V-Invexity in Nonlinear Multiobjective Programming

Hanson’s (1981) introduction of invex functions was motivated by the question of finding the widest class of functions for which weak duality hold for dual programs, such as the Wolfe and Mond-Weir duals, formulated from the necessary optimality conditions. Since then, various generalizations of invexity have been introduced in the literature e.g. Craven and Glover (1985), Egudo (1989), Hanson and Mond (1982), Kaul and Kaur (1985), Martin (1983), Kaul, Suneja and Srivastava (1994), Jeyakumar and Mond (1992), Mond and Hanson (1984, 1989), Nanda and Das (1993, 1994), Smart (1990), Weir (1988), Rueda and Hanson (1988), Mond and Husain (1989), Mond, Chandra and Husain (1988), Mishra and Mukherjee (1994a, 1994b, 1995, 1996, 1996a, 1996b). optimality and duality properties are concerned.

In this Chapter, we consider the role of invexity and its generalizations, namely V-pseudo-invexity and V-quasi-invexity in standard multiobjective programming; in particular, the replacement is made of invexity in results related to necessary and sufficient optimality conditions, duality theorems symmetric duality results and vector valued constrained games. A vast number of theorems developed during the evolution of nonlinear programming theory were stated with assumptions of invexity. In most cases it has been possible to generalize these results under the assumptions of Vinvexity. However, this has not been a direct process. Intermediate and overlapping results have been achieved using the various notions of generalized convexity discussed in Chapter 1.