A Barzilai Borwein Adaptive Trust-Region Method for Solving Systems of Nonlinear Equation

Document Type: Research Paper


1 Department of Mathematics, Payame Noor University, PO BOX 19395--3697, Tehran, Iran.

2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.


In this paper, we introduce a new adaptive trust-region approach to solve systems of nonlinear equations. In order to improve the efficiency of adaptive radius strategy proposed by Esmaeili and Kimiaei [8], Barzilai Borwein technique (BB) [3] with low memory is used which can truly control the trust-region radius. In addition, the global convergence of the new approach is proved. Computational experience suggests that the new approach is more effective in practice in comparison with other adaptive trust-region algorithms.


Main Subjects

[1]      Ahookhosh, M., & Amini, K. (2010). A nonmonotone trust region method with adaptive radius for unconstrained optimization problems. Computers & mathematics with applications60(3), 411-422.

[2]      Ahookhosh, M., Amini, K., & Peyghami, M. R. (2012). A nonmonotone trust-region line search method for large-scale unconstrained optimization. Applied mathematical modelling36(1), 478-487.

[3]      Barzilai, J., & Borwein, J. M. (1988). Two-point step size gradient methods. IMA journal of numerical analysis8(1), 141-148.

[4]      Bouaricha, A., & Schnabel, R. B. (1998). Tensor methods for large sparse systems of nonlinear equations. Mathematical programming82(3), 377-400.

[5]      Broyden, C. G. (1971). The convergence of an algorithm for solving sparse nonlinear systems. Mathematics of computation25(114), 285-294.

[6]      Conn, A. R., Gould, N. I., & Toint, P. L. (2000). Trust region methods. Society for Industrial and Applied Mathematics.

[7]      Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical programming91(2), 201-213.

[8]      Esmaeili, H., & Kimiaei, M. (2014). A new adaptive trust-region method for system of nonlinear equations. Applied mathematical modelling38(11), 3003-3015.

[9]      Fan, J. (2006). Convergence rate of the trust region method for nonlinear equations under local error bound condition. Computational optimization and applications34(2), 215-227.

[10]  Fan, J., & Pan, J. (2011). An improved trust region algorithm for nonlinear equations. Computational optimization and applications48(1), 59-70.

[11]  Fan, J., & Pan, J. (2010). A modified trust region algorithm for nonlinear equations with new updating rule of trust region radius. International journal of computer mathematics87(14), 3186-3195.

[12]  Grippo, L., & Sciandrone, M. (2007). Nonmonotone derivative-free methods for nonlinear equations. Computational optimization and applications37(3), 297-328.

[13]  La Cruz, W., Martínez, J., & Raydan, M. (2006). Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Mathematics of computation75(255), 1429-1448.

[14]  Li, D. H., & Fukushima, M. (2000). A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optimization methods and software13(3), 181-201.

[15]  Lukšan, L., & Vlček, J. (1999). Sparse and partially separable test problems for unconstrained and equality constrained optimization. Technical report, 767.

[16]  Moré, J. J., Garbow, B. S., & Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM transactions on mathematical software (TOMS)7(1), 17-41.

[17]  Nesterov, Y. (2007). Modified Gauss–Newton scheme with worst case guarantees for global performance. Optimisation methods and software22(3), 469-483.

[18]  Bonnans, J. F., Gilbert, J. C., Lemaréchal, C., & Sagastizábal, C. A. (2006). Numerical optimization: Theoretical and practical aspects. Springer Science & Business Media.

[19]  Toint, P. L. (1986). Numerical solution of large sets of algebraic nonlinear equations. Mathematics of computation46(173), 175-189.

[20]  Yuan, G., Lu, S., & Wei, Z. (2011). A new trust-region method with line search for solving symmetric nonlinear equations. International journal of computer mathematics88(10), 2109-2123.

[21]  Yuan, G., Wei, Z., & Lu, X. (2011). A BFGS trust-region method for nonlinear equations. Computing92(4), 317-333.

[22]  Yuan, Y. X. (1994). Trust region algorithms for nonlinear equations. Hong Kong Baptist University, Department of Mathematics.

[23]  Zhang, J. L., & Wang, Y. (2003). A new trust region method for nonlinear equations. Mathematical methods of operations research58(2), 283-298.