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Numerical Algorithms

25.04.2024 | Original Paper

A two-step Broyden-like method for nonlinear equations

verfasst von: Jingyong Tang, Jinchuan Zhou

Erschienen in: Numerical Algorithms

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Abstract

In this paper, based on a nonmonotone derivative-free line search, we propose a two-step Broyden-like method (denoted by TS-BLM) for solving the nonlinear equations. TS-BLM computes one quasi-Newton step at the beginning iterations. When the iterations are close to the solution set of the nonlinear equations, an additional approximate quasi-Newton step is calculated by solving a linear system formed by using the previous Broyden-like matrix. We prove that TS-BLM converges globally under appropriate conditions. Moreover, we analyze the convergence rate of TS-BLM depending on the approximation of the Broyden-like matrix to the Jacobian. Some numerical results are reported to show the superior numerical performances of TS-BLM compared with the traditional BLM.

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Literatur
1.
Amini, K., Rostami, F.: A new modified two steps Levenberg-Marquardt method for nonlinear equations. J. Comput. Appl. Math. 288, 341–350 (2015)MathSciNetCrossRef
2.
Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965)MathSciNetCrossRef
3.
Chen, B., Ma, C.: A new smoothing Broyden-like method for solving nonlinear complementarity problem with a \(P_0\)-function. J. Global Optim. 51, 473–495 (2011)MathSciNetCrossRef
4.
Chen, B., Ma, C.: Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem. Nonlinear Anal. Real World Appl. 12, 1250–1263 (2011)MathSciNetCrossRef
5.
6.
Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems, Vol. I and II. Springer, New York (2003)
7.
Fan, B.: A smoothing Broyden-like method with a nonmonotone derivative-free line search for nonlinear complementarity problems. J. Comput. Appl. Math. 290, 641–655 (2015)MathSciNetCrossRef
8.
Fan, J.Y.: The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence. Math. Comput. 81, 447–466 (2012)MathSciNetCrossRef
9.
Fan, J.Y.: Accelerating the modified Levenberg-Marquardt method for nonlinear equations. Math. Comput. 83, 1173–1187 (2014)MathSciNetCrossRef
10.
Kelley, C.T.: Iterative methods for optimization, Frontiers in Applied Mathematics, 18. SIAM, Philadelphia (1999)CrossRef
11.
Li, D.H., Fukushima, M.: A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optim. Methods Softw. 13(3), 181–201 (2000)MathSciNetCrossRef
12.
Li, D.H., Fukushima, M.: Globally convergent Broyden-Like methods for semismooth equations and applications to VIP. NCP and MCP. Ann. Oper. Res. 103, 71–97 (2001)MathSciNetCrossRef
13.
14.
Li, Y.F., Sun, G., Wang, Y.J.: A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numer. Algebra. Control. Optim. 1(3), 529–537 (2011)CrossRef
15.
La Cruz, W., Martinez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75(255), 1429–1448 (2006)MathSciNetCrossRef
16.
17.
Mannel, F.: Convergence properties of the Broyden-like method for mixed linear-nonlinear systems of equations. Numer. Algor. 88, 853–881 (2021)MathSciNetCrossRef
18.
Mannel, F.: On the convergence of Broyden’s method and some accelerated schemes for singular problems. IMA J. Numer. Anal. 43, 414–442 (2023)MathSciNetCrossRef
19.
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)
20.
Powell, M.J.D.: A Fortran subroutine for solving systems of nonlinear algebraic equations. In: Rabinowitz, P. (Ed.), Numerical Methods for Nonlinear Algebraic Equations. Gordon and Breach, London (Chapter 7) (1970)
21.
Tang, J.Y., Zhou, J.C.: A smoothing quasi-Newton method for solving general second-order cone complementarity problems. J. Global Optim. 80, 415–438 (2021)MathSciNetCrossRef
22.
Tang, J.Y., Zhou, J.C.: Smoothing inexact Newton method based on a new derivative-free nonmonotone line search for the NCP over circular cones. Ann. Oper. Res. 295, 787–808 (2020)MathSciNetCrossRef
23.
Tang, J.Y., Zhou, J.C., Sun, Z.F.: A derivative-free line search technique for Broyden-like method with applications to NCP, wLCP and SI. Ann. Oper. Res. 321, 541–564 (2023)MathSciNetCrossRef
24.
Tang, J.Y., Zhou, J.C., Zhang, H.C.: An accelerated smoothing Newton method with cubic convergence for weighted complementarity problems. J. Optim. Theory Appl. 196, 641–665 (2023)MathSciNetCrossRef
25.
Wang, H.Y.: Convergence rate of the accelerated modified Levenberg-Marquardt method under Hölderian local error bound. Pac. J. Optim. 8(3), 661–677 (2022)
26.
Zheng, L., Chen, L., Ma, Y.F.: Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound. Open Math. 20, 998–1012 (2022)MathSciNetCrossRef
27.
Zhou, W.J.: On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order Armijo type line search. J. Comput. Appl. Math. 239, 152–161 (2013)MathSciNetCrossRef
Metadaten
Titel
A two-step Broyden-like method for nonlinear equations
verfasst von
Jingyong Tang
Jinchuan Zhou
Publikationsdatum
25.04.2024
Verlag
Springer US
Erschienen in
Numerical Algorithms
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-024-01827-7

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