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

where An is the n-fold product of the σ-algebra A of subsets of a given set X , Fi, Gi, i∈ p ≡ {1,2, . . . , p}, and Hj , j ∈ q, are real-valued functions defined on An, and for each i∈ p, Gi(S) > 0 for all S∈An such that Hj(S) ≤ 0, j ∈ q. Optimization problems of this type in which the functions Fi, Gi, i∈ p, and Hj , j ∈ q, are defined on a subset of Rn (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.