2016, 6(4): 505-519. doi: 10.3934/naco.2016023

Convergence analysis of a parallel projection algorithm for solving convex feasibility problems

1. 

Business School, University of Shanghai for Science and Technology, Shanghai, China

2. 

Department of Health Services and Outcomes Research, National Healthcare Group, 138543

3. 

Department of Mathematics and Statistics, Curtin University, Perth,WA 6845

Received  January 2016 Revised  November 2016 Published  December 2016

The convex feasibility problem (CFP) is a classical problem in nonlinear analysis. In this paper, we propose an inertial parallel projection algorithm for solving CFP. Different from the previous algorithms, the proposed method introduces a sequence of parameters and uses the information of last two iterations at each step. To prove its convergence in a simple way, we transform the parallel algorithm to a sequential one in a constructed product space. Preliminary experiments are conducted to demonstrate that the proposed approach converges faster than the general extrapolated algorithms.
Citation: Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023
References:
[1]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems,, \emph{SIAM Rev.}, 38 (1996), 367.  doi: 10.1137/S0036144593251710.  Google Scholar

[2]

H. H. Bauschke, J. M.Borwein, and A. S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space,, in \emph{Proceedings of the Special Session on Optimization and Nonlinear Analysis}, (1995).   Google Scholar

[3]

S. Boyd, L. EI Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[4]

Y. Censor and A. Lent, Cyclic subgradient projections,, \emph{Math. Program.}, 24 (1982), 233.  doi: 10.1007/BF01585107.  Google Scholar

[5]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy,, \emph{Math. Program.}, 42 (1998), 307.  doi: 10.1007/BF01589408.  Google Scholar

[6]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs,, \emph{INFORMS J. Comput.}, 16 (2004), 255.  doi: 10.1287/ijoc.1030.0046.  Google Scholar

[7]

P. L. Combeties and S. G. Kruk, Extroplation algorithm for affine-convex feasibility problems,, \emph{Numer. Algorithms}, 41 (2006), 239.  doi: 10.1007/s11075-005-9010-6.  Google Scholar

[8]

Y. Dang and Y. Gao, Non-monotonous accelerated parallel subgradient projection algorithm for convex feasibility problem,, \emph{Optimization}, 63 (2014), 571.  doi: 10.1080/02331934.2012.677447.  Google Scholar

[9]

F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications,, Kluwer Academic Publishers, (1992).   Google Scholar

[10]

Y. Gao, Viability criteria for differential inclusions,, \emph{J. Syst. Sci. Complex.}, 24 (2011), 825.  doi: 10.1007/s11424-011-9056-6.  Google Scholar

[11]

U. Garcia-palomares, Parallel projected aggregation methods for solving the convex feasibility problem,, \emph{SIAM J. Optim.}, 3 (1993), 882.  doi: 10.1137/0803046.  Google Scholar

[12]

U. M. Garcia-Palomares and F. J. Gonzalez-Castano, Incomplete projection algorithms for solving the convex feasibility problem,, \emph{Numer. Algorithms}, 18 (1998), 177.  doi: 10.1023/A:1019165330848.  Google Scholar

[13]

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory,, Cambridge Studies in Advanced Mathematics 28, (1990).  doi: 10.1017/CBO9780511526152.  Google Scholar

[14]

N. I. M. Gould, How good are projection methods for convex feasibility problems,, \emph{Comput. Optim. Appl.}, 40 (2008), 1.  doi: 10.1007/s10589-007-9073-5.  Google Scholar

[15]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography,, Academic Press, (1980).   Google Scholar

[16]

K. C. Kiwiel, Block-iterative surrogate projection methods for convex feasibility problem,, \emph{Linear Algebra Appl.}, 215 (1995), 225.  doi: 10.1016/0024-3795(93)00089-I.  Google Scholar

[17]

L. Li and Y. Gao, Approximate subgradient projection algorithm for a convex feasibility problem,, \emph{J. Syst. Eng. Electron.}, 21 (2010), 527.   Google Scholar

[18]

T. Lucio, A parallel subgradient projections method for the convex feasibility problem,, \emph{J. Comput. Appl. Math.}, 18 (1987), 307.  doi: 10.1016/0377-0427(87)90004-5.  Google Scholar

[19]

P. E. Mainge, Convergence theorem for inertial KM-type algorithms,, \emph{J. Comput. Appl. Math.}, 219 (2008), 223.  doi: 10.1016/j.cam.2007.07.021.  Google Scholar

[20]

M. Moudafi, Convergence of a splitting inertial proximal method for monotone operators,, \emph{J. Comput. Appl. Math.}, 155 (2008), 447.  doi: 10.1016/S0377-0427(02)00906-8.  Google Scholar

[21]

A. Moudafi and E. Elisabeth, An approximate inertial proximal method using enlargement of a maximal monotone operator,, \emph{Int. J. Pure Appl. Math.}, 5 (2003), 283.   Google Scholar

[22]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, \emph{Bull. Am. Math. Soc.}, 73 (1967), 591.   Google Scholar

[23]

G. Pierra, Decompasition through formalization in a product space,, \emph{Math. Program.}, 28 (1984), 96.  doi: 10.1007/BF02612715.  Google Scholar

[24]

B. T. Polyak, Some methods of speeding up the convergence of iteration method,, \emph{Zh. Vych. Math.}, 4 (1964), 791.   Google Scholar

show all references

References:
[1]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems,, \emph{SIAM Rev.}, 38 (1996), 367.  doi: 10.1137/S0036144593251710.  Google Scholar

[2]

H. H. Bauschke, J. M.Borwein, and A. S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space,, in \emph{Proceedings of the Special Session on Optimization and Nonlinear Analysis}, (1995).   Google Scholar

[3]

S. Boyd, L. EI Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Society for Industrial and Applied Mathematics, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[4]

Y. Censor and A. Lent, Cyclic subgradient projections,, \emph{Math. Program.}, 24 (1982), 233.  doi: 10.1007/BF01585107.  Google Scholar

[5]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy,, \emph{Math. Program.}, 42 (1998), 307.  doi: 10.1007/BF01589408.  Google Scholar

[6]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs,, \emph{INFORMS J. Comput.}, 16 (2004), 255.  doi: 10.1287/ijoc.1030.0046.  Google Scholar

[7]

P. L. Combeties and S. G. Kruk, Extroplation algorithm for affine-convex feasibility problems,, \emph{Numer. Algorithms}, 41 (2006), 239.  doi: 10.1007/s11075-005-9010-6.  Google Scholar

[8]

Y. Dang and Y. Gao, Non-monotonous accelerated parallel subgradient projection algorithm for convex feasibility problem,, \emph{Optimization}, 63 (2014), 571.  doi: 10.1080/02331934.2012.677447.  Google Scholar

[9]

F. Deutsch, The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications,, Kluwer Academic Publishers, (1992).   Google Scholar

[10]

Y. Gao, Viability criteria for differential inclusions,, \emph{J. Syst. Sci. Complex.}, 24 (2011), 825.  doi: 10.1007/s11424-011-9056-6.  Google Scholar

[11]

U. Garcia-palomares, Parallel projected aggregation methods for solving the convex feasibility problem,, \emph{SIAM J. Optim.}, 3 (1993), 882.  doi: 10.1137/0803046.  Google Scholar

[12]

U. M. Garcia-Palomares and F. J. Gonzalez-Castano, Incomplete projection algorithms for solving the convex feasibility problem,, \emph{Numer. Algorithms}, 18 (1998), 177.  doi: 10.1023/A:1019165330848.  Google Scholar

[13]

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory,, Cambridge Studies in Advanced Mathematics 28, (1990).  doi: 10.1017/CBO9780511526152.  Google Scholar

[14]

N. I. M. Gould, How good are projection methods for convex feasibility problems,, \emph{Comput. Optim. Appl.}, 40 (2008), 1.  doi: 10.1007/s10589-007-9073-5.  Google Scholar

[15]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography,, Academic Press, (1980).   Google Scholar

[16]

K. C. Kiwiel, Block-iterative surrogate projection methods for convex feasibility problem,, \emph{Linear Algebra Appl.}, 215 (1995), 225.  doi: 10.1016/0024-3795(93)00089-I.  Google Scholar

[17]

L. Li and Y. Gao, Approximate subgradient projection algorithm for a convex feasibility problem,, \emph{J. Syst. Eng. Electron.}, 21 (2010), 527.   Google Scholar

[18]

T. Lucio, A parallel subgradient projections method for the convex feasibility problem,, \emph{J. Comput. Appl. Math.}, 18 (1987), 307.  doi: 10.1016/0377-0427(87)90004-5.  Google Scholar

[19]

P. E. Mainge, Convergence theorem for inertial KM-type algorithms,, \emph{J. Comput. Appl. Math.}, 219 (2008), 223.  doi: 10.1016/j.cam.2007.07.021.  Google Scholar

[20]

M. Moudafi, Convergence of a splitting inertial proximal method for monotone operators,, \emph{J. Comput. Appl. Math.}, 155 (2008), 447.  doi: 10.1016/S0377-0427(02)00906-8.  Google Scholar

[21]

A. Moudafi and E. Elisabeth, An approximate inertial proximal method using enlargement of a maximal monotone operator,, \emph{Int. J. Pure Appl. Math.}, 5 (2003), 283.   Google Scholar

[22]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, \emph{Bull. Am. Math. Soc.}, 73 (1967), 591.   Google Scholar

[23]

G. Pierra, Decompasition through formalization in a product space,, \emph{Math. Program.}, 28 (1984), 96.  doi: 10.1007/BF02612715.  Google Scholar

[24]

B. T. Polyak, Some methods of speeding up the convergence of iteration method,, \emph{Zh. Vych. Math.}, 4 (1964), 791.   Google Scholar

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