# 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  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] 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 [2] 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 [3] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066 [4] Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $k$-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056 [5] Zheng Chang, Haoxun Chen, Farouk Yalaoui, Bo Dai. Adaptive large neighborhood search Algorithm for route planning of freight buses with pickup and delivery. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1771-1793. doi: 10.3934/jimo.2020045 [6] Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781 [7] Ashkan Ayough, Farbod Farhadi, Mostafa Zandieh, Parisa Rastkhadiv. Genetic algorithm for obstacle location-allocation problems with customer priorities. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1753-1769. doi: 10.3934/jimo.2020044 [8] Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 [9] Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 [10] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [11] Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier. Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks & Heterogeneous Media, 2021, 16 (2) : 257-281. doi: 10.3934/nhm.2021006 [12] Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040 [13] Tadeusz Kaczorek, Andrzej Ruszewski. Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021007 [14] Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063 [15] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [16] Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053 [17] Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021065 [18] Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021059 [19] Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080 [20] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

2019 Impact Factor: 1.366