AN EFFICIENT HYBRID DERIVATIVE-FREE PROJECTION ALGORITHM FOR CONSTRAINT NONLINEAR EQUATIONS
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Published:
Oct 31, 2021
Keywords:
Nonlinear equations, Conjugate gradient method, Projection method, Global convergence.
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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.
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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
Licensee MJS, Universiti Malaya, Malaysia. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
References
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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.