# A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

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## Abstract

**:**

## 1. Introduction

**Algorithm**

**1.**

**Algorithm**

**2.**

**Algorithm**

**3.**

## 2. Preliminaries

- (i)
- $\langle x-{P}_{C}x,z-{P}_{C}x\rangle \le 0$ for all $z\in C$;
- (ii)
- ${\parallel {P}_{C}x-{P}_{C}y\parallel}^{2}\le \langle {P}_{C}x-{P}_{C}y,x-y\rangle $ for all $x,y\in H$; and
- (iii)
- ${\parallel {P}_{C}x-z\parallel}^{2}\le {\parallel x-z\parallel}^{2}-{\parallel {P}_{C}x-x\parallel}^{2}$ for all $z\in C$.

**Lemma**

**1.**

- (i)
- $\underset{n\to \infty}{lim}\parallel {x}_{n}-x\parallel$ exists for each $x\in S$; and
- (ii)
- ${\omega}_{w}\left({x}_{n}\right)\subset S$.

## 3. Main Results

**Algorithm**

**4.**

**Lemma**

**2.**

**Theorem**

**1.**

- (a1)
- $\underset{n\to \infty}{lim}{\theta}_{n}=0$; and
- (a2)
- $\underset{n\to \infty}{lim\; inf}{\beta}_{n}(4-{\beta}_{n})>0$.

**Proof.**

## 4. Numerical Experiments

## 5. Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**From top to bottom: original signal, observation data and recovered signal by Algorithms 1–4, respectively.

**Figure 3.**The objective function value versus number of iterations when $N=1024$, $M=512$ and $\kappa ={10}^{-5}$.

**Figure 4.**From top to bottom: original signal, observation data and recovered signal by Algorithms 1–4, respectively.

**Figure 6.**The objective function value versus number of iterations when $N=4096$, $M=2048$ and $\kappa ={10}^{-5}$.

m-Sparse | Method | $\mathit{\kappa}={10}^{-4}$ | $\mathit{\kappa}={10}^{-5}$ | ||
---|---|---|---|---|---|

CPU | Iter | CPU | Iter | ||

$m=10$ | Algorithm 1 | 0.7801 | 93 | 0.5931 | 83 |

Algorithm 2 | 0.0962 | 187 | 0.1000 | 158 | |

Algorithm 3 | 0.1416 | 257 | 0.0605 | 74 | |

Algorithm 4 | 0.0271 | 33 | 0.0592 | 39 | |

$m=15$ | Algorithm 1 | 0.6345 | 93 | 0.6778 | 101 |

Algorithm 2 | 0.1020 | 196 | 0.1001 | 195 | |

Algorithm 3 | 0.1087 | 170 | 0.0823 | 97 | |

Algorithm 4 | 0.0251 | 35 | 0.0430 | 51 | |

$m=20$ | Algorithm 1 | 1.1535 | 161 | 1.1177 | 156 |

Algorithm 2 | 0.1661 | 308 | 0.1573 | 296 | |

Algorithm 3 | 0.3557 | 500 | 0.1139 | 134 | |

Algorithm 4 | 0.0516 | 55 | 0.0695 | 78 | |

$m=25$ | Algorithm 1 | 0.7380 | 103 | 2.9774 | 443 |

Algorithm 2 | 0.0990 | 196 | 0.4746 | 940 | |

Algorithm 3 | 0.0623 | 115 | 0.7258 | 1308 | |

Algorithm 4 | 0.0354 | 42 | 0.0922 | 165 | |

$m=30$ | Algorithm 1 | 1.1423 | 168 | 3.7280 | 92 |

Algorithm 2 | 0.1568 | 321 | 1.7980 | 666 | |

Algorithm 3 | 0.1219 | 164 | 0.4119 | 111 | |

Algorithm 4 | 0.0704 | 70 | 0.1335 | 38 |

m-sparse | Method | $\mathit{\kappa}={10}^{-4}$ | $\mathit{\kappa}={10}^{-5}$ | ||
---|---|---|---|---|---|

CPU | Iter | CPU | Iter | ||

$m=20$ | Algorithm 1 | 53.4863 | 28 | 77.8192 | 40 |

Algorithm 2 | 3.1953 | 43 | 4.7627 | 62 | |

Algorithm 3 | 1.5285 | 19 | 2.3102 | 28 | |

Algorithm 4 | 1.0771 | 13 | 1.6199 | 20 | |

$m=40$ | Algorithm 1 | 74.6456 | 38 | 106.3420 | 54 |

Algorithm 2 | 4.5607 | 60 | 6.1862 | 83 | |

Algorithm 3 | 2.0701 | 26 | 2.9406 | 37 | |

Algorithm 4 | 1.4418 | 18 | 2.1713 | 27 | |

$m=60$ | Algorithm 1 | 86.1752 | 45 | 137.6885 | 70 |

Algorithm 2 | 5.2204 | 70 | 8.1821 | 110 | |

Algorithm 3 | 2.3965 | 30 | 3.6434 | 46 | |

Algorithm 4 | 1.7580 | 22 | 2.6908 | 34 | |

$m=80$ | Algorithm 1 | 133.5504 | 67 | 219.4587 | 112 |

Algorithm 2 | 7.8185 | 104 | 13.3599 | 178 | |

Algorithm 3 | 3.4220 | 43 | 5.9392 | 75 | |

Algorithm 4 | 2.4207 | 30 | 3.7902 | 47 | |

$m=100$ | Algorithm 1 | 148.3098 | 75 | 327.4775 | 163 |

Algorithm 2 | 8.7840 | 118 | 19.7221 | 258 | |

Algorithm 3 | 3.8024 | 48 | 16.0518 | 202 | |

Algorithm 4 | 2.6962 | 34 | 5.3538 | 66 |

${\mathit{\beta}}_{\mathit{n}}$ | CPU | Iter | |
---|---|---|---|

N = 1024 | 0.1 | 0.4585 | 139 |

M = 512 | 0.5 | 0.1976 | 73 |

m = 20 | 1.0 | 0.1632 | 55 |

1.5 | 0.1272 | 44 | |

2.0 | 0.1187 | 38 | |

2.5 | 0.1048 | 35 | |

3.0 | 0.1065 | 32 | |

3.5 | 0.1298 | 29 | |

3.9 | 0.0954 | 28 | |

N = 4096 | 0.1 | 4.4547 | 58 |

M = 2048 | 0.5 | 3.6075 | 39 |

m = 20 | 1.0 | 2.2021 | 29 |

1.5 | 1.8119 | 24 | |

2.0 | 1.6024 | 21 | |

2.5 | 1.5748 | 29 | |

3.0 | 1.4055 | 17 | |

3.5 | 1.3297 | 16 | |

3.9 | 1.3172 | 15 |

$\mathit{\rho}$ | CPU | Iter | |
---|---|---|---|

N = 1024 | 0.1 | 0.0634 | 27 |

M = 512 | 0.3 | 0.0981 | 26 |

m = 20 | 0.5 | 0.1065 | 26 |

0.7 | 0.1773 | 27 | |

0.9 | 0.5421 | 27 | |

N = 4096 | 0.1 | 0.7554 | 17 |

M = 2048 | 0.3 | 1.2094 | 17 |

m = 20 | 0.5 | 1.7697 | 17 |

0.7 | 3.1876 | 17 | |

0.9 | 10.1536 | 18 |

$\mathit{\sigma}$ | CPU | Iter | |
---|---|---|---|

N = 1024 | 1 | 0.2985 | 53 |

M = 512 | 2 | 0.2974 | 53 |

m = 20 | 3 | 0.2636 | 56 |

4 | 0.2478 | 53 | |

5 | 0.2584 | 52 | |

6 | 0.2816 | 56 | |

N = 4096 | 1 | 1.9105 | 16 |

M = 2048 | 2 | 1.9990 | 16 |

m = 20 | 3 | 2.0937 | 16 |

4 | 2.1371 | 16 | |

5 | 2.2449 | 17 | |

6 | 2.3816 | 16 |

$\mathit{\kappa}={10}^{-4}$ | $\mathit{\kappa}={10}^{-5}$ | |||
---|---|---|---|---|

CPU | Iter | CPU | Iter | |

$M=1024$ | 0.9967 | 13 | 1.4998 | 21 |

$N=2048$ | ||||

$M=2048$ | 3.8625 | 11 | 5.6119 | 16 |

$N=4096$ | ||||

$M=3072$ | 5.0449 | 6 | 6.5788 | 8 |

$N=6144$ | ||||

$M=4096$ | 7.3689 | 5 | 10.1838 | 7 |

$N=8192$ |

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**MDPI and ACS Style**

Suantai, S.; Kesornprom, S.; Cholamjiak, P.
A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing. *Mathematics* **2019**, *7*, 789.
https://doi.org/10.3390/math7090789

**AMA Style**

Suantai S, Kesornprom S, Cholamjiak P.
A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing. *Mathematics*. 2019; 7(9):789.
https://doi.org/10.3390/math7090789

**Chicago/Turabian Style**

Suantai, Suthep, Suparat Kesornprom, and Prasit Cholamjiak.
2019. "A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing" *Mathematics* 7, no. 9: 789.
https://doi.org/10.3390/math7090789