Generalized (ℱ, b, ϕ, ρ, θ)-univex n-set functions and parametric duality models 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. This problem was considered previously in the companion paper [7] where a survey of currently available optimality and duality results for minmax fractional susbset programming problems was presented, a fairly comprehensive list of references dealing with different aspects of these problems was provided, several new classes of generalized convex n-set functions were defined, and numerous sets of global parametric sufficient optimality conditions under various generalized ( ,b,φ,ρ,θ)-univexity assumptions were established. In the present study, we construct some dual problems for (P) and prove appropriate duality theorems utilizing most of the new classes of generalized n-set convex functions that were introduced in [7]. The rest of this paper is organized as follows. In Section 2, we recall the definitions of differentiability and certain types of generalized convexity for n-set functions which will be used frequently throughout the sequel. We begin our discussion of duality for (P) in Section 3 where we formulate a simple dual problem and prove weak, strong, and strict converse duality theorems. In Section 4, we consider another dual problem