Generalized -Type I Univex Functions in Multiobjective Optimization

A new class of generalized functions -type I univex is introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established. Weak, strong, converse, and strict converse duality theorems are also derived for Mond-Weir-type multiobjective dual program. 1. Introduction Generalizations of convexity related to optimality conditions and duality for nonlinear single objective or multiobjective optimization problems have been of much interest in the recent past and thus explored the extent of optimality conditions and duality applicability in mathematical programming problems. Invexity theory was originated by Hanson [1]. Many authors have then contributed in this direction. For a nondifferentiable multiobjective programming problem, there exists a generalisation of invexity to locally Lipschitz functions with gradients replaced by the Clarke generalized gradient. Zhao [2] extended optimality conditions and duality in nonsmooth scalar programming assuming Clarke generalized subgradients under type I functions. However, Antczak [3] used directional derivative in association with a hypothesis of an invex kind following Ye [4]. On the other hand, Bector et al. [5] generalized the notion of convexity to univex functions. Rueda et al. [6] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [7] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions which were further generalized to univex type I-vector-valued functions by Mishra et al. [8]. Jayswal [9] introduced new classes of generalized -univex type I vector valued functions and established sufficient optimality conditions and various duality results for Mond-Weir type dual program. Generalizing the work of Antczak [3], recently Nahak and Mohapatra [10] obtained duality results for multiobjective programming problem under invexity assumptions. In this paper, by combining the concepts of Mishra et al. [8] and Nahak and Mohapatra [10], we introduce a new generalized class of -type I univex functions and establish weak, strong, converse, and strict converse duality results for Mond-Weir type dual. 2. Preliminaries and Definitions The following convention of vectors in will be followed throughout this paper: , ; , ; , . Let be a nonempty subset of , , be an arbitrary point of and ,？？ ,？？ . Also, we denote and and and . Definition 2.1 (Ben-Israel and Mond [11]). Let be an invex set. A