# American Institute of Mathematical Sciences

• Previous Article
Predicting non-life insurer's insolvency using non-kernel fuzzy quadratic surface support vector machines
• JIMO Home
• This Issue
• Next Article
Predicting 72-hour reattendance in emergency departments using discriminant analysis via mixed integer programming with electronic medical records
April  2019, 15(2): 963-984. doi: 10.3934/jimo.2018080

## A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications

 1 Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel 2 The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838 Haifa, Israel 3 Department of Mathematics, University of Transport and Communications, 3 Cau Giay Street, Hanoi, Vietnam

* Corresponding author: avivg@braude.ac.il

Received  November 2017 Revised  January 2018 Published  April 2019 Early access  June 2018

Fund Project: The first author work is supported by the EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669. The research of the third author was partially supported by University of Transport and Communications (UTC) [grant number T2018-CB-002] and Vietnam Institute for Advanced Study in Mathematics (VIASM).

Inspired by the works of López et al. [21] and the recent paper of Dang et al. [15], we devise a new inertial relaxation of the CQ algorithm for solving Split Feasibility Problems (SFP) in real Hilbert spaces. Under mild and standard conditions we establish weak convergence of the proposed algorithm. We also propose a Mann-type variant which converges strongly. The performances and comparisons with some existing methods are presented through numerical examples in Compressed Sensing and Sparse Binary Tomography by solving the LASSO problem.

Citation: Aviv Gibali, Dang Thi Mai, Nguyen The Vinh. A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. Journal of Industrial & Management Optimization, 2019, 15 (2) : 963-984. doi: 10.3934/jimo.2018080
##### References:

show all references

##### References:
Numerical results for m = 210; n = 212; k = 20
Numerical results for m = 210; n = 212; k = 40
Numerical results for m = 212; n = 213; k = 50
Parallelbeam geometry set-up: a set of parallel rays is shot through the object from different directions. These are typically coined as one projection. Two projections are illustrated above. (Left) Illustration of a single projection corresponding to a measurement along one ray. The image domain $\Omega$ is tiled into pixels or mathematically Haar-basis functions. Hence, a single projection corresponds to the line integral over a piecewise constant function
] (left). We illustrate how such a $32\times 32$ image is sampled (right) along $45$ parallel and equidistant lines that are all perpendicular to $\theta: = (\cos(30^\circ),\sin(30^\circ))^T$">Figure 5.  Vessel test image from [4] (left). We illustrate how such a $32\times 32$ image is sampled (right) along $45$ parallel and equidistant lines that are all perpendicular to $\theta: = (\cos(30^\circ),\sin(30^\circ))^T$
] representing a vascular system containing larger and smaller vessels. The results are for the recover $u$ from a $15$ (limited number of) tomographic projections">Figure 6.  Reconstructing a binary test image $u\in \mathbb{R}^{64\times 64}$ from [4] representing a vascular system containing larger and smaller vessels. The results are for the recover $u$ from a $15$ (limited number of) tomographic projections
Numerical results obtained by Algorithm 1 compared with Lopez et al. algorithm [21] ((8)-(10)) and Zhou and Wang [39,Algorithm 3.1]
 K and m, n Methods $\epsilon$=10-6 Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$ K = 10 Algorithm 1 3310 2.23e - 5 3.3438 0.0033 m = 210 Lopez et al. [21] 3491 3.23e - 5 3.8125 0.0045 n = 212 Zhou and Wang [39] 3991 2.79e - 4 4.7438 0.0012 K = 20 Algorithm 1 7180 9.6601e - 13 5.08281 3.380e - 5 m = 210 Lopez et al. [21] 8901 7.6601e - 13 5.28281 3.271e - 5 n = 212 Zhou and Wang [39] 8180 8.4375e - 12 6.478 3.260e - 5 K = 50 Algorithm 1 8780 5.7301e - 10 12.3312 4.276e - 4 m = 212 Lopez et al. [21] 9901 6.6601e - 9 11.2761 4.238e - 4 n = 213 Zhou and Wang [39] 9180 7.251e - 10 11.453 3.457e - 4
 K and m, n Methods $\epsilon$=10-6 Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$ K = 10 Algorithm 1 3310 2.23e - 5 3.3438 0.0033 m = 210 Lopez et al. [21] 3491 3.23e - 5 3.8125 0.0045 n = 212 Zhou and Wang [39] 3991 2.79e - 4 4.7438 0.0012 K = 20 Algorithm 1 7180 9.6601e - 13 5.08281 3.380e - 5 m = 210 Lopez et al. [21] 8901 7.6601e - 13 5.28281 3.271e - 5 n = 212 Zhou and Wang [39] 8180 8.4375e - 12 6.478 3.260e - 5 K = 50 Algorithm 1 8780 5.7301e - 10 12.3312 4.276e - 4 m = 212 Lopez et al. [21] 9901 6.6601e - 9 11.2761 4.238e - 4 n = 213 Zhou and Wang [39] 9180 7.251e - 10 11.453 3.457e - 4
Numerical results obtained by Algorithm 1 compared with Lopez et al. algorithm [21] ((8)-(9)) and Zhou and Wang [39,Algorithm 3.1]
 K and m, n Methods $\epsilon$ = 10-6 Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$ K = 916 Algorithm 1 87 37.6590 0.6406 65.7008 m = 1365 Lopez et al. [21] 100 49.3198 0.3541 38.2665 n = 4096 Zhou and Wang [39] 100 48.5597 0.6565 68.3321
 K and m, n Methods $\epsilon$ = 10-6 Iter fk (10) CPU time (sec.) $\frac{\|x^*-x^k\|}{\|x^0-x^{k+1}\|}$ K = 916 Algorithm 1 87 37.6590 0.6406 65.7008 m = 1365 Lopez et al. [21] 100 49.3198 0.3541 38.2665 n = 4096 Zhou and Wang [39] 100 48.5597 0.6565 68.3321
 [1] Ya-Zheng Dang, Zhong-Hui Xue, Yan Gao, Jun-Xiang Li. Fast self-adaptive regularization iterative algorithm for solving split feasibility problem. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1555-1569. doi: 10.3934/jimo.2019017 [2] Guash Haile Taddele, Poom Kumam, Habib ur Rehman, Anteneh Getachew Gebrie. Self adaptive inertial relaxed $CQ$ algorithms for solving split feasibility problem with multiple output sets. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021172 [3] Suthep Suantai, Nattawut Pholasa, Prasit Cholamjiak. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1595-1615. doi: 10.3934/jimo.2018023 [4] Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082 [5] Yazheng Dang, Jie Sun, Honglei Xu. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1383-1394. doi: 10.3934/jimo.2016078 [6] Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020152 [7] 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 [8] Hatim Tayeq, Amal Bergam, Anouar El Harrak, Kenza Khomsi. Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400 [9] Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial & Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187 [10] Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial & Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255 [11] Xuefeng Zhang, Hui Yan. Image enhancement algorithm using adaptive fractional differential mask technique. Mathematical Foundations of Computing, 2019, 2 (4) : 347-359. doi: 10.3934/mfc.2019022 [12] Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial & Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749 [13] 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, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104 [14] Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021011 [15] Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021046 [16] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [17] Bin Feng, Lixin Wei, Ziyu Hu. An adaptive large neighborhood search algorithm for Vehicle Routing Problem with Multiple Time Windows constraints. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021197 [18] Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459 [19] Ali Fuat Alkaya, Dindar Oz. An optimal algorithm for the obstacle neutralization problem. Journal of Industrial & Management Optimization, 2017, 13 (2) : 835-856. doi: 10.3934/jimo.2016049 [20] David Bourne, Howard Elman, John E. Osborn. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence. Communications on Pure & Applied Analysis, 2009, 8 (1) : 143-160. doi: 10.3934/cpaa.2009.8.143

2020 Impact Factor: 1.801