Vector Variational-Like Inequalities with Constraints: Separation and Alternative

作者：Chen, Jiawei; Li, Shengjie^*; Wan, Zhongping; Yao, Jen-Chih

来源：Journal of Optimization Theory and Applications, 2015, 166(2): 460-479.

DOI：10.1007/s10957-015-0736-6

• Vector variational-like inequalities with constraints
• Image space analysis
• Saddle point
• Alternative
• Optimality condition
• Oriented distance function

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.

• 出版日期2015-8
• 单位中国医科大学; 武汉大学; 重庆大学