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Forward-Backward Extragradient Methods for Quasimonotone Variational Inequalities
Published in Anurag Jayswal, Tadeusz Antczak, Continuous Optimization and Variational Inequalities, 2023
Habib ur Rehman, Poom Kumam, Ioannis K. Argyros
The mathematical model of the variational inequality problem is a crucial problem in nonlinear analysis. It is a significant mathematical model that unites a number of key topics such as a system of nonlinear equations, optimization problems, complementarity problems, network equilibrium problems and finance (for more details see [11, 13–16, 25]). This notion has numerous applications in mathematical programming, engineering, transportation analysis, network economics, game theory and computer science. Regularized methods and projection methods are two well-known and comprehensive methods for approximating a solution to variational inequalities. It is also worth noting that the first approach is frequently used to solve variational inequalities accompanied by the monotone operator class. In this method, the regularized sub-problem is strongly monotone, and its unique solution exists and is easier to find than the initial problem.
General Wiener-Hopf equation technique for extended Verma general variational inequalities
Published in Amir Hussain, Mirjana Ivanovic, Electronics, Communications and Networks IV, 2015
Xiaomin Wang, Yanyan Zhang, Ying Liu, Xiuyan Fan
By the above list, we see that the extended Verma general variational inequality (1) is the most general class of variational inequalities. Several known classes of variational inequalities are the special cases of it. These variational inequalities have wide and important applications in economics, mathematical programming, physics and engineering sciences.
Convex Analysis and Subdifferential Operators
Published in Behzad Djafari-Rouhani, Hadi Khatibzadeh, Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces, 2019
Behzad Djafari-Rouhani, Hadi Khatibzadeh
Convexity has an essential role in optimization, variational inequalities, evolution equations and many other branches of nonlinear analysis. This chapter is a quick review of convex analysis. Our aim is to provide the reader with the essential facts on convex sets and convex functions that will be needed in the subsequent chapters of the book.
Mixed vector equilibrium-like problems on Hadamard manifolds: error bound analysis
Published in Applicable Analysis, 2023
Nguyen Van Hung, Vo Minh Tam, Donal O'Regan
In 1976, Auslender [1] introduced the notion of (non-differentiable) gap function for finite-dimensional variational inequalities. It is known as a bridge between variational inequalities and optimization problems, i.e. a variational inequality can be reformulated as an optimization problem. To overcome the non-differentiability of gap functions considered in [1], Fukushima [2] developed the concept of a differentiable gap function which is called the regularized gap function. Furthermore, Yamashita et al. [3] investigated the regularized function of Moreau–Yosida type for variational inequalities and in terms of regularized gap functions in [2] and regularized functions of Moreau–Yosida type, they also provided error bounds for variational inequalities. An error bound is an expression which proposes an upper estimate of the distance of an arbitrary feasible point to the solution set of a certain problem. It plays a major role in convergence analysis of iterative algorithms to a solution of the variational inequality. Moreover, the theory of equilibrium problems and variational inequalities has applications in the fields of mechanics, economics, transportation, optimization and operations research. For details, we refer the reader to [4–13] and the references therein.
A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets
Published in Optimization, 2023
Liang Lu, Lijie Li, Jiangfeng Han
The notion of differential variational inequalities was introduced in [20] by Aubin and Cellina. Differential variational inequalities are systems which couple differential or partial differential equations with a time-dependent variational inequality. Various mathematical models arising in the study of contact and impact problems lead to differential variational inequalities. Since a systematic study was carried out by Pang and Stewart [21], there is a number of papers have been dedicated to the development of the theory of differential variational inequalities and their applications [1,22–27]. Furthermore, differential hemivariational inequalities were firstly introduced by Liu et al. [2]. Interest in differential hemivariational inequalities and, more generally, differential variational–hemivariational inequalities, represent an important extension, originated, similarly as in differential variational inequalities. Over the past decade, the theory of this topic grew rapidly. We refer the reader to some recent references [2,3,28–32] and their references.
Generalized well-posedness results for a class of new mixed variational inequalities
Published in Optimization, 2023
Jinxia Cen, Chao Min, Guo-ji Tang, Van Thien Nguyen
The theory of variational inequalities as a powerful mathematical tool has been applied widely to investigate various comprehensive problems arising in partial differential equations, economics, mechanics, control and optimization, mathematical physics and so forth. Particularly, in mechanics, variational inequalities express the principle of virtual work or power in their inequality form. Historically, variational inequalities stem from the study of the classical result in Contact Mechanics concerns the problem posed by Signorini in 1933 [1], which is an elastostatic problem, that in simplest terms consists in determining the equilibrium configuration of an anisotropic non-homogeneous linearly elastic body, resting on a rigid frictionless horizontal plane and subject only to its mass forces. Whereas, a complete proof of the existence and uniqueness of a solution for the Signorini problem was based on the principle of virtual work and provided by Signorini's student Fichera in 1964. It should be mentioned that the solution of the Signorini problem symbols the birth of the field of variational inequalities. Thereafter, it was introduced at about the same time by Stampacchia [2] in 1964 who proved the generalization of the Lax-Milgram theorem in order to study the regularity problem for elliptic partial differential equations. Because of the widely applications of variational inequalities, over the years, more and more scholars are attracted to boost the development of the theory and applications for variational inequalities, see e.g. [3–26].