Abstract
We first define a new class of generalized convex n‐set functions, called (𝔉, b, ϕ, ρ, θ)‐univex functions, and then establish a fairly large number of global parametric sufficient optimality conditions under a variety of generalized (𝔉, b, ϕ, ρ, θ)‐univexity assumptions for a discrete minmax fractional subset programming problem.
Highlights
We will present a number of global parametric sufficient optimality conditions under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses for the following discrete minmax fractional subset programming problem: Minimize max Fi(S) 1≤i≤p Gi(S)
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
Summary
We will present a number of global parametric sufficient optimality conditions under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses for the following discrete minmax fractional subset programming problem: Minimize max Fi(S) 1≤i≤p Gi(S). More general sets of sufficiency conditions are formulated and discussed in Section 4 with the help of two partitioning schemes The first of these schemes was originally used in [27] for constructing generalized dual problems for nonlinear programs with point functions, whereas the second appears to be new and leads to a number of different sufficiency criteria for generalized fractional programming problems. All these optimality results are applicable, when appropriately specialized, to the following three classes of problems with discrete max, fractional, and conventional objective functions, which are particular cases of (1.1): Minimize max Fi(S), S∈F. The optimality results established for (1.1) can be modified and restated for each one of the above problems, we will not explicitly state these results
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