一类新的推广非凸变分不等式的平行投影算法(英文)
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摘要
对定义在一致临近正则集上的一类新的推广的非凸变分不等式,本文提出了一个平行投影算法,算法的收敛点既是该变分不等式的解,又是两个Lipschitz映像的不动点。进一步,本文在适当条件下证明了该算法的收敛性。本文所得结论改进并推广了有关变分不等式和相关最优化问题的一些结果。
For a system of general nonconvex variational inequalities defined on uniformly prox-regular sets,we propose a parallel projection algorithm which converges to its solution and common fixed points of two Lipschitzian mappings.We further consider the convergence of the algorithm under some suitable conditions.Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems.
For a system of general nonconvex variational inequalities defined on uniformly prox-regular sets,we propose a parallel projection algorithm which converges to its solution and common fixed points of two Lipschitzian mappings.We further consider the convergence of the algorithm under some suitable conditions.Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems.
引文
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[4]Noor M A.Projection methods for nonconvex variational inequalities[J].Optim Lett,2009,3(3):411-418.
[5]Noor M A.Iterative methods for general nonconvex variational inequalities[J].Albanian J Math,2009,3(1):117-127.
[6]Verma R U.General Convergence analysis for two-step projection methods and applications to variational problems[J].Appl Math Lett,2005,18(14):1286-1292.
[7]Noor M A.On implicit methods for nonconvex variational inequalities[J].J Optim Theory Appl,2010,147(2):411-417.
[8]Thakur B S,Khan M S,Kang S M.Existence and approximation of solutions for system of generalized mixed variational inequalities[J].Fixed Point Theory and Applications,2013,2013(1):1-15.
[9]Plubtieng S,Thammathiwat T.Existence of solutions of a new system of generalized variational inequalities in Banach spaces[J].Journal of Inequalities and Applications,2012,2012(1):1-10.
[10]Wen D J,Long X J,Gong Q F.Convergence analysis of projection methods for a new system of general nonconvex variational inequalities[J].Fixed Point Theory and Applications,2012,2012(1):1-10.
[11]Noor M A,Noor K I.Projection algorithms for solving system of general variational inequalities[J].Nonlinear Anal,2009,70(7):2700-2706.
[12]Noor M A.Some iterative methods for non-convex variational inequalities[J].Comput Math Model,2010,21(1):97-108.
[13]Weng X L.Fixed point iteration for local strictly pseudocontractive mapping[J].Proc Am Math Soc,1994,113(4):727-731.
[2]Clarke F H,Ledyaev Y S,Wolenski P R.Nonsmooth Analysis and Control Theory[M].Berlin:Springer-Verlag New York Inc,1998.
[3]Poliquin R A,Rockafellar R T,Thibault L.Local differentiability of distance functions[J].Trans Am Math Soc,2000,352(11):5231-5249.
[4]Noor M A.Projection methods for nonconvex variational inequalities[J].Optim Lett,2009,3(3):411-418.
[5]Noor M A.Iterative methods for general nonconvex variational inequalities[J].Albanian J Math,2009,3(1):117-127.
[6]Verma R U.General Convergence analysis for two-step projection methods and applications to variational problems[J].Appl Math Lett,2005,18(14):1286-1292.
[7]Noor M A.On implicit methods for nonconvex variational inequalities[J].J Optim Theory Appl,2010,147(2):411-417.
[8]Thakur B S,Khan M S,Kang S M.Existence and approximation of solutions for system of generalized mixed variational inequalities[J].Fixed Point Theory and Applications,2013,2013(1):1-15.
[9]Plubtieng S,Thammathiwat T.Existence of solutions of a new system of generalized variational inequalities in Banach spaces[J].Journal of Inequalities and Applications,2012,2012(1):1-10.
[10]Wen D J,Long X J,Gong Q F.Convergence analysis of projection methods for a new system of general nonconvex variational inequalities[J].Fixed Point Theory and Applications,2012,2012(1):1-10.
[11]Noor M A,Noor K I.Projection algorithms for solving system of general variational inequalities[J].Nonlinear Anal,2009,70(7):2700-2706.
[12]Noor M A.Some iterative methods for non-convex variational inequalities[J].Comput Math Model,2010,21(1):97-108.
[13]Weng X L.Fixed point iteration for local strictly pseudocontractive mapping[J].Proc Am Math Soc,1994,113(4):727-731.