Iterative Methods for Variational Inequalities

Noor, Muhammad Aslam; Noor, Khalida İnayat; Rassias, Themistocles M.

doi:10.1007/978-3-030-27407-8_23

Cited by 15 publications

(29 citation statements)

References 35 publications

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“…Inequalities of type equations (2) are known as variational inequalities, which were introduced by Lions and Stampacchia [20] and have been studied extensively in recent years, see [20,28,29,32,[35][36][37][38].…”

Section: Formulations and Basic Resultsmentioning

confidence: 99%

“…Complementarity problem was introduced by Lemke [19]. The equivalence between variational inequalities and complementarity problem has been used in suggesting many iterative algorithms for solving complementarity problems, see [17,26,27,37]. Absolute value complementarity problem is an important and useful generalization of the complementarity problem.…”

Section: Introductionmentioning

confidence: 99%

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Absolute Value Variational Inequalities and Dynamical Systems

Batool

Noor

2020

ijaa

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In this paper, we consider the absolute value variational inequalities. We propose and analyze the projected dynamical system associated with absolute value variational inequalities by using the projection method. We suggest different iterative algorithms for solving absolute value variational inequalities by discretizing the corresponding projected dynamical system. The convergence of the suggested methods is proved under suitable constraints. Numerical examples are given to illustrate the efficiency and implementation of the methods. Results proved in this paper continue to hold for previously known classes of absolute value variational inequalities.

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Section: Formulations and Basic Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Absolute Value Variational Inequalities and Dynamical Systems

Batool

Noor

2020

ijaa

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“…For the convergence analysis and other aspects of Algorithm 4, see Noor et al [149]. It is obvious that Algorithm 4 and Algorithm 6 have been suggested using different variants of the fixed point formulations (25).…”

Section: Algorithmmentioning

confidence: 99%

New Trends in General Variational Inequalities

Noor

Rassias

2020

Acta Appl Math

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It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, the auxiliary principle and the penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.

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“…It is remarkable that a wide class of unrelated problems, which arises in every branch of pure and applied sciences can be studied in the unified and general framework of variational inequalities and their variant forms. For the applications, formulation, dynamical systems, sensitivity analysis, numerical methods, error bounds and other aspects of the variational inequalities and optimization see [3,4,8,[16][17][18][19][20][21]35].…”

Section: Introductionmentioning

confidence: 99%

“…These generalized convex functions are nonconvex functions. For recent developments, see [5,6,7,14,15,16,17,18,20,21,22] and the references therein.…”

Section: Introductionmentioning

confidence: 99%