Schemes for finding minimum-norm solutions of variational inequalities

Yao, Yonghong; Chen, Rudong; Xu, Hong–Kun

doi:10.1016/j.na.2009.12.029

Cited by 76 publications

(29 citation statements)

References 13 publications

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“…Variational inequalities have extensively been investigated, see [1,2,15,17,18,20,22,24,[26][27][28] for more details. In 1976, Korpelevich [14] proposed an iterative algorithm for solving VIP (1.1) in Euclidean space R n : y n = P C (x n − τAx n ), x n+1 = P C (x n − τAy n ), ∀n 0, with τ > 0 a given number, which is known as the extragradient method.…”

Section: A Mapping F : C → H Is Called L-lipschitz Continuous If Thermentioning

confidence: 99%

Hybrid steepest-descent methods for systems of variational inequalities with constraints of variational inclusions and convex minimization problems

Rong¹,

Chuan²,

Cheng³

et al. 2017

J. Nonlinear Sci. Appl.

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Two hybrid steepest-descent schemes (implicit and explicit) for finding a solution of the general system of variational inequalities (in short, GSVI) with the constraints of finitely many variational inclusions for maximal monotone and inversestrongly monotone mappings and a minimization problem for a convex and continuously Fréchet differentiable functional (in short, CMP) have been presented in a real Hilbert space. We establish the strong convergence of these two hybrid steepestdescent schemes to the same solution of the GSVI, which is also a common solution of these finitely many variational inclusions and the CMP. Our results extend, improve, complement and develop the corresponding ones given by some authors recently in this area.

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Section: A Mapping F : C → H Is Called L-lipschitz Continuous If Thermentioning

confidence: 99%

Hybrid steepest-descent methods for systems of variational inequalities with constraints of variational inclusions and convex minimization problems

Rong¹,

Chuan²,

Cheng³

et al. 2017

J. Nonlinear Sci. Appl.

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“…This problem is a fundamental problem in the variational analysis; in particular, in the optimization theory and mechanics; see e.g., [13,[18][19][20][21][33][34][35][36][37][38] and the references therein. A popular algorithm for solving this problem is extragradient method introduced by Korpelevich [22].…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

Ceng¹,

Wen²

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce and analyze implicit and explicit iteration methods for solving a variational inequality problem over the set of common fixed points of an infinite family of nonexpansive mappings on a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Strong convergence results are given. Our results improve and extend the corresponding results in the literature.

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“…This indicates that one can use projection technique to find the minimum norm solution. In this respect, there are a large number references in which the projection technique is applied to find the minimum norm solution of some nonlinear operators, see e.g., [2,3,19,20,21].…”

Section: Minimization Problemmentioning

confidence: 99%