Abstract
In this paper, a new class of nonconvex nonsmooth multiobjective programming problems with directionally differentiable functions is considered. The so-called G-V-type I objective and constraint functions and their generalizations are introduced for such nonsmooth vector optimization problems. Based upon these generalized invex functions, necessary and sufficient optimality conditions are established for directionally differentiable multiobjective programming problems. Thus, new Fritz John type and Karush-Kuhn-Tucker type necessary optimality conditions are proved for the considered directionally differentiable multiobjective programming problem. Further, weak, strong and converse duality theorems are also derived for Mond-Weir type vector dual programs.
Highlights
It is well known that the convexity notion plays a vital role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems
This paper represents the study concerning the new class of directionally differentiable multiobjective programming problems with nonconvex functions
The so-called class of semi-G-V -type I objective and constraint functions and its various generalizations are introduced in the case of directional differentiability of the functions constituting the considered nonconvex multiobjective programming problem
Summary
It is well known that the convexity notion plays a vital role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems. Assume that there exist vectors λ ∈ Rk and ξ ∈ Rm such that the G-Karush-Kuhn-Tucker type necessary optimality conditions (23)-(25) are satisfied at x with functions Gf and Gg. assume that (f, g) is semi-G-V -type I objective and constraint functions at x on D with respect to η and with respect to Gf and Gg. If the Lagrange multiplier λ is assumed to satisfy λ > 0, x is an efficient solution in problem (VP). Assume that there exist a differentiable real-valued strictly increasing function Gfi, i ∈ I, defined on Ifi(D), a differentiable real-valued strictly increasing function Ggj , j ∈ J , defined on Igj (D) with Ggj (0) = 0, j ∈ J , and vectors λ ∈ Rk and ξ ∈ Rm such that the G-Karush-KuhnTucker type necessary optimality conditions (23)(25) are satisfied at x. Proof of this theorem is similar to that one for Theorem 4 and, it is omitted in the paper
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