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A Derivative-Free Multivariate Spectral Projection Algorithm for Constrained NonLinear Monotone Equations

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Abstract

In this paper, we present a derivative-free multivariate spectral projection algorithm for convex constrained nonlinear monotone equations. The search direction is a product of a convex combination of two different spectral diagonal matrices and the residual vector. Moreover, to ensure positive definiteness of the diagonal matrix associated with the search direction, suitable safeguard is formulated. Some of the remarkable properties of the algorithm include: Jacobian free approach, capacity to solve large-scale problems and the search direction generated by the algorithm, satisfy the descent property independent on the line search employed. Under appropriate assumptions, the global convergence of the algorithm is given. Numerical experiments show that the algorithm has advantages over the recently proposed multivariate derivative-free projection algorithm by Liu and Li (J Ind Manag Optim 13(1):283–295, 2017) and also compete with another algorithm having the standard choice of the Barzilai-Borwein step as the search direction.

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References

  1. Abubakar, A.B., Kumam, P.: An improved three-term derivative-free method for solving nonlinear equations. Comput. Appl. Math. 37, 6760–6773 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Abubakar, A.B., Kumam, P., Mohammad, H.: A note on the spectral gradient projection method for nonlinear monotone equations with applications. Comput. Appl. Math. 39, 129 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Ahookhosh, M., Amini, K., Bahrami, S.: Two derivative-free projection approaches for systems of large-scale nonlinear monotone equations. Numer. Algorithms 64(1), 21–42 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Awwal, A.M., Kumam, P., Mohammad, H., Watthayu, W., Abubakar, A.B.: A perry-type derivative-free algorithm for solving nonlinear system of equations and minimizing \(ell_1\) regularized problem. Optimization (2020a). https://doi.org/10.1080/02331934.2020.1808647

  5. Awwal, A.M., Wang, L., Kumam, P., Mohammad, H.: A two-step spectral gradient projection method for system of nonlinear monotone equations and image deblurring problems. Symmetry 12(6), 874 (2020b)

    Google Scholar 

  6. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Bellavia, S., Bertaccini, D., Morini, B.: Nonsymmetric preconditioner updates in Newton-Krylov methods for nonlinear systems. SIAM J. Sci. Comput. 33(5), 2595–2619 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Bing, Y., Lin, G.: An efficient implementation of merrill’s method for sparse or partially separable systems of nonlinear equations. SIAM J. Optim. 1(2), 206–221 (1991). https://doi.org/10.1137/0801015

    Article  MathSciNet  MATH  Google Scholar 

  9. Dai, Y.H., Al-Baali, M., Yang, X.: A positive Barzilai-Borwein-like stepsize and an extension for symmetric linear systems. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds.) Numerical Analysis and Optimization, pp. 59–75. Springer International Publishing, Cham (2015a)

    MATH  Google Scholar 

  10. Dai, Z., Chen, X., Wen, F.: A modified perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Han, L., Yu, G., Guan, L.: Multivariate spectral gradient method for unconstrained optimization. Appl. Math. Comput. 201(1–2), 621–630 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Hu, Y., Wei, Z.: Wei-Yao-Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints. Int. J. Comput. Math. 92(11), 2261–2272 (2015)

    MathSciNet  MATH  Google Scholar 

  14. La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18(5), 583–599 (2003)

    MathSciNet  MATH  Google Scholar 

  15. La Cruz, W., Martínez, J., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75(255), 1429–1448 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Landi, G., Piccolomini, E.L.: A feasible direction method for image restoration. Optim. Lett. 6(8), 1795–1817 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Leong, W.J., Hassan, M.A., Yusuf, M.W.: A matrix-free quasi-Newton method for solving large-scale nonlinear systems. Comput. Math. Appl. 62(5), 2354–2363 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Li, M.: An Liu-Storey-type method for solving large-scale nonlinear monotone equations. Numer. Funct. Anal. Optim. 35(3), 310–322 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Li, Q., Li, D.H.: A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 31(4), 1625–1635 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Liu, J., Li, S.: Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. J. Ind. Manag. Optim. 13(1), 283–295 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Liu, J., Li, S.J.: A projection method for convex constrained monotone nonlinear equations with applications. Comput. Math. Appl. 70(10), 2442–2453 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Ma, F., Wang, C.: Modified projection method for solving a system of monotone equations with convex constraints. J. Appl. Math. Comput. 34(1), 47–56 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Mohammad, H.: Barzilai-Borwein-like method for solving large-scale nonlinear systems of equations. J. Niger. Math. Soc. 36(1), 71–83 (2017)

    Google Scholar 

  24. Mohammad, H.: A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. J. Ind. Manag. Optim. 17(1), 101–116 (2021). https://doi.org/10.3934/jimo.2019101

    Article  MathSciNet  Google Scholar 

  25. Mohammad, H., Abubakar, A.B.: A positive spectral gradient-like method for large-scale nonlinear monotone equations. Bull. Comput. Appl. Math. 5(1), 97–113 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Mohammad, H., Abubakar, A.B.: A descent derivative-free algorithm for nonlinear monotone equations with convex constraints. RAIRO-Oper. Res. 54(2), 489–505 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Mohammad, H., Santos, S.A.: A structured diagonal Hessian approximation method with evaluation complexity analysis for nonlinear least squares. Comput. Appl. Math. (2018). https://doi.org/10.1007/s40314-018-0696-1

    Article  MathSciNet  MATH  Google Scholar 

  28. Mohammad, H., Waziri, M.Y.: On Broyden-like update via some quadratures for solving nonlinear systems of equations. Turk. J. Math. 39(3), 335–345 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Nikan, O., Avazzadeh, Z., Machado, J.T.: Numerical investigation of fractional nonlinear Sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics. Eng. Anal. Bound. Elem. 120, 223–237 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Nikan, O., Avazzadeh, Z., Machado, J.T.: An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model. J. King Saud Univ.-Sci. 33(1), 101243 (2020)

    Google Scholar 

  31. Nikan, O., Machado, J.T., Avazzadeh, Z., Jafari, H.: Numerical evaluation of fractional tricomi-type model arising from physical problems of gas dynamics. J. Adv. Res. 25, 205–216 (2020)

    Google Scholar 

  32. Nikan, O., Machado, J.T., Golbabai, A.: Numerical solution of time-fractional fourth-order reaction-diffusion model arising in composite environments. Appl. Math. Model. 89, 819–836 (2021)

    MathSciNet  Google Scholar 

  33. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Science, New York (2006)

    MATH  Google Scholar 

  34. Ou, Y., Liu, Y.: Supermemory gradient methods for monotone nonlinear equations with convex constraints. Comput. Appl. Math. 36(1), 259–279 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Solodov, M.V., Svaiter, B.: A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima, M., Liqun, Q. (eds.) Reformulation Nonsmooth Piecewise Smooth, Semismooth and Smoothing Methods, pp. 355–369. New York, Springer (1998)

    Google Scholar 

  36. Sun, W., Yuan, Y.: Optimization Theory and Methods: Nonlinear Programming, vol. 1. Springer Science & Business Media, New York (2006)

    MATH  Google Scholar 

  37. Wan, Z., Chen, Y., Huang, S., Feng, D.: A modified nonmonotone BFGS algorithm for solving smooth nonlinear equations. Optim. Lett. 8(6), 1845–1860 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Wang, C., Wang, Y., Xu, C.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 66(1), 33–46 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Waziri, M.Y., Ahmed, K., Sabi’u, J.: A family of Hager-Zhang conjugate gradient methods for system of monotone nonlinear equations. Appl. Math. Comput. 361, 645–660 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Xiao, Y., Zhu, H.: A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405(1), 310–319 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Yan, Q.-R., Peng, X.-Z., Li, D.-H.: A globally convergent derivative-free method for solving large-scale nonlinear monotone equations. J. Comput. Appl. Math. 234(3), 649–657 (2010)

    MathSciNet  MATH  Google Scholar 

  42. Yu, G., Niu, S., Ma, J.: Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. J. Ind. Manag. Optim. 9(1), 117–129 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Yu, Z., Lin, J., Sun, J., Xiao, Y.H., Liu, L.Y., Li, Z.H.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59(10), 2416–2423 (2009)

    MathSciNet  MATH  Google Scholar 

  44. Zhang, L., Zhou, W.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196(2), 478–484 (2006)

    MathSciNet  MATH  Google Scholar 

  45. Zhou, W.J., Li, D.H.: A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77(264), 2231–2240 (2008)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Associate Professor Jinkui Liu affiliated at the Chongqing Three Gorges University, Chongqing, China, for providing us with access to the MSDY MATLAB codes. We thanked the anonymous reviewers for their useful comments that lead to improvements in the quality and presentation of this paper. The third author acknowledge with thanks, the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Physical Science, Bayero University Kano, Kano, 700241, Kano, Nigeria

    Hassan Mohammad, Mohammed Yusuf Waziri & Auwal Bala Abubakar

  2. Numerical Optimization Research Group, Bayero University, Kano, Nigeria

    Hassan Mohammad & Mohammed Yusuf Waziri

  3. Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa, 0204, South Africa

    Auwal Bala Abubakar

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  1. Hassan Mohammad
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Mohammad, H., Waziri, M.Y. & Abubakar, A.B. A Derivative-Free Multivariate Spectral Projection Algorithm for Constrained NonLinear Monotone Equations. Int. J. Appl. Comput. Math 7, 55 (2021). https://doi.org/10.1007/s40819-021-00995-7

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