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Strongly convergent inertial projection and contraction methods for split variational inequality problem

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO 2024年第5期73卷 2069-2106页

作者： Mewomo, O. T. Ogwo, G. N. Alakoya, T. O. Izuchukwu, C. Univ KwaZulu Natal Sch Math Stat & Comp Sci Durban South Africa

In the literature, several methods have been proposed for solving the split variational inequality problems and most of these methods require that the underlying operators are co-coercive while some of them require that the problem is transformed into a product space. These restrictive conditions affect the feasibility of these existing methods. In order to overcome these setbacks, we propose two new inertial projection and contraction methods for solving the split variational inequality problem in real Hilbert spaces without the co-coercive condition and without the product space formulation, which does not fully exploit the attractive splitting structure of the split variational inequality problem. The sequences generated by these methods converge strongly to the solution of the split varitional inequality problems in real Hilbert spaces under the assumptions that the operators are pseudomonotone, Lipschitz continuous and without the sequentially weakly continuity condition. Furthermore, we present several numerical experiments for the proposed methods and compare their performance with other related methods in the literature.

关键词： Split variational inequality problems Pseudomonotone operators Lipschitz continuous projection and contraction methods Inertial extrapolation Minimum-norm solutions Product space formulation

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Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity

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NUMERICAL ALGORITHMS 2021年第3期88卷 1419-1456页

作者： Ogwo, G. N. Izuchukwu, C. Mewomo, O. T. Univ KwaZulu Natal Sch Math Stat & Comp Sci Durban South Africa

In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators to be co-coercive. This restrictive co-coercive assumption has been dispensed with in some methods, many of which require a product space formulation of the problem. However, it has been discovered that this product space formulation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting structure of the split variational inequality problem. In this paper, we present two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation. We prove that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous. Also, we provide several numerical experiments of the proposed methods in comparison with other related methods in the literature.

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从变分不等式的投影收缩算法到凸优化的分裂收缩算法

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高等学校计算数学学报 2016年第1期38卷 74-96页

作者：何炳生南京大学数学系

1引言由于互补问题(变分不等式的一种特殊情形)能用来描述经济与管理方面的平衡问题,我们对变分不等式的求解给予了极大关注.实际生活中的问题,函数只是一种对应关系,通常没有显式表达式,对一个给定的点,要获得相应的函数值叉往往要通... 详细信息

1引言由于互补问题(变分不等式的一种特殊情形)能用来描述经济与管理方面的平衡问题,我们对变分不等式的求解给予了极大关注.实际生活中的问题,函数只是一种对应关系,通常没有显式表达式,对一个给定的点,要获得相应的函数值叉往往要通过一次实验,代价不菲.求解这种"黑箱函数"的问题,需要只用函数值且少用函数值的方法.从管理科学中来的变分不等式问题,函数往往是黑箱的.受华罗庚先生推广优选法的影响,对单调

关键词： variational inequality projection and contraction methods convex optimization splitting contraction methods

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A new criterion for an inexact parallel splitting augmented Lagrangian method

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2015年第1期2015卷 1-18页

作者： Kou, Xipeng Li, Shengjie Wang, Xueyong Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

In this paper, we study the computational method for solving the variational inequality problem with the separable structure and linear constraints. We propose a new relaxed inexact criterion and a prediction-correction approach in the inexact splitting parallel augmented Lagrangian methods, which make it easier to solve the resulting subproblems. Under a mild condition, we prove the global convergence and establish a worst-case convergence rate for the new inexact algorithm. Some numerical experiments show the effectiveness and feasibility of the new inexact method.

关键词： variational inequalities proximal point method decomposition splitting method projection and contraction methods

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An Inexact Alternating Direction Method for Structured Variational Inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第2期163卷 439-459页

作者： Chen, Zhongming Wan, Li Yang, Qingzhi Nankai Univ Sch Math Sci Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China

Recently, the alternating direction method of multipliers has attracted great attention. For a class of variational inequalities (VIs), this method is efficient, when the subproblems can be solved exactly. However, the subproblems could be too difficult or impossible to be solved exactly in many practical applications. In this paper, we propose an inexact method for structured VIs based on the projection and contraction method. Instead of solving the subproblems exactly, we use the simple projection to get a predictor and correct it to approximate the subproblems' real solutions. The convergence of the proposed method is proved under mild assumptions and its efficiency is also verified by some numerical experiments.

关键词： Variational inequalities Alternating direction methods projection and contraction methods

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A new class of projection and contraction methods for solving variational inequality problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2006年第6-7期51卷 937-950页

作者： Han, D Nanjing Normal Univ Sch Math & Comp Sci Nanjing 210097 Peoples R China

This paper presents a new class of projection and contraction methods for solving monotone variational inequality problems. The methods can be viewed as combinations of some existing projection and contraction methods and the method of shortest residuals, a special case of conjugate gradient methods for solving unconstrained nonlinear programming problems. Under mild assumptions, we show the global convergence of the methods. Some preliminary computational results are reported to show the efficiency of the methods. (C) 2006 Elsevier Ltd. All rights reserved.

关键词： variational inequality problems projection and contraction methods conjugate gradient methods global convergence

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Comparison of two approximal proximal point algorithms for monotone variational inequalities

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2007年第6期8卷 969-977页

作者： TAO Min School of Applied Mathematics and Physics Nanjing University of Posts and Telecommunications Nanjing 210003 China

Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.

关键词： projection and contraction methods Proximal point algorithm （PPA） Approximate PPA （APPA） Monotone variational inequality （MVI） Prediction and correction

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Prediction-correction alternating direction method for a class of constrained min-max problems

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Frontiers of Mathematics in China 2007年第1期2卷 103-121页

作者： Li, Min He, Bingsheng Department of Mathematics Nanjing University Nanjing 210093 China

The problems concerned in this paper are a class of constrained min-max problems. By introducing the Lagrange multipliers to the linear constraints, such problems can be solved by some projection type prediction-correction methods. However, to obtain components of the predictor one by one, we use an alternating direction method. And then the new iterate is generated by a minor correction. Global convergence of the proposed method is proved. Finally, numerical results for a constrained single-facility location problem are provided to verify that the new method is effective for some practical problems. © Higher Education Press 2007.

关键词： Constrained minimum distance problem Linear variational inequality Nonlinear programming projection and contraction methods

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Prediction-correction alternating direction method for a class of constrained rain-max problems

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中国高等学校学术文摘·数学 2007年第1期2卷 103-121页

作者： LI Min HE Bingsheng Department of Mathematics Nanjing University Nanjing 210093 China

The problems concerned in this paper are a class of constrained min-max problems. By introducing the Lagrange multipliers to the linearconstraints, such problems can be solved by some projection type prediction-correction methods. However, to obtain components of the predictor one by one, we use an alternating direction method. And then the new iterate is generated by a minor correction. Global convergence of the proposed method is proved. Finally, numerical results for a constrained single-facility location problem are provided to verify that the new method is effective for some practical problems.

关键词： nonlinear programming constrained minimum distance problem linear variational inequality projection and contraction methods

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Solving linear variational inequality problems by a self-adaptive projection method

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期182卷 1765-1771页

作者： Han, Deren Nanjing Normal Univ Sch Math & Comp Sci Nanjing 210097 Peoples R China

We propose a self-adaptive projection method for solving linear variational inequality problems and show its global convergence under mild conditions. Some numerical results are also addressed which indicate that the ... 详细信息

关键词： linear variational inequality problems projection and contraction methods global convergence

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