AN EFFICIENT HYBRID DERIVATIVE-FREE PROJECTION ALGORITHM FOR CONSTRAINT NONLINEAR EQUATIONS
Article Sidebar
Published:
Oct 31, 2021
Keywords:
Nonlinear equations, Conjugate gradient method, Projection method, Global convergence.
Main Article Content
Abstract
In this paper, by combining the Solodov and Svaiter projection technique with the conjugate gradient method for unconstrained optimization proposed by Mohamed et al. (2020), we develop a derivative-free conjugate gradient method to solve nonlinear equations with convex constraints. The proposed method involves a spectral parameter which satisfies the sufficient descent condition. The global convergence is proved under the assumption that the underlying mapping is Lipschitz continuous and satisfies a weaker monotonicity condition. Numerical experiment shows that the proposed method is efficient.
Downloads
Download data is not yet available.
Article Details
How to Cite
Muangchoo, K. (2021). AN EFFICIENT HYBRID DERIVATIVE-FREE PROJECTION ALGORITHM FOR CONSTRAINT NONLINEAR EQUATIONS. Malaysian Journal of Science, 40(3), 64–75. https://doi.org/10.22452/mjs.vol40no3.6
Section
Original Articles
Transfer of Copyrights
- In the event of publication of the manuscript entitled [INSERT MANUSCRIPT TITLE AND REF NO.] in the Malaysian Journal of Science, I hereby transfer copyrights of the manuscript title, abstract and contents to the Malaysian Journal of Science and the Faculty of Science, University of Malaya (as the publisher) for the full legal term of copyright and any renewals thereof throughout the world in any format, and any media for communication.
Conditions of Publication
- I hereby state that this manuscript to be published is an original work, unpublished in any form prior and I have obtained the necessary permission for the reproduction (or am the owner) of any images, illustrations, tables, charts, figures, maps, photographs and other visual materials of whom the copyrights is owned by a third party.
- This manuscript contains no statements that are contradictory to the relevant local and international laws or that infringes on the rights of others.
- I agree to indemnify the Malaysian Journal of Science and the Faculty of Science, University of Malaya (as the publisher) in the event of any claims that arise in regards to the above conditions and assume full liability on the published manuscript.
Reviewer’s Responsibilities
- Reviewers must treat the manuscripts received for reviewing process as confidential. It must not be shown or discussed with others without the authorization from the editor of MJS.
- Reviewers assigned must not have conflicts of interest with respect to the original work, the authors of the article or the research funding.
- Reviewers should judge or evaluate the manuscripts objective as possible. The feedback from the reviewers should be express clearly with supporting arguments.
- If the assigned reviewer considers themselves not able to complete the review of the manuscript, they must communicate with the editor, so that the manuscript could be sent to another suitable reviewer.
Copyright: Rights of the Author(s)
- Effective 2007, it will become the policy of the Malaysian Journal of Science (published by the Faculty of Science, University of Malaya) to obtain copyrights of all manuscripts published. This is to facilitate:
- Protection against copyright infringement of the manuscript through copyright breaches or piracy.
- Timely handling of reproduction requests from authorized third parties that are addressed directly to the Faculty of Science, University of Malaya.
- As the author, you may publish the fore-mentioned manuscript, whole or any part thereof, provided acknowledgement regarding copyright notice and reference to first publication in the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers) are given. You may produce copies of your manuscript, whole or any part thereof, for teaching purposes or to be provided, on individual basis, to fellow researchers.
- You may include the fore-mentioned manuscript, whole or any part thereof, electronically on a secure network at your affiliated institution, provided acknowledgement regarding copyright notice and reference to first publication in the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers) are given.
- You may include the fore-mentioned manuscript, whole or any part thereof, on the World Wide Web, provided acknowledgement regarding copyright notice and reference to first publication in the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers) are given.
- In the event that your manuscript, whole or any part thereof, has been requested to be reproduced, for any purpose or in any form approved by the Malaysian Journal of Science and Faculty of Science, University of Malaya (as the publishers), you will be informed. It is requested that any changes to your contact details (especially e-mail addresses) are made known.
Copyright: Role and responsibility of the Author(s)
- In the event of the manuscript to be published in the Malaysian Journal of Science contains materials copyrighted to others prior, it is the responsibility of current author(s) to obtain written permission from the copyright owner or owners.
- This written permission should be submitted with the proof-copy of the manuscript to be published in the Malaysian Journal of Science
References
Abubakar, A. B., Ibrahim, A. H., Muhammad, A. B., and Tammer, C. (2020a). A modified descent dai-yuan conjugate gradient method for constraint nonlinear monotone operator equations. Applied Analysis and Optimization, 4(1):1–24.
Abubakar, A. B., Rilwan, J., Yimer, S. E., Ibrahim, A. H., and Ahmed, I. (2020b). Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints. Thai Journal of Mathematics, 18(1):501–517.
Berry, M. W., Browne, M., Langville, A. N., Pauca, V. P., and Plemmons, R. J. (2007). Algorithms and applications for approximate nonnegative matrix factorization. Computational statistics & data analysis, 52(1):155–173.
Bing, Y. and Lin, G. (1991). An efficient implementation of merrill’s method for sparse or partially separable systems of nonlinear equations. SIAM Journal on Optimization, 1(2):206–221.
Blumensath, T. (2013). Compressed sensing with nonlinear observations and related nonlinear optimization problems. IEEE Transactions on Information Theory, 59(6):3466–3474.
Dai, Z., Dong, X., Kang, J., and Hong, L. (2020). Forecasting stock market returns: New technical indicators and two-step economic constraint method. The North American Journal of Economics and Finance, page 101216.
Dennis, J. E. and Moré, J. J. (1974). A characterization of superlinear convergence and its application to quasi-newton methods. Mathematics of computation, 28(126):549–560.
Dennis, Jr, J. E. and Moré, J. J. (1977). Quasi-newton methods, motivation and theory. SIAM review, 19(1):46–89.
Dennis Jr, J. E. (1983). Rb schnabel numerical methods for unconstrained optimization and nonlinear equations.
Ding, Y., Xiao, Y., and Li, J. (2017). A class of conjugate gradient methods for convex constrained monotone equations. Optimization, 66(12):2309–2328.
Dirkse, S. P. and Ferris, M. C. (1995). Mcplib: A collection of nonlinear mixed complementarity problems. Optimization Methods and Software, 5(4):319–345.
Djordjevic´, S. S. (2019). New hybrid conjugate gradient method as a convex combination of ls and fr methods. Acta Mathematica Scientia, 39(1):214–228.
Dolan, E. D. and Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical programming, 91(2):201–213.
Feng, D., Sun, M., and Wang, X. (2017). A family of conjugate gradient methods for large-scale nonlinear equations. Journal of inequalities and applications, 2017(1):1–8.
Hassan Ibrahim, A., Kumam, P., Abubakar, A. B., Abubakar, J., and Muhammad, A. B. (2020). Least-square-based three-term conjugate gradient projection method for `1-norm problems with application to compressed sensing. Mathematics, 8(4):602.
Huang, N., Ma, C., and Xie, Y. (2016). The derivative-free double newton step methods for solving system of nonlinear equations. Mediterranean Journal of Mathematics, 13(4):2253–2270.
Ibrahim, A. H., Garba, A. I., Usman, H., Abubakar, J., and Abubakar, A. B. (2019a). Derivative-free rmil conjugate gradient algorithm for convex constrained equations. Thai Journal of Mathematics, 18(1).
Ibrahim, A. H., Garba, A. I., Usman, H., Abubakar, J., and Abubakar, A. B. (2019b). Derivative-free rmil conjugate gradient algorithm for convex constrained equations. Thai Journal of Mathematics, 18(1).
Ibrahim, A. H., Kumam, P., Abubakar, A. B., Jirakitpuwapat, W., and Abubakar, J. (2020). A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing. Heliyon, 6(3):e03466.
La Cruz, W. (2017). A spectral algorithm for large-scale systems of nonlinear monotone equations. Numerical Algorithms, 76(4):1109–1130.
La Cruz, W., Martínez, J., and Raydan, M. (2006). Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Mathematics of Computation, 75(255):1429–1448.
Lee, D. D. and Seung, H. S. (2001). Algorithms for non-negative matrix factorization. In Advances in neural information processing systems, pages 556–562.
Liu, J. and Feng, Y. (2018). A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numerical Algorithms, pages 1–18.
Mohamed, N. S., Mamat, M., Rivaie, M., and Shaharudin, S. M. (2020). A new hyhbrid coefficient of conjugate gradient method. Indonesian Journal of Electrical Engineering and Computer Science, 18(3):1454–1463.
Mohammad, H. and Abubakar, A. B. (2020). A descent derivative-free algorithm for nonlinear monotone equations with convex constraints. RAIRO-Operations Research, 54(2):489–505.
Mohammad, H. and Waziri, M. Y. (2015). On broyden-like update via some quadratures for solving nonlinear systems of equations. Turkish Journal of Mathematics, 39(3):335–345.
Qi, L. and Sun, J. (1993). A nonsmooth version of newton’s method. Mathematical programming, 58(1-3):353–367.
Solodov, M. V. and Svaiter, B. F. (1999). A new projection method for variational inequality problems. SIAM Journal on Control and Optimization, 37(3):765–776.
Yamashita, N. and Fukushima, M. (2001). On the rate of convergence of the levenberg-marquardt method. In Topics in numerical analysis, pages 239–249. Springer.
Yu, Z., Lin, J., Sun, J., Xiao, Y. H., Liu, L., and Li, Z. H. (2009). Spectral gradient projection method for monotone nonlinear equations with convex constraints. Applied Numerical Mathematics, 59(10):2416–2423.
Zhang, L. and Zhou, W. (2006). Spectral gradient projection method for solving nonlinear monotone equations. Journal of Computational and Applied Mathematics, 196(2):478–484.
Abubakar, A. B., Rilwan, J., Yimer, S. E., Ibrahim, A. H., and Ahmed, I. (2020b). Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints. Thai Journal of Mathematics, 18(1):501–517.
Berry, M. W., Browne, M., Langville, A. N., Pauca, V. P., and Plemmons, R. J. (2007). Algorithms and applications for approximate nonnegative matrix factorization. Computational statistics & data analysis, 52(1):155–173.
Bing, Y. and Lin, G. (1991). An efficient implementation of merrill’s method for sparse or partially separable systems of nonlinear equations. SIAM Journal on Optimization, 1(2):206–221.
Blumensath, T. (2013). Compressed sensing with nonlinear observations and related nonlinear optimization problems. IEEE Transactions on Information Theory, 59(6):3466–3474.
Dai, Z., Dong, X., Kang, J., and Hong, L. (2020). Forecasting stock market returns: New technical indicators and two-step economic constraint method. The North American Journal of Economics and Finance, page 101216.
Dennis, J. E. and Moré, J. J. (1974). A characterization of superlinear convergence and its application to quasi-newton methods. Mathematics of computation, 28(126):549–560.
Dennis, Jr, J. E. and Moré, J. J. (1977). Quasi-newton methods, motivation and theory. SIAM review, 19(1):46–89.
Dennis Jr, J. E. (1983). Rb schnabel numerical methods for unconstrained optimization and nonlinear equations.
Ding, Y., Xiao, Y., and Li, J. (2017). A class of conjugate gradient methods for convex constrained monotone equations. Optimization, 66(12):2309–2328.
Dirkse, S. P. and Ferris, M. C. (1995). Mcplib: A collection of nonlinear mixed complementarity problems. Optimization Methods and Software, 5(4):319–345.
Djordjevic´, S. S. (2019). New hybrid conjugate gradient method as a convex combination of ls and fr methods. Acta Mathematica Scientia, 39(1):214–228.
Dolan, E. D. and Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical programming, 91(2):201–213.
Feng, D., Sun, M., and Wang, X. (2017). A family of conjugate gradient methods for large-scale nonlinear equations. Journal of inequalities and applications, 2017(1):1–8.
Hassan Ibrahim, A., Kumam, P., Abubakar, A. B., Abubakar, J., and Muhammad, A. B. (2020). Least-square-based three-term conjugate gradient projection method for `1-norm problems with application to compressed sensing. Mathematics, 8(4):602.
Huang, N., Ma, C., and Xie, Y. (2016). The derivative-free double newton step methods for solving system of nonlinear equations. Mediterranean Journal of Mathematics, 13(4):2253–2270.
Ibrahim, A. H., Garba, A. I., Usman, H., Abubakar, J., and Abubakar, A. B. (2019a). Derivative-free rmil conjugate gradient algorithm for convex constrained equations. Thai Journal of Mathematics, 18(1).
Ibrahim, A. H., Garba, A. I., Usman, H., Abubakar, J., and Abubakar, A. B. (2019b). Derivative-free rmil conjugate gradient algorithm for convex constrained equations. Thai Journal of Mathematics, 18(1).
Ibrahim, A. H., Kumam, P., Abubakar, A. B., Jirakitpuwapat, W., and Abubakar, J. (2020). A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing. Heliyon, 6(3):e03466.
La Cruz, W. (2017). A spectral algorithm for large-scale systems of nonlinear monotone equations. Numerical Algorithms, 76(4):1109–1130.
La Cruz, W., Martínez, J., and Raydan, M. (2006). Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Mathematics of Computation, 75(255):1429–1448.
Lee, D. D. and Seung, H. S. (2001). Algorithms for non-negative matrix factorization. In Advances in neural information processing systems, pages 556–562.
Liu, J. and Feng, Y. (2018). A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numerical Algorithms, pages 1–18.
Mohamed, N. S., Mamat, M., Rivaie, M., and Shaharudin, S. M. (2020). A new hyhbrid coefficient of conjugate gradient method. Indonesian Journal of Electrical Engineering and Computer Science, 18(3):1454–1463.
Mohammad, H. and Abubakar, A. B. (2020). A descent derivative-free algorithm for nonlinear monotone equations with convex constraints. RAIRO-Operations Research, 54(2):489–505.
Mohammad, H. and Waziri, M. Y. (2015). On broyden-like update via some quadratures for solving nonlinear systems of equations. Turkish Journal of Mathematics, 39(3):335–345.
Qi, L. and Sun, J. (1993). A nonsmooth version of newton’s method. Mathematical programming, 58(1-3):353–367.
Solodov, M. V. and Svaiter, B. F. (1999). A new projection method for variational inequality problems. SIAM Journal on Control and Optimization, 37(3):765–776.
Yamashita, N. and Fukushima, M. (2001). On the rate of convergence of the levenberg-marquardt method. In Topics in numerical analysis, pages 239–249. Springer.
Yu, Z., Lin, J., Sun, J., Xiao, Y. H., Liu, L., and Li, Z. H. (2009). Spectral gradient projection method for monotone nonlinear equations with convex constraints. Applied Numerical Mathematics, 59(10):2416–2423.
Zhang, L. and Zhou, W. (2006). Spectral gradient projection method for solving nonlinear monotone equations. Journal of Computational and Applied Mathematics, 196(2):478–484.