Conic Scalarizations for Approximate Efficient Solutions in Nonconvex Vector Optimization Problems
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Conic Scalarizations for Approximate Efficient Solutions in Nonconvex Vector Optimization Problems Hui Guo1
· Wan-li Zhang2
Received: 25 March 2016 / Revised: 17 July 2016 / Accepted: 8 October 2016 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg 2016
Abstract Nonlinear scalarization is a very important method to deal with the vector optimization problems. In this paper, some conic nonlinear scalarization characterizations of E-optimal points, weakly E-optimal points, and E-Benson properly efficient points proposed via improvement sets are established by a new scalarization function, respectively. These results improved and generalized some previously known results. As a special case, the scalarization of Benson properly efficient points is also given. Some examples are given to illustrate the main results. Keywords Improvement set · E-optimal points · Weakly E-optimal points · E-Benson properly efficient points · Nonlinear scalarization Mathematics Subject Classification 90C29 · 90C30
This research was supported by the National Natural Science Foundation of China (No. 11301574), Chongqing Municipal Education Commission (No. KJ1500310), and the Doctor Startup Fund of Chongqing Normal University (No.16XLB010).
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Hui Guo [email protected] Wan-li Zhang [email protected]
1
Department of Mathematics, Chongqing Normal University, Chongqing 401331, China
2
College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China
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H. Guo, W. Zhang
1 Introduction Everyone knows that the concepts of approximate solutions have been playing an important role in vector optimization theory and applications. In recent years, there are a lot of works related to vector optimization and some concepts of approximate solutions for vector optimization problems are proposed and some scalarization characterizations of these approximate solutions are studied; see [1–7] and the references therein. In 2011, Chicco et al. [5] introduced E-optimal point concepts based on comprehensive sets and investigated some properties of improvement sets in Euclidean space. Gutiérre et al. [6] extended the definition of improvement sets to a general real locally convex topological linear space and obtained the scalar characterization for E-efficient solutions. Based on improvement sets, Zhao et al. [8] proposed E-Benson properly efficient solutions which unified some proper efficiency and approximate proper efficiency and obtained some linear scalarization characterizations under the nearly E-subconvexlikeness. It is well known that both theory and practice of vector optimization have always been closely related to scalarization procedures. In general, scalarization means the replacement of a vector optimization problem by a suitable scalar optimization problem. Since the scalar optimization theory is widely developed, scalarization turns out to be of great importance for the vector optimization theory. [9–11] Especially, in 20
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