October  2006, 2(4): 451-466. doi: 10.3934/jimo.2006.2.451

Some remarks on a successive projection sequence

1. 

Department of Economics and Management Sciences, University of Perpignan, 52 Avenue Paul Alduy, Perpignan, F-66860, France, France

Received  March 2006 Revised  August 2006 Published  October 2006

The successive projection algorithms were introduced by von Neuman [11] and Brègman [3]. These algorithms have a broad applicability in medical imaging, computerized tomography or signal processing. In this paper we focus on a particular projection sequence in which the points are successively projected onto the convex set from which they are the most distant. Among other things, we analyze the convergence of the algorithm and propose a finite termination criterion allowing for the analysis of the complexity of the algorithm. In line with the seminal work by Brègman, an application to linear optimization problems is proposed. This issue illustrates that successive projection algorithms may have some interest for very large finite dimensional problems.
Citation: Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial & Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451
[1]

H. D. Scolnik, N. E. Echebest, M. T. Guardarucci. Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 175-191. doi: 10.3934/jimo.2009.5.175

[2]

Jutamas Kerdkaew, Rabian Wangkeeree. Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019074

[3]

Luigi C. Berselli, Franco Flandoli. Remarks on determining projections for stochastic dissipative equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 197-214. doi: 10.3934/dcds.1999.5.197

[4]

Tran Ninh Hoa, Ta Duy Phuong, Nguyen Dong Yen. Linear fractional vector optimization problems with many components in the solution sets. Journal of Industrial & Management Optimization, 2005, 1 (4) : 477-486. doi: 10.3934/jimo.2005.1.477

[5]

Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193

[6]

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

[7]

Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial & Management Optimization, 2020, 16 (1) : 469-493. doi: 10.3934/jimo.2018163

[8]

Yunhai Xiao, Junfeng Yang, Xiaoming Yuan. Alternating algorithms for total variation image reconstruction from random projections. Inverse Problems & Imaging, 2012, 6 (3) : 547-563. doi: 10.3934/ipi.2012.6.547

[9]

Robert Azencott, Bernhard G. Bodmann, Tasadduk Chowdhury, Demetrio Labate, Anando Sen, Daniel Vera. ROI reconstruction from truncated cone-beam projections. Inverse Problems & Imaging, 2018, 12 (1) : 29-57. doi: 10.3934/ipi.2018002

[10]

Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077

[11]

Yan Gu, Nobuo Yamashita. A proximal ADMM with the Broyden family for convex optimization problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020091

[12]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415

[13]

Chengxiang Wang, Li Zeng. Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections. Inverse Problems & Imaging, 2016, 10 (3) : 829-853. doi: 10.3934/ipi.2016023

[14]

Jie Sun. On methods for solving nonlinear semidefinite optimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 1-14. doi: 10.3934/naco.2011.1.1

[15]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163

[16]

Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022

[17]

Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial & Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647

[18]

Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems & Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29

[19]

Roland Hildebrand. Barriers on projective convex sets. Conference Publications, 2011, 2011 (Special) : 672-683. doi: 10.3934/proc.2011.2011.672

[20]

Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020104

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]