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It is well known that the nonlinear conjugate gradient algorithm is one of the effective algorithms for optimization problems since it has low storage and simple structure properties. This motivates us to make a further study to design a modified conjugate gradient formula for the optimization model, and this proposed conjugate gradient algorithm possesses several properties: (1) the search direction possesses not only the gradient value but also the function value; (2) the presented direction has both the sufficient descent property and the trust region feature; (3) the proposed algorithm has the global convergence for nonconvex functions; (4) the experiment is done for the image restoration problems and compression sensing to prove the performance of the new algorithm.

Consider the following model defined by

We all know that the sufficient descent property designed by

The sufficient property and the trust region feature are obtained

The new direction possesses not only the gradient value but also the function value

The given algorithm has the global convergence under the Armijo line search for nonconvex functions

The experiments for image restoration problems and compression sensing are done to test the performance of the new algorithm

The next section states the given algorithm. The convergence analysis is given in Section

Based on the discussions of the above section, the CG algorithm is listed in Algorithm

Initial step: given any initial point

Step 1: stop if

Step 2: find

where

Step 3: set

Step 4: stop if

Step 5: compute

Step 6: set

The direction

By (

The relation (

For the nonconvex functions, the global convergence of Algorithm

Assume that the function

The function

Now, we prove the global convergence of Algorithm

Let Assumption

Using (

This implies that

Suppose that (

By (

We can see that the proof process of the global convergence is very simple since the defined direction (

The numerical experiments for image restoration problems and compression sensing will be done by Algorithm

Setting

We choose Barbara

Restoration of Barbara, man, Baboon, and Lena by Algorithm

Restoration of Barbara, man, Baboon, and Lena by Algorithm

Figures

PSNR Algorithm

20% noise | Barbara | Man | Baboon | Lena | Average |
---|---|---|---|---|---|

Algorithm | 31.115 | 38.0355 | 29.4393 | 41.0674 | 34.9143 |

PRP algorithm | 31.1118 | 37.9583 | 29.4534 | 41.356 | 34.969 |

40% noise | Barbara | Man | Baboon | Lena | Average |

Algorithm | 27.5415 | 34.0063 | 25.8947 | 36.6496 | 31.0230 |

PRP algorithm | 27.6153 | 34.5375 | 25.8571 | 36.701 | 31.1777 |

In this section, the following compressive sensing images are tested: Phantom

Figures

Phantom: (a) the general images, (b) the recovered images by Algorithm

Fruits: (a) the general images, (b) the recovered images by Algorithm

Boat: (a) the general images, (b) the recovered images by Algorithm

This paper, by designing a CG algorithm, studies the unconstrained optimization problems. The given method possesses not only the sufficient descent property but also the trust region feature. The global convergence is proved by a simple way. The image restoration problems and compressive sensing problems are tested to show that the proposed algorithm is better than the normal algorithm. In the future, we will focus on the following aspects to be paid attention: (i) we believe there are many perfect CG algorithms which can be successfully used for image restoration problems and compressive sensing; (ii) more experiments will be done to test the performance of the new algorithm.

All data are included in the paper.

There are no potential conflicts of interest.

The authors would like to thank the support of the funds. This work was supported by the National Natural Science Foundation of China under Grant no. 61772006, the Science and Technology Program of Guangxi under Grant no. AB17129012, the Science and Technology Major Project of Guangxi under Grant no. AA17204096, the Special Fund for Scientific and Technological Bases and Talents of Guangxi under Grant no. 2016AD05050, and the Special Fund for Bagui Scholars of Guangxi.