AIMS Mathematics, 2021, 6(1): 235-260. doi: 10.3934/math.2021016

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An efficient gradient-free projection algorithm for constrained nonlinear equations and image restoration

1 KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2 Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4 Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano. Kano, Nigeria
5 Department of Mathematics and Statistics, College of Science and Technology, Hassan Usman Katsina Polytechnic, Dutsin-Ma Road, Katsina, Nigeria
6 Department of Mathematics, College of Computation and Natural Science, Debre Berhan University, P. O. Box 445, Debre Berhan, Ethiopia
7 Department of Mathematics, Usmanu Danfodiyo University, Sokoto 840004, Nigeria

Motivated by the projection technique, in this paper, we introduce a new method for approximating the solution of nonlinear equations with convex constraints. Under the assumption that the associated mapping is Lipchitz continuous and satisfies a weaker assumption of monotonicity, we establish the global convergence of the sequence generated by the proposed algorithm. Applications and numerical example are presented to illustrate the performance of the proposed method.
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1. M. V. Solodov, B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.

2. Y. Xiao, H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.

3. J. Liu, S. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453.

4. J. Liu, Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numer. Algorithms, 82 (2019) 245-262.

5. A. H. Ibrahim, A. I. Garba, H. Usman, J. Abubakar, A. B. Abubakar, Derivative-free projection algorithm for nonlinear equations with convex constraints, Thai Journal of Mathematics, 18 (2020), 212-232.

6. P. Kaelo, M. Koorapetse, A globally convergent projection method for a system of nonlinear monotone equations, Int. J. Comput. Math., 2020.

7. A. B. Abubakar, J. Rilwan, S. E. Yimer, A. H. Ibrahim, I. Ahmed, Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints, Thai Journal of Mathematics, 18 (2020) 501-517.

8. A. H. Ibrahim, P. Kumam, W. Kumam, A Family of Derivative-Free Conjugate Gradient Methods for Constrained Nonlinear Equations and Image Restoration, IEEE Access, 8 (2020), 162714- 162729.

9. A. B. Abubakar, A. H. Ibrahim, A. B. Muhammad, C. Tammer, A modified descent Dai-Yuan conjugate gradient method for constraint nonlinear monotone operator equations, Applied Analysis and Optimization, 4 (2020), 1-24.

10. K. Meintjes, A. P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361.

11. K. Meintjes, A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM T. Math. Software, 16 (1990), 143-151.

12. M. W. Berry, M. Browne, A. N. Langville, V. P. Pauca, R. J. Plemmons, Algorithms and applications for approximate nonnegative matrix factorization, Comput. stat. data an., 52 (2007), 155-173.

13. S. P. Dirkse, M. C. Ferris, Mcplib: A collection of nonlinear mixed complementarity problems, Optim. method. softw., 5 (1995), 319-345.

14. J. E. Dennis, J. J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. comput., 28 (1974), 549-560.

15. R. Fletcher, C. M. Reeves, Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154.

16. R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, 2013.

17. Y. H. Dai, Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. optimiz., 10 (1999), 177-182.

18. M. R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of research of the National Bureau of Standards, 49 (1952), 409-436.

19. E. Polak, G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, ESAIM-Math. Model. Num., 3 (1969), 35-43.

20. B. T. Polyak, The conjugate gradient method in extremal problems, U. S. S. R. Comput. Math. Math. Phys., 9 (1969), 94-112.

21. Y. Liu, C. Storey, Efficient generalized conjugate gradient algorithms, part 1: theory, J. optimiz. theory app., 69 (1991), 129-137.

22. S. S. Djordjević, New Hybrid Conjugate Gradient Method As A Convex Combination of Ls and Fr Methods, Acta Math. Sci., 39 (2019), 214-228.

23. A. H. Ibrahim, P. Kumam, A. B. Abubakar, W. Jirakitpuwapat, J. Abubakar, A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing, Heliyon, 6 (2020), 1-17.

24. D. H. Li, M. Fukushima, A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.

25. D. H. Li, M. Fukushima, On the global convergence of the BFGS method for nonconvex unconstrained optimization problems, SIAM J. Optimiz., 11 (2001), 1054-1064.

26. E. G. Birgin, J. M. Martinez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. opt., 43 (2001), 117-128.

27. G. Yuan, T. Li, W. Hu, A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems, Appl. Numer. Math., 147 (2020), 129-141.

28. D. H. Li, X. L. Wang, A modified Fletcher-Reeves-type derivative-free method for symmetric nonlinear equations, Numerical Algebra, Control & Optimization, 1 (2011), 71-82.

29. E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. program., 91 (2002), 201-213.

30. A. B. Abubakar, P. Kumam, H. Mohammad, A. M. Awwal, K, Sitthithakerngkiet, A modified fletcher-reeves conjugate gradient method for monotone nonlinear equations with some applications, Mathematics, 7 (2019), 1-25.

31. W. La Cruz, J. Martínez, M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput., 75 (2006), 1429-1448.

32. W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algorithms, 76 (2017), 1109-1130.

33. Y. Bing, G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM J. Optimiz., 1 (1991), 206-221.

34. Z. Yu, J. Lin, J. Sun, Y. Xiao, L. Liu, Z. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.

35. Y. Ding, Y. Xiao, J. Li, A class of conjugate gradient methods for convex constrained monotone equations, Optimization, 66 (2017), 2309-2328.

36. I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pur. Appl. Math., 57 (2004), 1413-1457.

37. A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.

38. E. T. Hale, W. Yin, Y. Zhang, A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing, CAAM TR07-07, Rice University, 2007, 1-45.

39. S. Huang, Z. Wan, A new nonmonotone spectral residual method for nonsmooth nonlinear equations, J. Comput. Appl. Math., 313 (2017), 82-101.

40. J. Abubakar, P. Kumam, A. H. Ibrahim, A. Padcharoen, Relaxed Inertial Tseng's Type Method for Solving the Inclusion Problem with Application to Image Restoration, Mathematics, 8 (2020), 1-19.

41. A. H. Ibrahim, P. Kumam, A. B. Abubakar, J. Abubakar, A. B. Muhammad, Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing, Mathematics, 8 (2020), 1-21.

42. A. B. Abubakar, K. MUangchoo, A. Muhammad, A. H. Ibrahim, A Spectral Gradient Projection Method for Sparse Signal Reconstruction in Compressive Sensing, Modern Applied Science, 14 (2020), 86-93.

43. A. C. Bovik, Handbook of Image and Video Processing, Academic press, 2010.

44. S. M. Lajevardi, Structural similarity classifier for facial expression recognition, Signal Image Video P., 8 (2014), 1103-1110.

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