Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application

Aliyu Muhammed Awwal; Poom Kumam; Kanokwan Sitthithakerngkiet; Abubakar Muhammad Bakoji; Abubakar S. Halilu; Ibrahim M. Sulaiman; Aliyu Muhammed Awwal; Poom Kumam; Kanokwan Sitthithakerngkiet; Abubakar Muhammad Bakoji; Abubakar S. Halilu; Ibrahim M. Sulaiman

doi:10.3934/math.2021510

AIMS Mathematics

2021, Volume 6, Issue 8: 8792-8814. doi: 10.3934/math.2021510

Previous Article Next Article

Research article

Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application

1.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), & KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.
Department of Mathematics, Faculty of Science, Gombe State University, Gombe 760214, Nigeria
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok (KMUTNB), Bangkok, Thailand
5.
Faculty of Natural Sciences II, Institute of Mathematics, Martin Luther University Halle–Wittenberg, 06099 Halle (Saale), Germany
6.
Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Phagwara 144411, India
7.
Faculty of informatics and Computing, Universiti Sultan Zainal Abidin (UniSZA), Terengganu 22200, Malaysia

Received: 15 March 2021 Accepted: 31 May 2021 Published: 10 June 2021
MSC : 90C30, 90C06, 90C56

In this paper, a new derivative-free approach for solving nonlinear monotone system of equations with convex constraints is proposed. The search direction of the proposed algorithm is derived based on the modified scaled Davidon-Fletcher-Powell (DFP) updating formula in such a way that it is sufficiently descent. Under some mild assumptions, the search direction is shown to be bounded. Subsequently, the convergence result of the proposed method is established. The performance of the proposed algorithm on a collection of some test problems as well as signal recovery problems is demonstrated in comparison with some existing algorithms with similar characteristics. The results of the numerical experiments confirm the efficiency as well as the robustness of the proposed algorithm by comparing it with some existing methods in the literature.
- derivative-free method,
- nonlinear monotone equations,
- hyperplane projection technique,
- DFP method,
- quasi-Newton method,
- signal reconstruction problem
Citation: Aliyu Muhammed Awwal, Poom Kumam, Kanokwan Sitthithakerngkiet, Abubakar Muhammad Bakoji, Abubakar S. Halilu, Ibrahim M. Sulaiman. Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application[J]. AIMS Mathematics, 2021, 6(8): 8792-8814. doi: 10.3934/math.2021510

Related Papers:

Abstract

In this paper, a new derivative-free approach for solving nonlinear monotone system of equations with convex constraints is proposed. The search direction of the proposed algorithm is derived based on the modified scaled Davidon-Fletcher-Powell (DFP) updating formula in such a way that it is sufficiently descent. Under some mild assumptions, the search direction is shown to be bounded. Subsequently, the convergence result of the proposed method is established. The performance of the proposed algorithm on a collection of some test problems as well as signal recovery problems is demonstrated in comparison with some existing algorithms with similar characteristics. The results of the numerical experiments confirm the efficiency as well as the robustness of the proposed algorithm by comparing it with some existing methods in the literature.

References

[1]	A. M. Awwal, P. Kumam, L. Wang, S. Huang, W. Kumam, Inertial-based derivative-free method for system of monotone nonlinear equations and application, IEEE Access, 8 (2020), 226921-226930. doi: 10.1109/ACCESS.2020.3045493
[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. doi: 10.1016/j.jmaa.2013.04.017
[3]	J. K. Liu, S. J. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453. doi: 10.1016/j.camwa.2015.09.014
[4]	S. Aji, P. Kumam, A. M. Awwal, M. M. Yahaya, W. Kumam, Two hybrid spectral methods with inertial effect for solving system of nonlinear monotone equations with application in robotics, IEEE Access, 9 (2021), 30918-30928. doi: 10.1109/ACCESS.2021.3056567
[5]	W. Sun, Y. X. Yuan, Optimization theory and methods: nonlinear programming, Volume 1, Springer Science & Business Media, 2006.
[6]	M. Zhu, J. L. Nazareth, H. Wolkowicz, The quasi-cauchy relation and diagonal updating, SIAM J. Optim., 9 (1999), 1192-1204. doi: 10.1137/S1052623498331793
[7]	N. Andrei, A diagonal quasi-newton updating method for unconstrained optimization, Numer. Algorithms, 81 (2019), 575-590. doi: 10.1007/s11075-018-0562-7
[8]	M. Y. Waziri, Z. Abdul Majid, An improved diagonal jacobian approximation via a new quasi-cauchy condition for solving large-scale systems of nonlinear equations, J. Appl. Math., 2013 (2013), 875935.
[9]	E. Polak, G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, Revue Française d'informatique et de recherche opérationnelle. Série rouge, 3 (1969), 35-43.
[10]	B. T. Polyak, The conjugate gradient method in extremal problems, USSR Comput. Math. Math. Phys., 9 (1969), 94-112. doi: 10.1016/0041-5553(69)90035-4
[11]	L. Zhang, W. Zhou, D. H. Li, A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal., 26 (2006), 629-640. doi: 10.1093/imanum/drl016
[12]	N. Andrei, A simple three-term conjugate gradient algorithm for unconstrained optimization, J. Comput. Appl. Math., 241 (2013), 19-29. doi: 10.1016/j.cam.2012.10.002
[13]	A. M. Awwal, P. Kumam, H. Mohammad, W. Watthayu, A. B. Abubakar, A Perry-type derivative-free algorithm for solving nonlinear system of equations and minimizing $\ell_1$ regularized problem, Optimization, (2020), 1-29.
[14]	A. Perry, A modified conjugate gradient algorithm, Oper. Res., 26 (1978), 1073-1078. doi: 10.1287/opre.26.6.1073
[15]	M. V. Solodov, B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, In: Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Springer, 1998,355-369.
[16]	A. B. Abubakar, J. Sabi'u, P. Kumam, A. Shah, Solving nonlinear monotone operator equations via modified sr1 update, J. Appl. Math. Comput., (2021), 1-31.
[17]	N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl., 38 (2007), 401-416. doi: 10.1007/s10589-007-9055-7
[18]	A. M. Awwal, L. Wang, P. Kumam, H. Mohammad, W. Watthayu, A projection Hestenes-Stiefel method with spectral parameter for nonlinear monotone equations and signal processing, Math. Comput. Appl., 25 (2020), 27.
[19]	A. M. Awwal, L. Wang, P. Kumam, H. Mohammad, A two-step spectral gradient projection method for system of nonlinear monotone equations and image deblurring problems, Symmetry, 12 (2020), 874. doi: 10.3390/sym12060874
[20]	J. Sabi'u, A. Shah, M. Y. Waziri, K. Ahmed, Modified Hager-Zhang conjugate gradient methods via singular value analysis for solving monotone nonlinear equations with convex constraint, Int. J. Comput. Methods, (2020), 2050043.
[21]	E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213. doi: 10.1007/s101070100263
[22]	M. A. T. Figueiredo, R. D. Nowak, S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., 1 (2007), 586-597. doi: 10.1109/JSTSP.2007.910281
[23]	Y. Xiao, Q. Wang, Q. Hu, Non-smooth equations based method for $\ell_1$-norm problems with applications to compressed sensing, Nonlinear Anal. Theory, Methods Appl., 74 (2011), 3570-3577. doi: 10.1016/j.na.2011.02.040
[24]	J. S. Pang, Inexact Newton methods for the nonlinear complementarity problem, Math. Program., 36 (1986), 54-71. doi: 10.1007/BF02591989
[25]	W. La Cruz, J. M. Martínez, M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems: theory and experiments, Citeseer, Technical Report RT-04-08, 2004.
[26]	W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algorithms, 76 (2017), 1109-1130. doi: 10.1007/s11075-017-0299-8
[27]	W. Zhou, D. Shen, An inexact PRP conjugate gradient method for symmetric nonlinear equations, Numer. Funct. Anal. Optim., 35 (2014), 370-388. doi: 10.1080/01630563.2013.871290
[28]	A. M. Awwal, P. Kumam, A. B. Abubakar, Spectral modified Polak-Ribiére-Polyak projection conjugate gradient method for solving monotone systems of nonlinear equations, Appl. Math. Comput., 362 (2019), 124514.
[29]	G. Yu, S. Niu, J. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, J. Ind. Manage. Optim., 9 (2013), 117-129.
[30]	P. Gao, C. He, Y. Liu, An adaptive family of projection methods for constrained monotone nonlinear equations with applications, Appl. Math. Comput., 359 (2019), 1-16. doi: 10.1016/j.cam.2019.03.025
[31]	J. Liu, Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numer. Algorithms, 82 (2019), 245-262. doi: 10.1007/s11075-018-0603-2

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)