# American Institute of Mathematical Sciences

March  2020, 16(2): 707-724. doi: 10.3934/jimo.2018174

## Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China 2 College of Computer Science, Chongqing University, Chongqing 400044, China 3 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 4 Center for General Education, China Medical University, Taichung 40402, Taiwan

* Corresponding author: Shengjie Li

Received  May 2017 Revised  May 2018 Published  December 2018

Fund Project: This research was supported by the Natural Science Foundation of China (Nos: 11171362, 11401487), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation (No: 2015M582512), National Scholarship under the Grant of China Scholarship Council, the Fundamental Research Funds for the Central Universities(XDJK2019C073) and the grant MOST 106-2923-E-039-001-MY3

In this paper, we consider a class of constrained vector optimization problems by using image space analysis. A class of vector-valued separation functions and a $\mathfrak{C}$-solution notion are proposed for the constrained vector optimization problems, respectively. Moreover, existence of a saddle point for the vector-valued separation function is characterized by the (regular) separation of two suitable subsets of the image space. By employing the separation function, we introduce a class of generalized vector-valued Lagrangian functions without involving any elements of the feasible set of constrained vector optimization problems. The relationships between the type-Ⅰ(Ⅱ) saddle points of the generalized Lagrangian functions and that of the function corresponding to the separation function are also established. Finally, optimality conditions for $\mathfrak{C}$-solutions of constrained vector optimization problems are derived by the saddle-point conditions.

Citation: Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174
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