Three-term derivative-free projection method for solving nonlinear monotone equations

Abstract

In this paper, a three-term derivative-free projection method is proposed for solving nonlinear monotone equations. Under some appropriate conditions, the global convergence and R-linear convergence rate of the proposed method are analyzed and proved. With no need of any derivative information, the proposed method is able to solve large-scale nonlinear monotone equations. Numerical comparisons show that the proposed method is effective.

Keywords

Nonlinear monotone equations / conjugate gradient method / derivative-free projection method / global convergence / R-linear convergence rate

References

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

[2] Andrei N. On three-term conjugate gradient algorithms for unconstrained optimization. Appl Math Comput 2013; 219: 6316–6317

[3] Cheng W Y. A PRP type method for systems of monotone equations. Math Comput Modelling 2009; 50: 15–20

[4] Dennis J E, More J J. A characterization of superlinear convergence and its application to quasi-Newton methods. Math Comp 1974; 28(126): 549–560

[5] Dennis J E, More J J. Quasi-Newton method, motivation and theory. SIAM Rev 1997; 19(1): 46–89

[6] Dirkse S P, Ferris M C. MCPLIB: A collection of nonlinear mixed complementarity problems. Optim Methods Softw 2002; 5(4): 319–345

[7] Hager W W, Zhang H. A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J Optim 2005; 16(1): 170–192

[8] Hestens M R, Stiefel E L. Methods of conjugate gradients for solving linear systems. J Research Nat Bur Standards 1952; 49(6): 409–436

[9] Iusem A N, Solodov M V. Newton-type methods with generalized distances for constrained optimization. Optimization 1997; 41(3): 257–278

[10] La Cruz M, Martinez J M, Raydan M. Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math Comp 2006; 75(255): 1429–1448

[11] Li D H, Fukushima M. A global and superlinear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM J Numer Anal 1999; 37(1): 152–172

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

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

[14] Meintjes K, Morgan A P. A methodology for solving chemical equilibrium systems. Appl Math Comput 1987; 22(4): 333–361

[15] Polyak B T. The conjugate gradient method in extremal problems. USSR Comput Math and Math Phys 1969; 9(4): 94–112

[16] Polak E, Ribière G. Note sur la convergence de méthodes de directions conjuguées. Rev Francaise Informat Recherche Opérationnelle 1969; 3(16): 35–43

[17] SolodovM VSvaiterB F. A globally convergent inexact Newton method for systems of monotone equations. In: Reformulation: Non-smooth, Piecewise Smooth, Semismooth and Smoothing Methods. Dordrecht: Springer, 1999, 355–369

[18] Wu X Y, Sai N E Z, Zhang H L. On three-term HS projection algorithm for solving nonlinear monotone equations. Southwest China Normal Univ Natur Sci Ed 2016; 41(5): 41–47

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

[20] Zhao Y B, Li D H. Monotonicity of fixed point and normal mapping associated with variational inequality and its application. SIAM J Optim 2000; 11(4): 962–973

[21] Zhou G, Toh K C. Superline convergence of a Newton-type algorithm for monotone equations. J Optim Theory Appl 2005; 125(1): 205–221

[22] Zhou W J, Li D H. Limited memory BFGS method for nonlinear monotone equations. J Comput Math 2007; 25(1): 89–96

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

RIGHTS & PERMISSIONS

Higher Education Press 2023