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

Jiawei Chen; Shengjie Li; Jen-Chih Yao

doi:10.3934/jimo.2018174

Article Contents

2020, Volume 16, Issue 2: 707-724. Doi: 10.3934/jimo.2018174

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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
* Corresponding author: Shengjie Li

Received: May 2017

Revised: May 2018

Early access: December 2018

Published: February 2020

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

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Abstract

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.

Keywords:

Constrained vector optimization problem,
image space analysis,
nonlinear vector-valued separation function,
saddle point,
optimality condition.

Mathematics Subject Classification: Primary: 49J40; Secondary: 65K10.

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