Nonlinear separation approach to inverse variational inequalities. (2nd July 2016)

• Record Type: Journal Article
• Authors: Xu, Y. D.
• Is Part Of: Optimization. Volume 65:Number 7(2016)
• Page Start: 1315
• Page End: 1335
• Publication Date: 2016-07-02
• Subjects: Inverse variational inequalities -- image space analysis -- nonlinear separation function -- gap function -- error bound; 49K05 -- 90C29; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1149584
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.