A descent derivative-free algorithm for nonlinear monotone equations with convex constraints
RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 2, pp. 489-505.

In this paper, we present a derivative-free algorithm for nonlinear monotone equations with convex constraints. The search direction is a product of a positive parameter and the negation of a residual vector. At each iteration step, the algorithm generates a descent direction independent from the line search used. Under appropriate assumptions, the global convergence of the algorithm is given. Numerical experiments show the algorithm has advantages over the recently proposed algorithms by Gao and He (Calcolo 55 (2018) 53) and Liu and Li (Comput. Math. App. 70 (2015) 2442–2453).

DOI : 10.1051/ro/2020008
Classification : 65K05, 90C06, 90C52, 90C56
Mots-clés : Derivative-free method, monotone equations, convex constraints, global convergence
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     title = {A descent derivative-free algorithm for nonlinear monotone equations with convex constraints},
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     publisher = {EDP-Sciences},
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Mohammad, Hassan; Bala Abubakar, Auwal. A descent derivative-free algorithm for nonlinear monotone equations with convex constraints. RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 2, pp. 489-505. doi : 10.1051/ro/2020008. http://www.numdam.org/articles/10.1051/ro/2020008/

[1] A.B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations. Comput. Appl. Math. 37 (2018) 6760–6773. | DOI | MR | Zbl

[2] A.B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations. Numer. Algorithms 81 (2019) 197–210. | DOI | MR

[3] A.B. Abubakar and M.Y. Waziri, A matrix-free approach for solving systems of nonlinear equations. J. Mod. Methods Numer. Math. 7 (2016) 1–9. | DOI | MR | Zbl

[4] A.B. Abubakar and P. Kumam, A.M. Awwal, P. Thounthong, A modified self-adaptive conjugate gradient method for solving convex constrained monotone nonlinear equations for signal reovery problems. Mathematics 7 (2019) 693. | DOI

[5] A.B. Abubakar, P. Kumam, H. Mohammad and A.M. Awwal, An efficient conjugate gradient method for convex constrained monotone nonlinear equations with applications. Mathematics 7 (2019) 767. | DOI

[6] A.B. Abubakar, P. Kumam, H. Mohammad, A.M. Awwal and S. Kanokwan, A modified Fletcher-Reeves conjugate gradient method for monotone nonlinear equations with some applications. Mathematics 7 (2019) 745. | DOI

[7] M. Ahookhosh, K. Amini and S. Bahrami, Two derivative-free projection approaches for systems of large-scale nonlinear monotone equations. Numer. Algorithms 64 (2013) 21–42. | DOI | MR | Zbl

[8] J. Barzilai and J.M. Borwein, Two-point step size gradient methods. IMA J. Numer. Anal. 8 (1988) 141–148. | DOI | MR | Zbl

[9] S. Bellavia, D. Bertaccini and B. Morini, Nonsymmetric preconditioner updates in Newton–krylov methods for nonlinear systems. SIAM J. Sci. Comput. 33 (2011) 2595–2619. | DOI | MR | Zbl

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

[11] E.D. Dolan and J.J. Moré, Benchmarking optimization software with performance profiles. Math. Program. 91 (2002) 201–213. | DOI | MR | Zbl

[12] P. Gao and C. He, An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo 55 (2018) 53. | DOI | MR | Zbl

[13] B. Ghaddar, J. Marecek and M. Mevissen, Optimal power flow as a polynomial optimization problem. IEEE Trans. Power Syst. 31 (2016) 539–546. | DOI

[14] Y. Hu and Z. Wei, Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints. Int. J. Comput. Math. 92 (2015) 2261–2272. | DOI | MR

[15] W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18 (2003) 583–599. | DOI | MR | Zbl

[16] W. La Cruz, J. Martnez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75 (2006) 1429–1448. | DOI | MR | Zbl

[17] W.J. Leong, M.A. Hassan and M.Y. Waziri, A matrix-free quasi-Newton method for solving large-scale nonlinear systems. Comput. Math. App. 62 (2011) 2354–2363. | MR | Zbl

[18] M. Li, An Liu-Storey-Type method for solving large-scale nonlinear monotone equations. Numer. Funct. Anal. Optim. 35 (2014) 310–322. | DOI | MR | Zbl

[19] Q. Li and D.H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 31 (2011) 1625–1635. | DOI | MR | Zbl

[20] J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algorithms 82 (2018) 245–262. | DOI | MR

[21] J. Liu and S.J. Li, A projection method for convex constrained monotone nonlinear equations with applications. Comput. Math. App. 70 (2015) 2442–2453. | MR

[22] J.K. Liu and S.J. Li, A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo 53 (2016) 427–450. | DOI | MR | Zbl

[23] H. Liu, Z. Liu and X. Dong, A new adaptive Barzilai and Borwein method for unconstrained optimization. Optim. Lett. 12 (2018) 845–873. | DOI | MR

[24] F. Ma and C. Wang, Modified projection method for solving a system of monotone equations with convex constraints. J. Appl. Math. Comput. 34 (2010) 47–56. | DOI | MR | Zbl

[25] H. Mohammad, Barzilai–Borwein-like method for solving large-scale non-linear systems of equations. J. Niger. Math. Soc. 36 (2017) 71–83. | MR

[26] H. Mohammad and A.B. Abubakar, A positive spectral gradient-like method for large-scale nonlinear monotone equations. Bull. Comput. Appl. Math. 5 (2017) 99–115. | MR | Zbl

[27] H. Mohammad and M.Y. Waziri, On Broyden-like update via some quadratures for solving nonlinear systems of equations. Turkish J. Math. 39 (2015) 335–345. | DOI | MR | Zbl

[28] H. Mohammad and M.Y. Waziri, Structured two-point stepsize gradient methods for nonlinear least squares. J. Optim. Theory Appl. 181 (2019) 298–317. | DOI | MR

[29] J. Nocedal and S.J. Wright. Numerical Optimization. Springer Science (2006). | MR | Zbl

[30] M. Raydan, On the Barzilai and Borwein choice of steplength for the gradient method. IMA J. Numer. Anal. 13 (1993) 321–326. | DOI | MR | Zbl

[31] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7 (1997) 26–33. | DOI | MR | Zbl

[32] M.V. Solodov and B.F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations. Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Springer (1998) 355–369. | MR | Zbl

[33] W. Sun and Y.X. Yuan, Optimization Theory and Methods: Nonlinear Programming. Springer Science & Business Media 1 (2006). | MR | Zbl

[34] C. Wang, Y. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 66 (2007) 33–46. | DOI | MR | Zbl

[35] X.Y. Wang, S.J. Li and X.P. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints. Calcolo 53 (2016) 133–145. | DOI | MR | Zbl

[36] M.Y. Waziri and J. Sabi’U, A derivative-free conjugate gradient method and its global convergence for solving symmetric nonlinear equations. Int. J. Math. Math. Sci. 2015 (2015) 961487. | DOI | MR | Zbl

[37] M.Y. Waziri, W.J. Leong, M.A. Hassan and M. Monsi, Jacobian computation-free Newton’s method for systems of non-linear equations. J. Numer. Math. Stochastic 2 (2010) 54–63. | MR

[38] A.J. Wood and B.F. Wollenberg. Power Generation, Operation, and Control. John Wiley & Sons (2012).

[39] Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. App. 405 (2013) 310–319. | DOI | MR | Zbl

[40] Q.R. Yan, X.Z. Peng and D.H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations. J. Comput. Appl. Math. 234 (2010) 649–657. | DOI | MR | Zbl

[41] Z. Yu, J. Lin, J. Sun, Y.H. Xiao, L. Liu and Z.H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59 (2009) 2416–2423. | DOI | MR | Zbl

[42] L. Zhang and W. Zhou, Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196 (2006) 478–484. | DOI | MR | Zbl

[43] W.J. Zhou and D.H. Li, A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77 (2008) 2231–2240. | DOI | MR | Zbl

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