General Inertial Mann Algorithms and Their Convergence Analysis for Nonexpansive Mappings

In this article, we introduce general inertial Mann algorithms for finding fixed points of nonexpansive mappings in Hilbert spaces, which includes some other algorithms as special cases. We reanalyze the accelerated Mann algorithm, which actually is an inertial type Mann algorithm. We investigate the convergence of the general inertial Mann algorithm, based on which, the strict convergence condition on the accelerated Mann algorithm is eliminated. Also, we apply the general inertial Mann algorithm to show the existence of solutions of the minimization problems by proposing a general inertial type gradient-projection algorithm. Finally, we give preliminary experiments to illustrate the advantage of the accelerated Mann algorithm.

[1]  É. Picard.  Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives , 1890 .

[2]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[3]  Boris Polyak.  Some methods of speeding up the convergence of iteration methods , 1964 .

[4]  B. Halpern.  Fixed points of nonexpanding maps , 1967 .

[5]  S. Ishikawa.  Fixed points by a new iteration method , 1974 .

[6]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[7]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[8]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[9]  H. Attouch,et al.  An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping , 2001 .

[10]  A. Moudafi,et al.  Convergence of a splitting inertial proximal method for monotone operators , 2003 .

[11]  Felipe Alvarez,et al.  Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space , 2003, SIAM J. Optim..

[12]  P. Maingé.  Convergence theorems for inertial KM-type algorithms , 2008 .

[13]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[14]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[15]  Hong-Kun Xu,et al.  Averaged Mappings and the Gradient-Projection Algorithm , 2011, J. Optim. Theory Appl..

[16]  C. Micchelli,et al.  Proximity algorithms for image models: denoising , 2011 .

[17]  Hideaki Iiduka,et al.  Iterative Algorithm for Triple-Hierarchical Constrained Nonconvex Optimization Problem and Its Application to Network Bandwidth Allocation , 2012, SIAM J. Optim..

[18]  P. Pardalos,et al.  Nonlinear analysis : stability, approximation, and inequalities : in honor of Themistocles M. Rassias on the occasion of his 60th birthday , 2012 .

[19]  Hideaki Iiduka,et al.  Fixed Point Optimization Algorithms for Distributed Optimization in Networked Systems , 2013, SIAM J. Optim..

[20]  Xiaoqun Zhang,et al.  A primal–dual fixed point algorithm for convex separable minimization with applications to image restoration , 2013 .

[21]  Dirk A. Lorenz,et al.  An Inertial Forward-Backward Algorithm for Monotone Inclusions , 2014, Journal of Mathematical Imaging and Vision.

[22]  R. Boţ,et al.  A Hybrid Proximal-Extragradient Algorithm with Inertial Effects , 2014, 1407.0214.

[23]  Boundary point algorithms for minimum norm fixed points of nonexpansive mappings , 2014 .

[24]  General alternative regularization methods for nonexpansive mappings in Hilbert spaces , 2014 .

[25]  L. Tóth,et al.  Topics in Mathematical Analysis and Applications , 2014 .

[26]  Kaito Sakurai,et al.  Acceleration of the Halpern algorithm to search for a fixed point of a nonexpansive mapping , 2014 .

[27]  Yu. V. Malitsky,et al.  Projected Reflected Gradient Methods for Monotone Variational Inequalities , 2015, SIAM J. Optim..

[28]  Y. Cho,et al.  A new hybrid algorithm and its numerical realization for two nonexpansive mappings , 2015 .

[29]  Q. Dong,et al.  Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping , 2015 .

[30]  Radu Ioan Bot,et al.  Inertial Douglas-Rachford splitting for monotone inclusion problems , 2014, Appl. Math. Comput..

[31]  A. Chambolle,et al.  On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm” , 2015, J. Optim. Theory Appl..

[32]  Xiaoqun Zhang,et al.  A primal-dual fixed point algorithm for minimization of the sum of three convex separable functions , 2015, 1512.09235.

[33]  Q. Dong,et al.  A new hybrid algorithm for a nonexpansive mapping , 2015 .

[34]  Paul-Emile Maingé,et al.  Convergence of One-Step Projected Gradient Methods for Variational Inequalities , 2016, J. Optim. Theory Appl..

[35]  P. Pardalos,et al.  Contributions in Mathematics and Engineering , 2016 .

[36]  Paul-Emile Maingé.  Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure , 2016, Comput. Math. Appl..

[37]  Q. Dong,et al.  The extragradient algorithm with inertial effects for solving the variational inequality , 2016 .