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Convex Analysisの二,三の進展について

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Nonlinear Functional Analysis And Its Applications

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

/pdf/the-subgradient-extragradient-method-for-solving-variational-1hy37t0r8l.pdf

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space.

Yair Censor and Stavros A. Zenios, Oxford University Press, New York, 1997, 539 pp., ISBN 0-19-510062-X, $85.00

Parallel Optimization: Theory, Algorithms and Applications

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

Algorithms for the Split Variational Inequality Problem

/pdf/algorithms-for-the-split-variational-inequality-problem-3a8vtbvp7c.pdf

We present two extensions of Korpelevich's extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevich's extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.

/pdf/extensions-of-korpelevich-s-extragradient-method-for-the-3xqnuze2fe.pdf

Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

We study two projection algorithms for solving the variational inequality problem in Hilbert space. One algorithm is a modified subgradient extragradient method in which an additional projection onto the intersection of two half-spaces is employed. Another algorithm is based on the shrinking projection method. We establish strong convergence theorems for both algorithms.

/pdf/strong-convergence-of-subgradient-extragradient-methods-for-3775yis03j.pdf

Aviv Gibali

Papers

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space.

Algorithms for the Split Variational Inequality Problem

Algorithms for the Split Variational Inequality Problem

Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space