一种求解分裂共同半压缩映射不动点问题的迭代算法
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摘要
分裂共同半压缩映射不动点问题是一类比较经典的问题模型,目前算法多是运用当前迭代点的信息构建新的迭代点,这类算法收敛比较慢,且仅具有线性收敛性。为构建快速有效算法,受惯性近似算法求解极大单调算子零点问题的启发,针对半压缩映射的共同分裂不动点问题,通过引入惯性因子,提出一种具有二次收敛性的惯性迭代算法,并在适当条件下证明了算法的渐近收敛性。惯性技术的应用能提高迭代序列的收敛速率,减少迭代步数,从而大大减少计算量。
Split common fixed-point problem for demicontractive mappings is a classic problem model. Currently,most algorithms employ existed point to obtain the next iterate which leads to slow convergence(only linear convergence). In order to construct fast and valid algorithm,the inertial proximal algorithm is inspired to find zero of a maximal monotone operator. In this paper,an inertial iteration algorithm is proposed to solve the split common fixed point problem for demicontractive mappings by introducing a inertial factor. The asymptotical convergence of the algorithm is also proved under some suitable conditions. The algorithm constructed by applying inertial technique can improve convergence rate of the iterative sequence.
Split common fixed-point problem for demicontractive mappings is a classic problem model. Currently,most algorithms employ existed point to obtain the next iterate which leads to slow convergence(only linear convergence). In order to construct fast and valid algorithm,the inertial proximal algorithm is inspired to find zero of a maximal monotone operator. In this paper,an inertial iteration algorithm is proposed to solve the split common fixed point problem for demicontractive mappings by introducing a inertial factor. The asymptotical convergence of the algorithm is also proved under some suitable conditions. The algorithm constructed by applying inertial technique can improve convergence rate of the iterative sequence.
引文
[1]HERMAN G T.Image reconstruction from projections:The fundamentals of computerized tomography[M].New York:Academic Press,1980.
[2]BAUSCHKE H H,BORWEIN J M.On projection algorithms for solving convex feasibility problems[J].SIAM Review,1996,38:367.
[3]CENSOR Y,ELFVING T,KOOF N,et al.The multiple-sets split feasibility problem and its applications for inverse problems[J].Inverse Problems,2005,21:2071.
[4]DANG Y,GAO Y.The strong convergence of a KMCQ-Like algorithm for split feasibility problem[J].Inverse Problems,2011,27:015007.
[5]BYRNE C.Iterative oblique projection onto convex sets and the split feasibility problem[J].Inverse Problems,2002,18:441.
[6]QU B,XIU N.A new halfspace-relaxation projection method for the split feasibility problem[J].Linear Algebra and Its Application,2008,428:1218.
[7]YANG Q.The relaxed CQ algorithm solving the split feasibility problem[J].Inverse Problems,2004,20:1261.
[8]ZHAO J L,YANG Q Z.Several solution methods for the split feasibility problem[J].Inverse Problems,2005,21:1791.
[9]CROMBEZ G.A geometrical look at iterative methods for operators with fixed points[J].Numerical Analysis and Optimization,2005,26:137.
[10]BYRNE C.An unifed treatment of some iterative algorithms in signal processing and image reconstruction[J].Inverse Problems,2004,20:103.
[11]ALVAREZ F,ATTOUCH H.An inertial proximal method for maximal monotone operators via Discretization of a nonlinear oscillator with damping[J].SetValued Analysis,2001,9:3.
[12]ALVAREZ F.Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space[J].SIAM Journal on Optimization,2004,3:773.
[13]MAINGE P E.Inertial iterative process for fixed points of certain quasi-nonexpansive mappings[J].Set-valued Analysis,2007,15,67.
[14]MOUDA A.The split common fixed-poiny problem for demicontractive mappings[J].Inverse Problem,2010,26:055007.doi:10.1088/0266-5611/26/5/055007
[15]OPIAL Z.Weak convergence of the sequence of successive approximations for nonexpansive mappings,Bull[J].American Mathematical Society,1967,73:591.
[2]BAUSCHKE H H,BORWEIN J M.On projection algorithms for solving convex feasibility problems[J].SIAM Review,1996,38:367.
[3]CENSOR Y,ELFVING T,KOOF N,et al.The multiple-sets split feasibility problem and its applications for inverse problems[J].Inverse Problems,2005,21:2071.
[4]DANG Y,GAO Y.The strong convergence of a KMCQ-Like algorithm for split feasibility problem[J].Inverse Problems,2011,27:015007.
[5]BYRNE C.Iterative oblique projection onto convex sets and the split feasibility problem[J].Inverse Problems,2002,18:441.
[6]QU B,XIU N.A new halfspace-relaxation projection method for the split feasibility problem[J].Linear Algebra and Its Application,2008,428:1218.
[7]YANG Q.The relaxed CQ algorithm solving the split feasibility problem[J].Inverse Problems,2004,20:1261.
[8]ZHAO J L,YANG Q Z.Several solution methods for the split feasibility problem[J].Inverse Problems,2005,21:1791.
[9]CROMBEZ G.A geometrical look at iterative methods for operators with fixed points[J].Numerical Analysis and Optimization,2005,26:137.
[10]BYRNE C.An unifed treatment of some iterative algorithms in signal processing and image reconstruction[J].Inverse Problems,2004,20:103.
[11]ALVAREZ F,ATTOUCH H.An inertial proximal method for maximal monotone operators via Discretization of a nonlinear oscillator with damping[J].SetValued Analysis,2001,9:3.
[12]ALVAREZ F.Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space[J].SIAM Journal on Optimization,2004,3:773.
[13]MAINGE P E.Inertial iterative process for fixed points of certain quasi-nonexpansive mappings[J].Set-valued Analysis,2007,15,67.
[14]MOUDA A.The split common fixed-poiny problem for demicontractive mappings[J].Inverse Problem,2010,26:055007.doi:10.1088/0266-5611/26/5/055007
[15]OPIAL Z.Weak convergence of the sequence of successive approximations for nonexpansive mappings,Bull[J].American Mathematical Society,1967,73:591.