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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
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##### 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
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 $\theta: = (\cos(30^\circ),\sin(30^\circ))^T$
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
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