基于投影交替方向法求解结构型单调变分不等式
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摘要
通过构造新的下降方向对孙敏等人给出的投影型交替方向法进行改进和推广,提出了改进投影型交替方向法.与前者相比较,该方法具有收敛速度快,迭代次数少的特点.在相同的假设条件下,证明了新方法的全局收敛性,并通过数值试验初步验证了该方法的有效性.
Alternating direction method plays an important role in solving structured monotone variational inequality problems.In this paper,we propose a modified projection alternating direction method by constructing a new descent direction,which improves and promotes the projection alternating direction method given by Sun,M.et al.Compared with Suns method,the proposed method has greater speed of convergence and fewer iterations.Under the same assumptions,we prove the global convergence property of the method,and then test its effectiveness with a numerical example.
Alternating direction method plays an important role in solving structured monotone variational inequality problems.In this paper,we propose a modified projection alternating direction method by constructing a new descent direction,which improves and promotes the projection alternating direction method given by Sun,M.et al.Compared with Suns method,the proposed method has greater speed of convergence and fewer iterations.Under the same assumptions,we prove the global convergence property of the method,and then test its effectiveness with a numerical example.
引文
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[2]YE C H,YUAN X M.A Descent Method for Structured Monotone Variational Inequalities[J].Optimization Methods and Software,2007,22(2):329-338.
[3]HAN D R.A Modified Alternating Direction Method for Variational Inequality Problems[J].Applied Mathematics and Optimization,2002,45(1):63-74.
[4]何炳生.一类广义线性变分不等式的求解与应用[J].中国科学:A辑,1995,38(9):939-945.
[5]孙敏,徐健腾,时贞军.一种新的投影型变分不等式交替方向方法[J].工程数学学报,2006,23(6):1101-1104.
[6]孙敏.求解结构型单调变分不等式的投影类交替方向法[J].安徽大学学报(自然科学版),2009,33(2):12-14.
[7]HE B S,LIAO L Z.Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities[J].Journal of Optimization Theory and Applications,2002,112(1):111-128.
[8]GAFNI E M,BERTSEKAS D P.Two-Metric Projection Methods for Constrained Optimization[J].SIAM Journal on Control and Optimization,1984,22(6):936-964.
[9]张敏,韩德仁,何洪津,等.解可分离结构变分不等式的一种新的交替方向法[J].中国科学:数学,2012,42(2):133-149.
[2]YE C H,YUAN X M.A Descent Method for Structured Monotone Variational Inequalities[J].Optimization Methods and Software,2007,22(2):329-338.
[3]HAN D R.A Modified Alternating Direction Method for Variational Inequality Problems[J].Applied Mathematics and Optimization,2002,45(1):63-74.
[4]何炳生.一类广义线性变分不等式的求解与应用[J].中国科学:A辑,1995,38(9):939-945.
[5]孙敏,徐健腾,时贞军.一种新的投影型变分不等式交替方向方法[J].工程数学学报,2006,23(6):1101-1104.
[6]孙敏.求解结构型单调变分不等式的投影类交替方向法[J].安徽大学学报(自然科学版),2009,33(2):12-14.
[7]HE B S,LIAO L Z.Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities[J].Journal of Optimization Theory and Applications,2002,112(1):111-128.
[8]GAFNI E M,BERTSEKAS D P.Two-Metric Projection Methods for Constrained Optimization[J].SIAM Journal on Control and Optimization,1984,22(6):936-964.
[9]张敏,韩德仁,何洪津,等.解可分离结构变分不等式的一种新的交替方向法[J].中国科学:数学,2012,42(2):133-149.