Vector Variational-Like Inequalities with Constraints: Separation and Alternative

Jiawei Chen (), Shengjie Li (), Zhongping Wan () and Jen-Chih Yao ()
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Jiawei Chen: Chongqing University
Shengjie Li: Chongqing University
Zhongping Wan: Wuhan University
Jen-Chih Yao: China Medical University

Journal of Optimization Theory and Applications, 2015, vol. 166, issue 2, No 6, 460-479

Abstract: Abstract Based on the oriented distance function, a linear weak separation function and three nonlinear regular weak separation functions are introduced in reflexive Banach spaces. Particularly, a nonlinear regular weak separation function does not involve any parameters. Moreover, theorems of the weak alternative for vector variational-like inequalities with constraints are derived by the separation functions without any convexity. Saddle-point conditions, which show the equivalence between the existence of a saddle point and a (linear) nonlinear separation of two suitable subsets of the image space, are established for the linear and nonlinear regular weak separation functions, respectively. Necessary and sufficient optimality conditions for vector variational-like inequalities with constraints are also obtained via the saddle-point conditions. Finally, two gap functions for vector variational-like inequalities with constraints and their continuity are derived by using the image space analysis.

Keywords: Vector variational-like inequalities with constraints; Image space analysis; Saddle point; Alternative; Optimality condition; Oriented distance function; 49J40; 65K10 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10957-015-0736-6

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