A line search trust-region algorithm with nonmonotone adaptive radius for a system of nonlinear equations.

*(English)*Zbl 1342.90178Summary: In this paper, a trust-region procedure is proposed for the solution of nonlinear equations. The proposed approach takes advantages of an effective adaptive trust-region radius and a nonmonotone strategy by combining both of them appropriately. It is believed that selecting an appropriate adaptive radius based on a suitable nonmonotone strategy can improve the efficiency and robustness of the trust-region frameworks as well as decrease the computational cost of the algorithm by decreasing the required number subproblems that must be solved. The global convergence and the local Q-quadratic convergence rate of the proposed approach are proved. Preliminary numerical results of the proposed algorithm are also reported which indicate the promising behavior of the new procedure for solving the nonlinear system.

##### MSC:

90C30 | Nonlinear programming |

93E24 | Least squares and related methods for stochastic control systems |

34A34 | Nonlinear ordinary differential equations and systems, general theory |

##### Keywords:

nonlinear equations; trust-region; adaptive radius; nonmonotone technique; Armijo-type line search##### Software:

STRSCNE
Full Text:
DOI

##### References:

[1] | Ahookhosh, M; Amini, K, A nonmonotone trust-region method with adaptive radius for unconstrained optimization, Comput Math Appl, 60, 411-422, (2010) · Zbl 1201.90184 |

[2] | Ahookhosh, M; Amini, K; Peyghami, MR, A nonmonotone trust-region line search method for large-scale unconstrained optimization, Appl Math Model, 36, 478-487, (2012) · Zbl 1236.90077 |

[3] | Bellavia, S; Macconi, M; Morini, B, STRSCNE: a scaled trust-region solver for constrained nonlinear equations, Comput Optim Appl, 28, 31-50, (2004) · Zbl 1056.90128 |

[4] | Conn AR, Gould NIM, Toint PhL (2000) Trust-region methods. Society for Industrial and Applied Mathematics SIAM, Philadelphia · Zbl 0958.65071 |

[5] | Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math Program, 91, 201-213, (2002) · Zbl 1049.90004 |

[6] | Esmaeili, H; Kimiaei, M, A new adaptive trust-region method for system of nonlinear equations, Appl Math Model, 38, 3003-3015, (2014) |

[7] | Esmaeili, H; Kimiaei, M, An efficient adaptive trust-region method for systems of nonlinear equations, Int J Comput Math, 92, 151-166, (2015) · Zbl 1308.90167 |

[8] | Fan, JY, Convergence rate of the trust region method for nonlinear equations under local error bound condition, Comput Optim Appl, 34, 215-227, (2005) · Zbl 1121.65054 |

[9] | Fan JY(2011)An improved trust region algorithmfor nonlinear equations.ComputOptimAppl 48(1):59-70 · Zbl 0616.65067 |

[10] | Fan, JY; Pan, JY, A modified trust region algorithm for nonlinear equations with new updating rule of trust region radius, Int J Comput Math, 87, 3186-3195, (2010) · Zbl 1207.65055 |

[11] | Fasano, G; Lampariello, F; Sciandrone, M, A truncated nonmonotone Gauss-Newton method for large-scale nonlinear least-squares problems, Comput Optim Appl, 34, 343-358, (2006) · Zbl 1122.90094 |

[12] | Fischer, A; Shukla, PK; Wang, M, On the inexactness level of robust Levenberg-Marquardt methods, Optimization, 59, 273-287, (2010) · Zbl 1196.65097 |

[13] | Gertz EM (1999) Combination trust-region line-search methods for unconstrained optimization. University of California San Diego, San Diego |

[14] | Grippo, L; Lampariello, F; Lucidi, S, A nonmonotone line search technique for newton’s method, SIAM J Numer Anal, 23, 707-716, (1986) · Zbl 0616.65067 |

[15] | Grippo, L; Lampariello, F; Lucidi, S, A truncated Newton method with nonmonotone linesearch for unconstrained optimization, J Optim Theory Appl, 60, 401-419, (1989) · Zbl 0632.90059 |

[16] | Grippo, L; Lampariello, F; Lucidi, S, A class of nonmonotone stabilization method in unconstrained optimization, Numer Math, 59, 779-805, (1991) · Zbl 0724.90060 |

[17] | Grippo, L; Sciandrone, M, Nonmonotone derivative-free methods for nonlinear equations, Comput Optim Appl, 37, 297-328, (2007) · Zbl 1180.90310 |

[18] | Cruz, W; Raydan, M, Nonmonotone spectral methods for large-scale nonlinear systems, Optim Methods Softw, 18, 583-599, (2003) · Zbl 1069.65056 |

[19] | La Cruz W, Venezuela C, Martínez JM, Raydan M (2004) Spectral residual method without gradient information for solving large-scale nonlinear systems of equations: theory and experiments. In: Technical report RT-04-08, July 2004 |

[20] | Li DH, Fukushima M (2000a) A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optim Methods Softw 13:181-201 · Zbl 0960.65076 |

[21] | Li DH, Fukushima M (2000b) A globally and superlinearly convergent Gauss-Newton-Based BFGS method for symmetric nonlinear equations. SIAM J Numer Anal 37(1):152-172 · Zbl 0946.65031 |

[22] | Lukšan L, Vlček J (1999) Sparse and partially separable test problems for unconstrained and equality constrained optimization. In: Technical report, no 767 · Zbl 0724.90060 |

[23] | Nocedal J, Yuan YX (1998) Combining Trust-region and line-search techniques. Optimization Technology Center mar OTC 98(04) 1998 · Zbl 0909.90243 |

[24] | Nocedal J, Wright SJ (2006) Numerical optimization. Springer, New York · Zbl 1104.65059 |

[25] | Sartenaer, A, Automatic determination of an initial trust region in nonlinear programming, SIAM J Sci Comput, 18, 1788-1803, (1997) · Zbl 0891.90151 |

[26] | Toint Ph L (1982) Towards an efficient sparsity exploiting Newton method for minimization. In: Sparse matrices and their uses. Academic Press, New York 1982 I. S. Duff 57-87 · Zbl 1073.90024 |

[27] | Toint, PhL, Numerical solution of large sets of algebraic nonlinear equations, Math. Comput., 46, 175-189, (1986) · Zbl 0614.65058 |

[28] | Yamashita, N; Fukushima, M, On the rate of convergence of the Levenberg-Marquardt method, Computing, 15, 239-249, (2001) · Zbl 1001.65047 |

[29] | Yuan, G; Lu, S; Wei, Z, A new trust-region method with line search for solving symmetric nonlinear equations, Int J Comput Math, 88, 2109-2123, (2011) · Zbl 1254.90181 |

[30] | Yuan, Y, Trust region algorithm for nonlinear equations, Information, 1, 7-21, (1998) |

[31] | Zhang, HC; Hager, WW, A nonmonotone line search technique for unconstrained optimization, SIAM J Optim, 14, 1043-1056, (2004) · Zbl 1073.90024 |

[32] | Zhang, J; Wang, Y, A new trust region method for nonlinear equations, Math Methods Oper Res, 58, 283-298, (2003) · Zbl 1043.65072 |

[33] | Zhang, XS; Zhang, JL; Liao, LZ, An adaptive trust region method and its convergence, Sci China, 45, 620-631, (2002) · Zbl 1105.90361 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.