Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities

Abstract

This paper deals with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities (in short, NVI) by using gap functions. Several characterizations of these two sufficiency properties are provided. We also discuss the error bound for nonsmooth variational inequalities. Two open questions are given at the end.

Some aspects of nonsmooth variational inequalities on Hadamard manifolds

“…Motivated by the above results, in the present paper, we extend the work of Alshahrani et al [6] in the setting of Hadamard manifolds. Hence, we defined the minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using a gap function in the Hadamard manifolds setting and provide several characterizations for these two properties.…”

“…We consider the following condition, which was first considered by Wu and Wu [35], to develop the weak sharpness of variational inequalities in Hilbert spaces and later it was considered by Alshahrani et al [6].…”

“…In this case, the relevant optimization problem can be considered by using a nonsmooth variational inequality with a bifunction, see [10]. In 2016, Alshahrani et al [6] defined the minimum and maximum principle sufficiency properties for nonsmooth variational inequalities (NVI) by using a gap function that is similar to that of Wu and Wu [35] and provided some characterizations for these two properties. They also discussed error bounds for nonsmooth variational inequalities.…”

Weak Sharp Type Solutions for Some Variational Integral Inequalities

“…Thereafter, various researchers, like Hu and Song [4], Liu and Wu [5], and Zhu [6] have continued this study by considering gap functions, extremization problems and an appropriate framework. Also, Alshahrani et al [7] studied nonsmooth variational inequalities by using gap-type functions.…”

On weak sharp solutions in $(\rho , \mathbf{b}, \mathbf{d})$-variational inequalities

“…Based on the works of Burke and Ferris [3], Patriksson [11] and following Marcotte and Zhu [10], the concept of weak sharp solution associated with variational-type inequalities has attracted the attention of many researchers (see, for instance, Hu and Song [7], Liu and Wu [9], Zhu [17] and Jayswal and Singh [8]). Recently, by using gap-type functions, in accordance with Ferris and Mangasarian [5] and following Hiriart-Urruty and Lemaréchal [6], Alshahrani et al [1] studied the minimum and maximum principle sufficiency properties associated with nonsmooth variational inequalities.…”