The Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Xu, Haiwen; Shao, Hu; Zhang, Qianchuan

doi:10.1109/etcs.2009.63

Cited by 2 publications

(5 citation statements)

References 17 publications

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Section: Introductionmentioning

confidence: 87%

“…In this paper, we will prove the strong convergence of PRH method under different and suitable restrictions imposed on parameters (Condition 12), which differs from that of [15]. Moreover, the proof of strong convergence is different from the previous proof in [15], which is not similar to that in [7] in Step 2. And more importantly, numerical experiments verify that the PRH method under Condition 12 is more efficient than that under Condition 10, and the PRH method under some descent directions is more slightly efficient than that of the MRHSD method [14,16].…”

Section: Introductionmentioning

confidence: 87%

“…Theorem 11 (see [15]). In Condition 10, the sequence { } converges strongly to * ∈ , and * is the unique solution of the ( , ).…”

Section: Correctionmentioning

confidence: 99%

“…Algorithm 8 (see [15]). Take three fixed numbers , , ∈ (0, 2 / 2 ), starting with arbitrarily chosen initial points 0 ∈ , compute the sequences { }, { }, {̃}, {̂} such that;…”

Section: Convergence Theoremmentioning

confidence: 99%

“…A set of practical numerical experiments in the literature [16] demonstrated that the MRHSD method has different efficiency under different conditions. Subsequently, the prediction-correction and relaxed hybrid steepest-descent method (PRH method) [15] makes more use of the history information and less decreases the loss of information than the methods [7,14]. The PRH method introduced more descent directions than the MRHSD method [14,16], and computing these descent directions only needs the history information.…”

Section: Introductionmentioning

confidence: 99%

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The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

2013

Abstract and Applied Analysis

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We establish the strong convergence of prediction-correction and relaxed hybrid steepest-descent method (PRH method) for variational inequalities under some suitable conditions that simplify the proof. And it is to be noted that the proof is different from the previous results and also is not similar to the previous results. More importantly, we design a set of practical numerical experiments. The results demonstrate that the PRH method under some descent directions is more slightly efficient than that of the modified and relaxed hybrid steepest-descent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 87%

“…Theorem 11 (see [15]). In Condition 10, the sequence { } converges strongly to * ∈ , and * is the unique solution of the ( , ).…”

Section: Correctionmentioning

confidence: 99%

“…Algorithm 8 (see [15]). Take three fixed numbers , , ∈ (0, 2 / 2 ), starting with arbitrarily chosen initial points 0 ∈ , compute the sequences { }, { }, {̃}, {̂} such that;…”

Section: Convergence Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

2013

Abstract and Applied Analysis

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Some Recent Trends in Variational Inequalities and Optimization Problems with Applications

Bnouhachem

Hamdi

2014

Abstract and Applied Analysis

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Variational inequalities theory, which was introduced in the sixties, has emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, finance, economics, social, and pure and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. The ideas and techniques of variational inequalities are being applied in a variety of diverse areas of sciences and prove to be productive and innovative. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems. Variational inequalities have been extended and generalized in several directions using novel and new techniques. In parallel, optimization methods based on proximal point and proximal-like type methods have attracted a large number of researchers in the last three decades. In the same spirit, we can cite, for instance, the alternating direction multipliers method, which is based on the augmented lagrangian algorithm, which itself can be seen as a direct application of the proximal point algorithm to the dual problem of a constrained optimization problem.The aim of this special issue is to present new approaches and theories for variational inequalities arising in mathematics and applied sciences. This special issue includes 14 highquality peer-reviewed papers that deal with different aspects of variational inequalities. These papers contain some new, novel, and innovative techniques and ideas. We hope that all the papers published in this special issue can motivate and foster further scientific works and development of the research in the area of theory, algorithms, and applications of variational inequalities.The summaries of the 14 papers in this issue are listed as follows.The paper of C. Chen et al. considers a class of linearly constrained separable convex programming problems without coupled variables. They weaken some conditions to obtain convergence of the alternating direction method of multipliers and they propose also a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions. The paper of H. Sun and Y. Wang revisits the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP) and sharpens the global error bound for the GNCP under weaker conditions, which improves the existing error bound estimation for the problem.The paper of M. Ma concerns the design and the convergence analysis of algorithms to split variational inequality and equilibrium problems.The paper of Y. Wang and C. Wang gives a new modified Ishikawa type iteration algorithm for common fixed points of total asymptotically strict pseudocontractive semigroups. Abstract and Applied AnalysisStrong and weak convergence are proved under mild conditions. Further...

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The Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Cited by 2 publications

References 17 publications

The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

Some Recent Trends in Variational Inequalities and Optimization Problems with Applications

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