Abstract
In this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. Using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. Furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. We further investigate Q-proper efficiency in a real vector space, where Q is some nonempty (not necessarily convex) set. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. The scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the Q-proper efficiency in vector optimization with and without constraints.
Highlights
Many works have been done with vector optimization problems under real linear spaces without any particular topology [1, 2, 4,5,6,7,8,9,10,11,12,13,14]
Only a few authors focus on nonconvex vector optimization problems [5,7,13,14,15]
The main purpose of this paper is to study some optimality conditions on Q-proper efficiency of general nonconvex optimization problems in a real linear vector space without topology, by using nonlinear scalarization functions
Summary
Many works have been done with vector optimization problems under real linear spaces without any particular topology [1, 2, 4,5,6,7,8,9,10,11,12,13,14]. Only a few authors focus on nonconvex vector optimization problems [5,7,13,14,15] Inspired by this fact, the main purpose of this paper is to study some optimality conditions on Q-proper efficiency of general nonconvex optimization problems in a real linear vector space without topology, by using nonlinear scalarization functions. Hernández and Rodríguez-Marín [30] presented an extension of the Gerstewitz function and characterized some topological properties to obtain a nonconvex scalarization and optimality conditions for set-valued optimization problems. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via scalarization by oriented distance function and nonlinear scalarization function in a vector space. The results of this paper can be used to develop a vector optimization on vector spaces, which can be applied to any numerical and theoretical scalar optimization
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