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

  • * Corresponding author: Shengjie Li

    * Corresponding author: Shengjie Li 
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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  • 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.

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


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