Optimality conditions and duality with new variants of generalized derivatives and convexity

Abstract

In this paper, a general vector optimization problem with inequality constraints is considered, this topic is very popular and important model with a long research history in optimization. The generality of setting is mainly expressed in the following three factors. The underlying spaces being linear spaces without topology (except the decision space being additionally equipped with this structure in some results). The “orderings” in both objective and constraint spaces are defined by arbitrary nonempty sets (not necessarily convex cones). The problem data are nonsmooth mappings, i.e., they are not Fréchet differentiable. For this problem, the optimality conditions and Wolfe and Mond-Weir duality properties are investigated , which lie at the heart of optimization theory. These results are established for the three main and typical optimal solutions: (Pareto) minimal, weak minimal, and strong minimal solutions in both local and global considerations. The research define a type of Gateaux variation to play the role of a derivative. For optimality conditions, and introduce the concepts of on-set differentiable quasiconvexity for global solutions and sequential differentiable quasiconvexity for local ones. Furthermore, each of them is separated into type 1 for sufficient optimality conditions and type 2 for necessary ones. After obtaining optimality conditions, applying them to derive weak and strong duality relations for the above types of solutions following our duality schemes of the Wolfe and Mon-Weir types. Due to the complexity of the research subject: considerations of duality are different from that of optimality conditions, we have to design two more appropriate types of generalized quasiconvexity: scalar quasiconvexity for the weak solution and scalar strict convexity for the Pareto solution. So all the results are in terms of the aforementioned Gateaux variation and various types of generalized quasiconvexity. The results are remarkably different from the related known ones with some clear advantages in particular cases of applications.