Generalized Convexity And Applications In Mathematical Programming Problems

Quasiconvex functions and various generalized convexity of functions play important roles in mathematical programming. Criteria were derived for functions to be quasiconvex under lower semicontinuity or upper semicontinuity conditions by Mukherjee and Yeddy(Ref.l) .In the first part of this paper, we generalized almost all of the results there. New sufficient conditions are also given.Hanson(Ref.l7) introduced the concept of invexity for scalar programming problems. Liu and Wu(Ref.26) presented sufficient conditions and proposed three dual models for the generalized fractional programming problem with η - pseudoinvexity and η - quasi-invexity in Ref.26. Preda(Ref.27) introduced generalized (F, ρ )-convexity, an extension of F-convexity and generalized ρ -convexiry(Ref.28-29), relative results are also given for multiobjective programming problems. In the frame work of(F, ρ )-convex functions, Liu and Wu(Ref3.4) derived results similar to the results of Ref.26.It is well known that Wolfe type and Mond-Weir type duals are important duals in mathematical programming. Professor Xu(Ref.35) firstly introduced a mixed type dual in multiobjective programming, which is more general and flexible than the above two duals.Recently, it was pointed out that if a function is invex, then it is F-convex by Ref.37 and vice versa. Thus, it is easy to see that a function is (F, ρ )-convex if it is invex.In view of these, the second part of this paper presents two sufficient conditions and two mixed type duals for the generalized fractional programming only under (F, ρ )-convexity assumptions. These sufficient conditions apply to a broader class of mathematical programming problems.The results about weak duality, strong duality and strictly reverse duality arealso presented under more suitable conditions.Martin(Ref.36) proved the equivalence between some invexity and the case that every stationary point or Kuhn-Tucker point is a global minimum point for unconstrained or constrained scalar programming. Similar results were generalized to multiobjective programming in Ref.37, and the equivalent conditions that every stationary point or Kuhn-Tucker point is a weakly efficient solution were also obtained. Motivated by the above results, the third part of this paper considers the equivalence problems that every stationary point or Kuhn-Tucker point is an efficient solution.We define I-quasi-invex vector function ., I-strictly quasi-invex vector function and KT-I-strictly quasi invex vector function, and derive the above equivalent condition for unconstrained or constrained multiobjective programming.