An explicit algorithm for monotone variational inequalities

Bello-Cruz, Yunier; Iusem, Alfredo N.

doi:10.1080/02331934.2010.536232

Cited by 26 publications

(2 citation statements)

References 22 publications

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“…Assumption (a) implies that projections onto C are well-defined. Condition (b) is slightly stronger than continuity of A; the same (or very similar) condition is also used, e.g., in [7,25]. Note that this assumption is automatically satisfied for continuous operators defined on finite-dimensional Hilbert spaces H = R n .…”

Section: Yekini Shehu and Olaniyi Iyiolamentioning

confidence: 99%

On a modified extragradient method for variational inequality problem with application to industrial electricity production

Shehu¹,

Iyiola²

2019

Journal of Industrial &Amp; Management Optimization

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In this paper, we present a modified extragradient-type method for solving the variational inequality problem involving uniformly continuous pseudomonotone operator. It is shown that under certain mild assumptions, this method is strongly convergent in infinite dimensional real Hilbert spaces. We give some numerical computational experiments which involve a comparison of our proposed method with other existing method in a model on industrial electricity production.

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Section: Yekini Shehu and Olaniyi Iyiolamentioning

confidence: 99%

On a modified extragradient method for variational inequality problem with application to industrial electricity production

Shehu¹,

Iyiola²

2019

Journal of Industrial &Amp; Management Optimization

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“…One of the most important and interesting topics in the theory of equilibria is to develop efficient and implementable algorithms for solving equilibrium problems and their generalizations (see, e.g., [8,26,27,47] and the references therein). Since the equilibrium problems have very close connections with both the fixed-point problems and the variational inequalities problems, finding the common elements of these problems has drawn many people's attention and has become one of the hot topics in the related fields in the past few years (see, e.g., [7,17,21,29,30,31,39,42,43,44,48] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Approximation of common solutions for system of equilibrium problems and fixed-point problems

Shehu

2014

Math Sci

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In this paper, we complement the result of Zhu and Chang (J Ineq Appl 2013:146, 2013) by proving strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces using the Halpern-Mann's iterations used in Zhu and Chang (J Ineq Appl 2013:146, 2013). We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

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Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

Millán

Lindstrom

Roshchina

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that-depending upon how parameters are chosen-resemble alternating projections, the Douglas-Rachford method, relaxed reflect-reflect, or the Peaceman-Rachford method. Such methods are frequently used to solve convex feasibility problems. The standard convergence analyses of projection algorithms are based on the firm nonexpansivity property of the relevant operators. However if the projections onto the constraint sets are replaced by cutters (projections onto separating hyperplanes), the firm nonexpansivity is lost. We provide a proof of convergence for a family of related averaged relaxed cutter methods under reasonable assumptions, relying on a simple geometric argument. This allows us to clarify fine details related to the allowable choice of the relaxation parameters, highlighting the distinction between the exact (firmly nonexpansive) and approximate (strongly quasi-nonexpansive) settings. We provide illustrative examples and discuss practical implementations of the method. IntroductionProjection and reflection methods are used for solving the feasibility problem of finding a point in the intersection of a finite collection of closed, convex sets in a Hilbert space. Such problems have a wide range of application in variational analysis, optimisation, physics and mathematics in general. One of the most successful methods from this class is the Douglas-Rachford method that uses a combination of reflections and averaging on each iteration. The idea first appeared in [33] as a numerical scheme for solving differential equations, and the convergence of a more general scheme for finding a zero of the sum of two maximally monotone operators was framed in [43] (also see [12, Chapter 26] for a modern treatment). Aragón Artacho and Campoy have recently introduced a modification of the Douglas-Rachford method for finding closest feasible points [7].Convergence rates for such methods are the subject of extensive research; we provide a brief sampling. Under appropriate conditions, the Douglas-Rachford method converges in finitely many steps [12]. Convergence rates may frequently be obtained through analysis of regularity conditions [38]. Additionally, semialgebraic structure admits further bounds on convergence rates for projection methods more generally [20,21,34] and for the Douglas-Rachford method in particular [40]. For a recent survey on the Douglas-Rachford method, see [41].The idea of replacing projections with their approximations, and specifically with the approximations constructed from the subdifferentials of the convex functions that describe the sets, was introduced by Fukushima [36]. It has been used in various contexts recently, including the numerical solution of variational arXiv:1810.02463v1 [math.OC] 4 Oct 2018inequalities; see, for example, [16,17]. In particular, Combettes has used relaxation parameters together with subgradient projections in the construction of his extrapolation method of parallel sub...

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An explicit algorithm for monotone variational inequalities

Cited by 26 publications

References 22 publications

On a modified extragradient method for variational inequality problem with application to industrial electricity production

On a modified extragradient method for variational inequality problem with application to industrial electricity production

Approximation of common solutions for system of equilibrium problems and fixed-point problems

Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

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