Generalized (ℱ, b, φ, ρ, θ)-univex n-set functions and global semiparametric sufficient efficiency conditions in multiobjective fractional subset programming

In this paper, we will present a multitude of global semiparametric sufficient efficiency conditions under various generalized ( ,b,φ,ρ,θ)-univexity hypotheses for the following multiobjective fractional subset programming problem: (P) Minimize(F1(S)/G1(S),F2(S)/G2(S), . . . ,Fp(S)/Gp(S)) subject to Hj(S) 0, j ∈ q, S∈An, 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. The point-function counterparts of (P) are known in the area of mathematical programming as multiobjective fractional programming problems. These problems have been the focus of intense interest in the past few years, which has resulted in numerous publications. A fairly extensive list of references concerning various aspects of these problems is given in [43]. For more information about general multiobjective problems with point functions, the reader may consult the recently published monograph by Miettinen [30]. In the area of subset programming, multiobjective problems have been investigated in [2, 4, 7, 14, 15, 16, 18, 23, 24, 26, 27, 29, 38, 42], and multiobjective fractional problems in [1, 5, 6, 19, 21, 22, 24, 36, 44, 46]. We next give a brief overview of the available results pertaining to the latter class of problems. A parametric dual problem for (P) was constructed in [6] and a number of weak and strong duality theorems involving generalized ρ-convexity assumptions were proved. In [18], two parametric dual problems, which are slightly different from the one considered in [6], were formulated and some weak, strong, and strict converse duality results were established using generalized ρ-convexity hypotheses. Some of these results are further extended in [22] by using generalized -convex n-set functions. A multiobjective fractional