Minimum and Maximum Principle Sufficiency for a Nonsmooth Variational Inequality
- PDF / 435,195 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 82 Downloads / 216 Views
Minimum and Maximum Principle Sufficiency for a Nonsmooth Variational Inequality Zili Wu1
· Yun Lu1
Received: 5 February 2020 / Revised: 7 August 2020 / Accepted: 11 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, the minimum and maximum principle sufficiency properties for a nonsmooth variational inequality problem (NVIP) are studied. We discuss the relationship among the solution set of an NVIP and those defined by its dual problem and relevant gap functions. For a pseudomonotone NVIP, the weaker sharpness of its solution set has been shown to be sufficient for it to have minimum principle sufficiency property. As special cases, pseudomonotonicity∗ and pseudomonotonicity+ of the relevant bifunction have been characterized, from which the minimum and maximum principle sufficiency properties have also been characterized. Keywords Nonsmooth variational inequality problem · Minimum principle sufficiency property · Maximum principle sufficiency property · Weaker sharpness Mathematics Subject Classification 90C33 · 49J52
1 Introduction Topics about a variational inequality problem are important and interesting in the study of optimization with equilibrium problems and algorithms for computational purposes. The concept of variational inequalities was firstly posed by Hartman and Stampacchia [22] and was connected with the study of solving partial differential equations. For solving variational inequality problems, most efforts were qualitatively analyzing the
Communicated by Anton Abdulbasah Kamil.
B
Yun Lu [email protected] Zili Wu [email protected]
1
Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren Ai Road, Suzhou 215123, Jiangsu, China
123
Z. Wu, Y. Lu
existence and uniqueness of solutions, iterative algorithms for a sequence to converge strongly to the solution and applications in economics and engineering. For related results, see [7–10,12,14–18,20,21,23–26,28,31,35,36] and references therein. The idea of minimum principle sufficiency (MinPS) property was firstly introduced by Ferris and Mangasarian in [19]. They established the MinPS property to solve a convex program and proved the existence of a weakly sharp minimum. Marcotte and Zhu [27] extended Ferris and Mangasarian’s result and explained that a variational inequality possessing the MinPS property is equivalent to its solutions being weakly sharp. It was further studied by Wu and Wu [34] who presented characterization results of the MinPS property under new conditions. Recently, it is recognized in [32] that the MinPS property can be characterized when the mapping in a variational inequality is constant and pseudomonotone+ . It is described in [5] that a variational inequality can be applied to solve any differentiable optimization problem over a convex feasible region. However, actually most of the functions we meet are not differentiable but have some kinds of directional derivative. The relevant optimization problem should be discussed
Data Loading...