In this chapter we introduce the notion of convexity and generalized convexity including invexity for vector valued functions. Some characterizations of these functions are provided. Then we study vector problems involving generalized convex functions. The major aspects of this study concern the existence of efficient solutions, optimality conditions using contingent derivatives and approximate Jacobians, scalarization for convex and quasiconvex problems, and topological properties of efficient solution sets of generalized convex problems.
- Generalized convex vector functions
- efficient solutions
- vector optimization problems
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Arrow, K., Barankin, E. and Blackwell, D., Admissible points of convex sets, in Contributions to the Theory of Games, eds. Kuhn H. W. and Tucker F. H., Princeton University Press, Princeton, NJ., 1953, pp. 87–92.
Avriel, M., Diewert, W.E., Schaible, S. and Zang, I., Generalized Concavity, Plenum Publishing Corporation, New York, 1988.
Aubin, J.P. and Frankowska, H., Set-Valued Analysis, Birkhauser, Boston, 1990.
Benoist, J., Connectedness of the efficient set for strictly quasiconcave sets, J. Optimization Theory and Applications, 96 (1998), pp. 627–654.
Benoist, J., Contractibility of the efficient set in strictly quasiconcave vector maximization, J. Optimization Theory and Applications, 110 (2001), pp. 325–336.
Benoist, J. and Popovici, N., Contractibility of the efficient frontier of three-dimensional simply-shaded sets, J. Optimization Theory and Applications, 111 (2001), pp. 81–116.
Benoist, J., Borwein, J.M., and Popovici, N., A characterization of quasiconvex vector-valued functions, Proceedings of the American Mathematical Society 131 (2003), pp. 1109–1113.
Borwein, J.M., The geometry of Pareto efficiency over cones, Mathematische Operationsforschung und Statistik, Serie Optimization, 11 (1980), pp. 235–248.
Borwein, J.M., and Zhuang, D., Supper efficiency in vector optimization, Transactions of the American Mathematical Society, 339 (1993), pp. 105–122.
Cambini, A., and Martein, L., Generalized concavity and optimality conditions in vector and scalar optimization, in “Generalized Convexity”, eds. Komlosi S., Rapcsak T. and Schaiblez S., Lecture Notes on Economics and Mathematical Systems 405, Springer-Verlag, Berlin, 1994, pp. 337–357.
Cambini, A., Luc, D.T. and Martein, L., Order preserving transformations and applications, J. Optimization Theory and Applications 118 (2003), pp. 275–293.
Cambini, R., Some new classes of generalized convex vector valued functions, Optimization 36 (1996), pp. 11–24.
Cambini, R., and Komlosi, S., On the scalarization of pseudoconcavity and pseudomonotonicity concepts for vector valued functions, in “Generalized Convexity, Generalized Monotonicity” (Proceedings of the Fifth Symposium on Generalized Convexity, Luminy, June 1996), eds. Crouzeix J.P., Martinez-Legas J.E. and Volle M., Kluwer Academic Publishers, London, 1998, pp. 277–290.
Cambini, R., and Komlosi, S., On polar generalized monotonicity in vector optimization, Optimization 47 (2000), pp. 111–121.
Chen, G.Y. and Jahn, J., Optimality conditions for set-valued optimization problems, Mathematical Methods of Operations Research, 48 (1998), pp. 187–200.
Choo, E.U. and Atkins, D.R., Connectedness in multiple linearfractional programming, Management Science, 29 (1983), pp. 250–255.
Clarke, F.H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York 1983.
Choo, E.U., Schaible, S. and Chew, S., Connectedness of the efficient set in three-objective quasiconcave programming, Cahiers du Centre d’Etudes de Recherche Operationnelle 27 (1985), pp. 213–220.
Corley, H.W., An existence result for maximization with respect to cones, J. Optimization Theorey and Applications 31 (1980), 277–281.
Craven, B.D., Nonsmooth multiobjective programming, Numerical Functional Analysis and Optimization, 10 (1989), pp. 49–64.
Craven, B.D., Nonsmooth multiobjective programming, Numerical Functional Analysis and Optimization, 10 (1989), pp. 49–64.
Daniilidis, A., Arrow-Barankin-Blackwell theorems and related results in cone duality: a survey, Optimization (Namur, 1998), Lecture Notes in Economics and Mathematical Systems, 481, Springer Verlag, Berlin, 2000, pp. 119–131.
Daniilidis, A., Hadjisavvas, N. and Schaible, S., Connectedness of the efficient set for three-objective quasiconcave maximization Problems, J. Optimization Theory and Applications, 93 (1997), pp. 517–524.
Durousseau, C., Contribution à l’analyse convexe et à l’optimization vectorielle, PhD. Thesis, Faculté des Sciences, Université de Limoges, 1999.
Giannessi, F., Maestroeni, G. and Pellegrini, L., On the theory of vector optimization and variational inequalities. Image space analysis and separation, in Vector Variational Inequalities and Vector Equilibria, Ed. F. Giannessi, Kluwer Academic Publishers, London, pp. 153–216, 2000.
Gong, X.H., Connectedness of the efficient solution set of a convex vector optimization problem in a normed space, Nonlinear Analysis 23 (1994), pp. 1105–1114.
Guerraggio, A., Molho, E. and Zaffaroni, A., On the notion of proper efficiency in vector optimization, J. Optimization Theory and Applications, 82 (1994), pp. 1–21.
Ha, T.X.D., Existence and density results for proper efficiency in cone compact sets, Optimization, 111 (2001), pp. 173–194.
Hanson, M.A., On sufficiency of the Kuhn-Tucker conditions, J. Mathematical Analysis and Applications, 80 (1981), pp. 545–550.
Henig, M.I., Proper efficiency with respect to cones, J. Optimization Theory and Applications, 36 (1982), pp. 387–407.
Huy, N.Q., Phuong, T.D., and Yen, N.D., On the contractibility of the efficient and weakly efficient sets in R2, in “Equilibrium Problems and Variational Models”, Maugeri A. (Ed.), Kluwer, 2001.
Ioffe, A.D., Nonsmooth analysis: Differential calculus of nondifferentiable mappings, Transactions of the American Mathematical Society, 266 (1981), pp. 1–56.
Isac, G., Pareto optimization in infinite dimensional spaces: The importance of nuclear cones, J. Math. Anal. Appl., 182 (1994), 393–404.
Jahn, J., Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986.
Jahn, J., A generalization of a theorem of Arrow, Barankin and Blackwell, SIAM J. Control and Optimization, 26 (1988), pp. 999–1005.
Jeyakumar, V. and Luc, D.T., Approximate Jacobian matrices for nonsmooth continuous maps and C1-optimization, SIAM Journal on Control and Optimization 36 (1998), pp. 1815–1832.
Jeyakumar, V. and Luc, D.T., Nonsmooth calculus, minimality and monotonicity of convexificators, J. Optimization Theory and Applications, 101 (1999), pp. 599–621.
Jeyakumar, V. and Luc, D.T., Open mapping theorem using unbounded generalized Jacobians, Nonlinear Analysis, 50 (2002), pp. 647–663.
Jeyakumar, V., Luc, D.T. and Schaible, S., Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians, J. Convex Analysis, 5 (1998), pp. 119–132.
Luc, D.T., On scalarizing method in vector optimization, in Large-Scale Modelling and Interactive Decision Analysis, Proceedings, Eisenach 1985, Eds. Fandel, Grauer, Kurzahanski and Wiersbicki, LNEMS 273 (1986), pp. 46–51.
Luc, D.T., Structure of the efficient point set, Proc. Amer. Math. Soc. 95 (1985), pp. 433–440.
Luc, D.T., Theory of Vector Optimization, Lecture Notes on Economics and Mathematical Systems 319, Springer-Verlag, Germany, 1989.
Luc, D.T., On three concepts of quasiconvexity in vector optimization, Acta Math. Vietnamica, 15 (1990), pp. 3–9.
Luc, D.T., Continuity properties of cone-convex functions, Acta Math.Hungar., 55 (1990), pp. 57–61.
Luc, D.T., Contractibility of efficient point setrs in normed spaces, J. Nonlinear Analysis and Applications 15 (1990), pp.3–9.; and Corrigendum, idem., 38 (1999), pp. 547.
Luc, D.T., Contingent derivative of set-valued maps and applications to vector optimization, Mathematical Programming, 50 (1991), pp. 99–111.
Luc, D.T., On generalized convex nonsmooth functions, Bulletin of the Austral. Math.Soc., 49 (1994), pp. 139–149.
Luc, D.T., On the properly efficient points of nonconvex sets, European Journal of Operational Research, 86 (1995), pp. 332–336.
Luc, D.T., Generalized monotone maps and bifunctions, Acta Math. Vietnamica, 21 (1996), pp. 213–253.
Luc, D.T., A multiplier rule for multiobjective programming problems with continuous data, SIAM J. Optimization, 13 (2002), pp. 168–178.
Luc, D.T. and Malivert, C., Invex optimization problems, Bulletin of the Australian Mathematical Society, 46 (1992), pp. 47–66.
Luc, D.T., and Schaible, S., Efficiency and generalized concavity, J. Optimization Theory and Applications, 94 (1997), pp. 147–153.
Luc, D.T., and Swaminathan, S., A characterization of convex functions, J.Nonlinear Analysis: TMA, 20 (1993), pp. 697–701.
Luc, D.T., Tan, N.X., and Tinh, P.N., Convex vector functions and their subdifferential, Acta Mathematica Vietnamica, 23 (1998), pp. 107–127.
Luc, D.T., and Volle, M., Level sets, infimal convolution and level addition, J. Optimization Theory and Application, 94 (1997), pp. 695–714.
Makarov, E.K. and Rachkovski, N.N., Density theorems for generalized Henig proper efficiency, J. Optimization Theory and Applications, 91 (1996), pp. 419–437.
Makarov, E.K. and Rachkovski, N.N., Efficient sets of convex compacta are arcwise connected, J. Optimization Theory and Applications, 110 (2001), pp. 159–172.
Makarov, E.K., Rachkovski, N.N. and SONG, W., Arcwise connectedness of closed point sets, J. Mathematical Analysis and Applications, 247 (2000), pp. 377–383.
Peleg, B., Topological properties of the efficient point set, Proceedings Amer. Math. Soc, 35 (1972), pp. 531–536.
Penot, J.-P., Differentiability of relations and differential stability of perturbed optimization problems, SIAM J. Control and Optimization, 22 (1984), pp. 529–551.
Petschke, M., On a theorem of Arrow, Barankin and Blackwell, SIAM J. Control and Optimization, 28 (1990), pp. 395–401.
Popovici, N., Contribution à l’ optimization vectorielle, PhD. Thesis, Faculté des Sciences, Université de Limoges, 1995.
Postolica, V., New existence results for efficient points in locally convex spaces ordered by supernormal cones, J. Global Optimization, 3 (1993), pp. 233–243.
Sach, P.H., Yen, N.D. and Craven, B.D., Generalized invexity and duality theories with multifunctions, Numerical Functional Analysis and Optimization, 15 (1994), pp. 131–153.
Schaible, S., Bicriteria quasiconvex programs, Cahiers du Centre d’Etudes de Recherche Opérationnelle, 25 (1983), pp. 93–101.
Sterna-Karwat, A., On existence of cone-maximal points in real topological linear spaces, Israel J.Math. 54 (1986), pp. 33–41.
Sun, E.J., On the connectedness of the efficient set for strictly quasiconvex vector minimization problem, J. Optimization Theory and Applications, 89 (1996), pp. 475–581.
Tinh, P.N., Luc, D.T., and Tan, N.X., Subdifferential characterization of quasiconvex and convex vector functions, Vietnam J. Mathematics, 26 (1998), pp. 53–69.
Valadier, M., Sous-différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné, Math. Scand., 30 (1972), pp. 65–74.
Warga, J., Fat homeomorphisms and unbounded derivate containers, J. Mathematical Analysis and Applications, 81 (1981), pp. 545–560.
Yen, N.D. and Phuong, T.D., Connectedness and stability of the solution set in linear fractional vector optimization inequalities, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theorems, Giannessi F., Ed., Kluwer Academic Publishers, Dordrecht, 2000, pp. 479–489.
Zalinescu, C., and Sonntag, Y., Comparison of existence results for efficient points, J. Optimization Theory and Applications, 105 (2000), pp. 161–188.
Zowe, J., Subdifferentiability of convex functions with values in an ordered vector space, Math. Scandi., 34 (1974), pp. 69–83.
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Dinh The, L. (2005). Generalized Convexity in Vector Optimization. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds) Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and Its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/0-387-23393-8_5
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