math Math Sci 15(1):65–81
Roberts AW, Varberg DE (1974) Another proof that convex functions are locally Lipschitz. Am Math Mon 81:1014–1016
Rudin W (1973) Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co, New York
Schaefer HH (1971) Topological vector spaces. Third printing corrected, Graduate Texts in Mathematics, vol 3. Springer, New York
Thibault L (1980) Subdifferentials of compactly Lipschitzian vector-valued functions. Ann Mat Pura Appl 4(125):157–192
Tyrrell Rockafellar R (1970) Convex analysis, Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ
Valadier M (1972) Sous-différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné. Math Scand 30:65–74
Zaffaroni A (2003) Degrees of efficiency and degrees of minimality. SIAM J Control Optim 42(3):1071–1086
Zălinescu C (2002) Convex analysis in general vector spaces. World Scientific, River Edge
The authors thank the anonymous referees for their very helpful comments. C. Zălinescu’s research was supported by the grant PN-II-ID-PCE-2011-3-0084, CNCS-UEFISCDI, Romania.
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Anh Tuan, V., Tammer, C. & Zălinescu, C. The Lipschitzianity of convex vector and set-valued functions. TOP 24, 273–299 (2016). https://doi.org/10.1007/s11750-015-0401-0