Symmetric Duality for Mathematical Programming in Complex Spaces with Second Order F-univexity

Bector et al., (1994) introduced the concept of univexity. Mishra (1998), Mishra and Reuda (2001) and Mishra et al., (2005) have applied and generalized these functions. Mathematical programming in complex spaces originated from Levinson’s discussion of linear problems (Levinson, 1966). Mishra and Reuda (2003) have generalized these upto first orders, second order with symmetric duality. Further Mishra and Rueda (2003) have given second order generalized invexity and it’s duality theorems in multiobjective programming problems. For more details reader may consult (Ferraro, 1992; Lai, 2000; Liu, 1997; Liu et al., 1997). Symmetric duality in real mathematical programming was introduced by Dorn (1961), who defined a program and its dual to be symmetric if the dual of the dual is original problem. Later, Mond and W eir (1981) presented a pair of symmetric dual nonlinear programs which allows the weakening of the convexity-concavity condition. For more work on symmetric duality in real spaces (Chandra et al., 1998; Gulati and A hmad, 1997; Mishra, 2000a; Mishra, 2000b). Mangasarian (1975) considered a nonlinear program and discussed second order duality under certain inequalities. Mond (1974) assumed simple inequalities respectively and indicated a possible computational advantage of the second order dual over the first order dual. Bector and Chandra (1987) defined the functions satisfying these inequalities (Mond, 1974) to be bonvex/boncave. Mishra (2000a) obtained second order duality results for a pair of Wolfe and Mon-Weir type second order symmetric dual nonlinear programming problems in real spaces under second order F-convexity, F-concavity and second order F-convexity is an extension of F-convex functions introduced by Hanson and Mond (1982). Mishra (2000b) formulated a pair of multiobjective second order symmetric dual programs for arbitrary cones in real space. The model considered in (Mishra, 2000b) unifies the Wolfe and the M ond-Weir type second order vector symmetric dual models. Gupta (1983) formulated a second order nonlinear symmetric dual program on the pattern of second order dual formulation as given by Mangasarian (1975) for the real case, but the constraints in the formulation (Gupta, 1983) is linear. Lai (2000) extended the concept of F-convex functions to the complex case and established sufficient optimality and duality theorems for a pair of nondifferentiable fractional complex programs. In this study, we defined F-univex in the complex space in two variables. Hence extend the concepts of F-pseudounivex, F-pseudounicave functions and study symmetric duality under the aforesaid assumptions for Wolfe and Mond-Weir type models with second order in complex spaces.