Generalized (𝔉,b,ϕ,ρ,θ)-univex n-set functions and global parametric sufficient optimality conditions in minmax fractional subset programming

G. J. Zalmai
2005 International Journal of Mathematics and Mathematical Sciences  
We first define a new class of generalized convex n-set functions, called (Ᏺ,b,φ,ρ,θ)univex functions, and then establish a fairly large number of global parametric sufficient optimality conditions under a variety of generalized (Ᏺ,b,φ,ρ,θ)-univexity assumptions for a discrete minmax fractional subset programming problem. where A n is the n-fold product of the σ-algebra A of subsets of a given set X, F i , G i , i ∈ p ≡ {1, 2,..., p}, and H j , j ∈ q, are real-valued functions defined on A n ,
more » ... nd for each Optimization problems of this type in which the functions F i , G i , i ∈ p, and H j , j ∈ q, are defined on a subset of R n (n-dimensional Euclidean space) are called generalized fractional programming problems. These problems have arisen in multiobjective programming [1], approximation theory [2, 3, 20, 34], goal programming [8, 19] , and economics [33] . The notion of duality for a generalized linear fractional programming problem with point functions was originally considered by von Neumann [33] in the context of an economic equilibrium problem. More recently, various optimality conditions, duality results, and computational algorithms for several classes of generalized fractional programs with point functions have appeared in the related literature. A fairly extensive list of references pertaining to different aspects of these problems is given in [40] . In the area of subset programming problems, minmax fractional programs like (1.1) were first discussed in [37, 38] . In [37] , necessary and sufficient optimality conditions and several duality results were established under generalized ρ-convexity assumptions.
doi:10.1155/ijmms.2005.1253 fatcat:dj32s6env5b5lg5hyvnb3jpyue