American Institute of Mathematical Sciences

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

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

Yazheng Dang 1, , Fanwen Meng 2, and  Jie Sun 3,

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.
Keywords: asymptotic convergence., Convex feasible problem, inertial parallel projection algorithm.
Mathematics Subject Classification: 49M37, 90C25, 90C9.
Citation: Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control and 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, SIAM Rev., 38 (1996), 367-426. doi: 10.1137/S0036144593251710.

[2]

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

[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, Philadlphia, PA, USA, 1994. doi: 10.1137/1.9781611970777.

[4]

Y. Censor and A. Lent, Cyclic subgradient projections, Math. Program., 24 (1982), 233-235. doi: 10.1007/BF01585107.

[5]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Math. Program., 42 (1998), 307-325. doi: 10.1007/BF01589408.

[6]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs, INFORMS J. Comput., 16 (2004), 255-265. doi: 10.1287/ijoc.1030.0046.

[7]

P. L. Combeties and S. G. Kruk, Extroplation algorithm for affine-convex feasibility problems, Numer. Algorithms, 41 (2006), 239-274. doi: 10.1007/s11075-005-9010-6.

[8]

Y. Dang and Y. Gao, Non-monotonous accelerated parallel subgradient projection algorithm for convex feasibility problem, Optimization, 63 (2014), 571-584. doi: 10.1080/02331934.2012.677447.

[9]

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

[10]

Y. Gao, Viability criteria for differential inclusions, J. Syst. Sci. Complex., 24 (2011), 825-834. doi: 10.1007/s11424-011-9056-6.

[11]

U. Garcia-palomares, Parallel projected aggregation methods for solving the convex feasibility problem, SIAM J. Optim., 3 (1993), 882-900. doi: 10.1137/0803046.

[12]

U. M. Garcia-Palomares and F. J. Gonzalez-Castano, Incomplete projection algorithms for solving the convex feasibility problem, Numer. Algorithms, 18 (1998), 177-193. doi: 10.1023/A:1019165330848.

[13]

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

[14]

N. I. M. Gould, How good are projection methods for convex feasibility problems, Comput. Optim. Appl., 40 (2008), 1-12. doi: 10.1007/s10589-007-9073-5.

[15]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[16]

K. C. Kiwiel, Block-iterative surrogate projection methods for convex feasibility problem, Linear Algebra Appl., 215 (1995), 225-260. doi: 10.1016/0024-3795(93)00089-I.

[17]

L. Li and Y. Gao, Approximate subgradient projection algorithm for a convex feasibility problem, J. Syst. Eng. Electron., 21 (2010), 527-530.

[18]

T. Lucio, A parallel subgradient projections method for the convex feasibility problem, J. Comput. Appl. Math., 18 (1987), 307-320. doi: 10.1016/0377-0427(87)90004-5.

[19]

P. E. Mainge, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236. doi: 10.1016/j.cam.2007.07.021.

[20]

M. Moudafi, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2008), 447-454. doi: 10.1016/S0377-0427(02)00906-8.

[21]

A. Moudafi and E. Elisabeth, An approximate inertial proximal method using enlargement of a maximal monotone operator, Int. J. Pure Appl. Math., 5 (2003), 283-299.

[22]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc., 73 (1967), 591-597.

[23]

G. Pierra, Decompasition through formalization in a product space, Math. Program., 28 (1984), 96-115. doi: 10.1007/BF02612715.

[24]

B. T. Polyak, Some methods of speeding up the convergence of iteration method, Zh. Vych. Math., 4 (1964), 791-803.

show all references

References:
[1]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426. doi: 10.1137/S0036144593251710.

[2]

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

[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, Philadlphia, PA, USA, 1994. doi: 10.1137/1.9781611970777.

[4]

Y. Censor and A. Lent, Cyclic subgradient projections, Math. Program., 24 (1982), 233-235. doi: 10.1007/BF01585107.

[5]

Y. Censor, Parallel application of block iterative methods in medical imaging and radiation therapy, Math. Program., 42 (1998), 307-325. doi: 10.1007/BF01589408.

[6]

J. W. Chinneck, The constraint consensus method for finding approximately feasible points in nonlinear programs, INFORMS J. Comput., 16 (2004), 255-265. doi: 10.1287/ijoc.1030.0046.

[7]

P. L. Combeties and S. G. Kruk, Extroplation algorithm for affine-convex feasibility problems, Numer. Algorithms, 41 (2006), 239-274. doi: 10.1007/s11075-005-9010-6.

[8]

Y. Dang and Y. Gao, Non-monotonous accelerated parallel subgradient projection algorithm for convex feasibility problem, Optimization, 63 (2014), 571-584. doi: 10.1080/02331934.2012.677447.

[9]

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

[10]

Y. Gao, Viability criteria for differential inclusions, J. Syst. Sci. Complex., 24 (2011), 825-834. doi: 10.1007/s11424-011-9056-6.

[11]

U. Garcia-palomares, Parallel projected aggregation methods for solving the convex feasibility problem, SIAM J. Optim., 3 (1993), 882-900. doi: 10.1137/0803046.

[12]

U. M. Garcia-Palomares and F. J. Gonzalez-Castano, Incomplete projection algorithms for solving the convex feasibility problem, Numer. Algorithms, 18 (1998), 177-193. doi: 10.1023/A:1019165330848.

[13]

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

[14]

N. I. M. Gould, How good are projection methods for convex feasibility problems, Comput. Optim. Appl., 40 (2008), 1-12. doi: 10.1007/s10589-007-9073-5.

[15]

G. T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

[16]

K. C. Kiwiel, Block-iterative surrogate projection methods for convex feasibility problem, Linear Algebra Appl., 215 (1995), 225-260. doi: 10.1016/0024-3795(93)00089-I.

[17]

L. Li and Y. Gao, Approximate subgradient projection algorithm for a convex feasibility problem, J. Syst. Eng. Electron., 21 (2010), 527-530.

[18]

T. Lucio, A parallel subgradient projections method for the convex feasibility problem, J. Comput. Appl. Math., 18 (1987), 307-320. doi: 10.1016/0377-0427(87)90004-5.

[19]

P. E. Mainge, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236. doi: 10.1016/j.cam.2007.07.021.

[20]

M. Moudafi, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2008), 447-454. doi: 10.1016/S0377-0427(02)00906-8.

[21]

A. Moudafi and E. Elisabeth, An approximate inertial proximal method using enlargement of a maximal monotone operator, Int. J. Pure Appl. Math., 5 (2003), 283-299.

[22]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc., 73 (1967), 591-597.

[23]

G. Pierra, Decompasition through formalization in a product space, Math. Program., 28 (1984), 96-115. doi: 10.1007/BF02612715.

[24]

B. T. Polyak, Some methods of speeding up the convergence of iteration method, Zh. Vych. Math., 4 (1964), 791-803.

[1]

Yazheng Dang, Marcus Ang, Jie Sun. An inertial triple-projection algorithm for solving the split feasibility problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022019
[2]

Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082
[3]

Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004
[4]

Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial and Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041
[5]

Ouafa Belguidoum, Hassina Grar. An improved projection algorithm for variational inequality problem with multivalued mapping. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022002
[6]

Le Thi Hoai An, Tran Duc Quynh, Kondo Hloindo Adjallah. A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors. Journal of Industrial and Management Optimization, 2014, 10 (1) : 243-258. doi: 10.3934/jimo.2014.10.243
[7]

Chengjin Li. Parameter-related projection-based iterative algorithm for a kind of generalized positive semidefinite least squares problem. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 511-520. doi: 10.3934/naco.2020048
[8]

Abdulkarim Hassan Ibrahim, Poom Kumam, Min Sun, Parin Chaipunya, Auwal Bala Abubakar. Projection method with inertial step for nonlinear equations: Application to signal recovery. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021173
[9]

Qingzhi Yang. The revisit of a projection algorithm with variable steps for variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (2) : 211-217. doi: 10.3934/jimo.2005.1.211
[10]

Guodong Ma, Jinbao Jian. A QP-free algorithm of quasi-strongly sub-feasible directions for inequality constrained optimization. Journal of Industrial and Management Optimization, 2015, 11 (1) : 307-328. doi: 10.3934/jimo.2015.11.307
[11]

Xin Yang, Nan Wang, Lingling Xu. A parallel Gauss-Seidel method for convex problems with separable structure. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 557-570. doi: 10.3934/naco.2020051
[12]

Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069
[13]

Luchuan Ceng, Qamrul Hasan Ansari, Jen-Chih Yao. Extragradient-projection method for solving constrained convex minimization problems. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 341-359. doi: 10.3934/naco.2011.1.341
[14]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial and Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117
[15]

Janosch Rieger. A learning-enhanced projection method for solving convex feasibility problems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 555-568. doi: 10.3934/dcdsb.2021054
[16]

Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems and Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055
[17]

Leyu Hu, Wenxing Zhang, Xingju Cai, Deren Han. A parallel operator splitting algorithm for solving constrained total-variation retinex. Inverse Problems and Imaging, 2020, 14 (6) : 1135-1156. doi: 10.3934/ipi.2020058
[18]

Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863
[19]

Jaakko Ketola, Lars Lamberg. An algorithm for recovering unknown projection orientations and shifts in 3-D tomography. Inverse Problems and Imaging, 2011, 5 (1) : 75-93. doi: 10.3934/ipi.2011.5.75
[20]

Suthep Suantai, Nattawut Pholasa, Prasit Cholamjiak. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1595-1615. doi: 10.3934/jimo.2018023

 Impact Factor: 

Tools

Metrics

  • PDF downloads (163)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy