Construction algorithms for a class of monotone variational inequalities

Self Cite

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.

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

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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

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.

“…For some related work, please refer to: Facchinei and Pang [8], Iusem [12], Glowinski [9], Korpelevich [13], Noor [1], Yao et al [20,22,24,25,27], Zegeye et al [28], and Zhang et al [29].…”

Section: Introductionmentioning

confidence: 99%

Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations

Yao¹,

Liou²,

Yao³

2017

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In the present paper, we consider the split variational inequality and fixed point problem that requires to find a solution of a generalized variational inequality in a nonempty closed convex subset C of a real Hilbert space H whose image under a nonlinear transformation is a fixed point of a pseudocontractive operator. An iterative algorithm is introduced to solve this split problem and the strong convergence analysis is given.