The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems
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چکیده
and Applied Analysis 3 open topic. For example, it is yet not clear whether the dual approach to (7) of [29] can be extended to the SFP. The original algorithm given in [15] involves the computation of the inverse A (assuming the existence of the inverse of A), and thus has not become popular. A seemingly more popular algorithm that solves the SFP is the CQ algorithm of Byrne [16, 21] which is found to be a gradient-projection method (GPM) in convex minimization. It is also a special case of the proximal forward-backward splitting method [30]. The CQ algorithm only involves the computation of the projections P C andP Q onto the setsC andQ, respectively, and is therefore implementable in the case where P C and P Q have closed-form expressions; for example, C and Q are closed balls or halfspaces. However, it remains a challenge how to implement the CQ algorithm in the case where the projections P C and/or P Q fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm. Very recently, Xu [20] gave a continuation of the study on the CQ algorithm and its convergence. He applied Mann’s algorithm to the SFP andpurposed an averagedCQ algorithm whichwas proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. Furthermore, Korpelevič [11] introduced the so-called extragradient method for finding a solution of a saddle point problem. He proved that the sequences generated by the proposed iterative algorithm converge to a solution of the saddle point problem. Throughout this paper, assume that the SFP is consistent; that is, the solution set Γ of the SFP is nonempty. Let f : H1 → R be a continuous differentiable function. The minimization problem
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تاریخ انتشار 2014