非凸变分不等式的三步投影算法及其收敛性分析
|
摘要
利用非凸变分不等式和不动点问题的等价关系,建立了一个新的求解非凸变分不等式的三步投影算法;该算法在现有的两步迭代算法基础上,利用校正方法建立了第三步迭代公式;最后在适当条件下证明了该算法的收敛性.
A new three-step projective algorithm is proposed for non-convex variational inequalities based on the equivalence between the non-convex variational inequalities and the fixed point problems,which included the known two-step iterative algorithms as a special case,and modified iterative algorithms with the third-step iteration scheme using the technique of updating. The convergence criteria of the algorithms are proved under some suitable conditions.
A new three-step projective algorithm is proposed for non-convex variational inequalities based on the equivalence between the non-convex variational inequalities and the fixed point problems,which included the known two-step iterative algorithms as a special case,and modified iterative algorithms with the third-step iteration scheme using the technique of updating. The convergence criteria of the algorithms are proved under some suitable conditions.
引文
[1]Stampacchia G.Formes bilinéaires coercitives sur les ensembles convexes[J].C R Acad Sci Paris,1964,258:4413-4416.
[2]Noor M A.Iterative schemes for nonconvex variational inequalities[J].J Optim Theory Appl,2004,121(2):385-395.
[3]Noor M A.Projection methods for nonconvex variational inequalities[J].Optim Lett,2009,3(3):411-418.
[4]Noor M A.Some iterative methods for nonconvex variational inequalities[J].Comput Math Model,2010,21(1):97-108.
[5]Clarke F H,Ledyaev Y S,Wolenski P R.Nonsmooth Analysis and Control Theory[M].New York,Berlin:Springer-Verlag,1998.
[6]Poliquin R A,Rockafellar R T,Thibault L.Local differentiability of distance functions[J].Trans Amer Math Soc,2000,352(11):5231-5249.
[7]Noor M A.Differentiable nonconvex functions and general variational inequalities[J].Appl Math Comput,2008,199(2):623-630.
[8]Wen D J.Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators[J].Nonlinear Anal,2010,73(7):2292-2297.
[9]Xu H K.Iterative algorithms for nonlinear operators[J].J Lond Math Soc,2002,66(1):240-256.
[10]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.
[11]Noor M A.On implicit methods for nonconvex variational inequalities[J].J Optim Theor Appl,2010,147(2):411-417.
[12]Lions J L.Parallel algorithms for the solution of variational inequalities[J].Interfaces and Free Boundaries,1999,1(1):3-16.
[13]Yang H,Zhou L,Li Q.A parallel projection method for a system of nonlinear variational inequalities[J].Appl Math Comput,2010,217(5):1971-1975.
[14]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 Theor Appl,2012,2012(1):1-10.
[15]Bnouhachem A,Noor M A,Russia T M.Three-step iterative algorithms for mixed variational inequalities[J].Appl Math Comput,2006,183(1):436-446.
[16]Plubtieng S,Thammathiwat T.Existence of solutions of a new system of generalized variational inequalities in Banach spaces[J].J Inequal Appl,2012,2012(8):1-10.
[17]王文惠,万波.广义混合似变分不等式组的两步迭代算法[J].数学的实践与认识,2009,39(14):126-131.
[18]艾艺红.混合变分不等式和非扩张映射解的迭代算法[J].四川师范大学学报:自然科学版,2012,35(1):21-26.
[19]李艳,夏福全.Banach空间中广义混合变分不等式解的迭代算法[J].四川师范大学学报:自然科学版,2011,34(1):13-19.
[20]闻道君,邓磊.一般变分不等式的三步迭代算法[J].四川师范大学学报:自然科学版,2009,32(4):436-438.
[2]Noor M A.Iterative schemes for nonconvex variational inequalities[J].J Optim Theory Appl,2004,121(2):385-395.
[3]Noor M A.Projection methods for nonconvex variational inequalities[J].Optim Lett,2009,3(3):411-418.
[4]Noor M A.Some iterative methods for nonconvex variational inequalities[J].Comput Math Model,2010,21(1):97-108.
[5]Clarke F H,Ledyaev Y S,Wolenski P R.Nonsmooth Analysis and Control Theory[M].New York,Berlin:Springer-Verlag,1998.
[6]Poliquin R A,Rockafellar R T,Thibault L.Local differentiability of distance functions[J].Trans Amer Math Soc,2000,352(11):5231-5249.
[7]Noor M A.Differentiable nonconvex functions and general variational inequalities[J].Appl Math Comput,2008,199(2):623-630.
[8]Wen D J.Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators[J].Nonlinear Anal,2010,73(7):2292-2297.
[9]Xu H K.Iterative algorithms for nonlinear operators[J].J Lond Math Soc,2002,66(1):240-256.
[10]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.
[11]Noor M A.On implicit methods for nonconvex variational inequalities[J].J Optim Theor Appl,2010,147(2):411-417.
[12]Lions J L.Parallel algorithms for the solution of variational inequalities[J].Interfaces and Free Boundaries,1999,1(1):3-16.
[13]Yang H,Zhou L,Li Q.A parallel projection method for a system of nonlinear variational inequalities[J].Appl Math Comput,2010,217(5):1971-1975.
[14]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 Theor Appl,2012,2012(1):1-10.
[15]Bnouhachem A,Noor M A,Russia T M.Three-step iterative algorithms for mixed variational inequalities[J].Appl Math Comput,2006,183(1):436-446.
[16]Plubtieng S,Thammathiwat T.Existence of solutions of a new system of generalized variational inequalities in Banach spaces[J].J Inequal Appl,2012,2012(8):1-10.
[17]王文惠,万波.广义混合似变分不等式组的两步迭代算法[J].数学的实践与认识,2009,39(14):126-131.
[18]艾艺红.混合变分不等式和非扩张映射解的迭代算法[J].四川师范大学学报:自然科学版,2012,35(1):21-26.
[19]李艳,夏福全.Banach空间中广义混合变分不等式解的迭代算法[J].四川师范大学学报:自然科学版,2011,34(1):13-19.
[20]闻道君,邓磊.一般变分不等式的三步迭代算法[J].四川师范大学学报:自然科学版,2009,32(4):436-438.