On Ginvex multiobjective programming. Part II. Duality
 Tadeusz Antczak
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This paper represents the second part of a study concerning the socalled Gmultiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On Ginvex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely Ginvexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The socalled GMond–Weir, GWolfe and Gmixed dual vector problems to the primal one are defined. Furthermore, various socalled Gduality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector Gdual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.
 Antczak T. (2004). (p, r)invexity in multiobjective programming. Eur. J. Oper. Res. 152: 72–87 CrossRef
 Antczak T. (2005). The notion of V–rinvexity in differentiable multiobjective programming. J. Appl. Anal. 11: 63–79 CrossRef
 Antczak T. (2007). New optimality conditions and duality results of Gtype in differentiable mathematical programming. Nonlinear 66: 1617–1632 CrossRef
 Antczak, T.: On Ginvex multiobjective programming. Part I. Optimality. J. Glob. Optim. (to be published)
 Bector, C.R., Bector, M.K., Gill, A., Singh, C.: Duality for vector valued Binvex programming, In: Proceedings Fourth International Workshop, Pecz, Hungary, pp. 358–373. Springer Verlag, Berlin (1994)
 Brumelle S. (1981). Duality for multiple objective convex programs. Math. Oper. Res. 6: 159–172 CrossRef
 Craven B.D. (1981). Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24: 357–366
 Craven B.D. and Glover B.M. (1985). Invex functions and duality. J. Aust. Math. Soc. Ser. A 39: 1–20 CrossRef
 Egudo R.R. and Hanson M.A. (1987). Multiobjective duality with invexity. J. Math. Anal. Appl. 126: 469–477 CrossRef
 Egudo R.R. (1989). Efficiency and generalized convex duality for multiobjective programs. J. Math. Anal. Appl. 138: 84–94 CrossRef
 Giorgi, G., Guerraggio, A.: The notion of invexity in vector optimization: smooth and nonsmooth case. In: Crouzeix, J.P., et al. (eds.) Generalized Convexity, Generalized Monotonicity. Kluwer Academic Publishers (1998)
 Hanson M.A. (1981). On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80: 545–550 CrossRef
 Jahn J. (1983). Duality in vector optimization. Math. Program. 25: 343–353 CrossRef
 Jeyakumar V. and Mond B. (1992). On generalized convex mathematical programming. J. Aust. Math. Soc. Ser. B 34: 43–53
 Kaul R.N., Suneja S.K. and Srivastava M.K. (1994). Optimality criteria and duality in multiple objective optimization involving generalized invexity. J. Optim. Theor. Appl. 80: 465–482 CrossRef
 Lin J.G. (1976). Maximal vectors and multiobjective optimization. J. Optim. Theor. Appl. 18: 41–64 CrossRef
 Li Z. (1993). Duality theorems for a class of generalized convex multiobjective programmings. Acta Sci. Nat. Univ. Neimenggu 24: 113–118
 Luc D.T. (1984). On duality theory in multiobjective programming. J. Optim. Theor. Appl. 43(4): 557–582 CrossRef
 Mond B. and Weir T. (1981). Generalized concavity and duality. In: Schaible, S. and Ziemba, W.T. (eds) Generalized Concavity in Optimization and economics, pp 263–279. Academic Press, New York
 Nakayama, H.: Duality theory in vector optimization: an overview, decision making with multiple objectives. In: Haimes, Y.Y., Chankong, V. (eds.) Lecture Notes in Economics and Mathematical Systems 337, pp. 86–93. SpringerVerlag (1989)
 Preda V. (1992). On efficiency and duality for multiobjective programs. J. Math. Anal. Appl. 166: 365–377 CrossRef
 Taninio T. and Sawaragi Y. (1987). Duality theory in multiobjective programming. J. Optim. Theor. Appl. 53: 115–123 CrossRef
 Weir T., Mond B. and Craven B.D. (1986). On duality for weakly minimized vector valued optimization problems. Optimization 17: 711–721 CrossRef
 Weir T. (1987). Proper efficiency and duality for vector valued optimization problems. J. Aust. Math. Soc. Ser. A 43: 24–34 CrossRef
 Weir T. (1988). A note on invex functions and duality in multipleobjective optimization. Opsearch 25: 98–104
 Weir T. and Jeyakumar V. (1981). A class of nonconvex functions and mathematical programming. Bull. Aust. Math. Soc. 38: 177–189
 Weir T. and Mond B. (1989). Generalized convexity and duality in multiobjective programming. Bull. Aust. Math. Soc. 39: 287–299
 Wolfe P. (1961). A duality theorem for nonlinear programming. Q. Appl. Math. 19: 239–244
 Title
 On Ginvex multiobjective programming. Part II. Duality
 Journal

Journal of Global Optimization
Volume 43, Issue 1 , pp 111140
 Cover Date
 20090101
 DOI
 10.1007/s1089800892986
 Print ISSN
 09255001
 Online ISSN
 15732916
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 (strictly) Ginvex vector function with respect to η
 GKarush–Kuhn–Tucker necessary optimality conditions
 GMond–Weir vector dual problems
 GWolfe vector dual problem
 Gmixed vector dual problem
 Industry Sectors
 Authors

 Tadeusz Antczak ^{(1)}
 Author Affiliations

 1. Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90238, Lodz, Poland