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An efficient line search trust-region for systems of nonlinear equations. (English) Zbl 07372039

Summary: An improved derivative-free trust-region method to solve systems of nonlinear equations in several variables is presented, combined with the Wolfe conditions to update the trust-region radius. We believe that producing step-sizes by the Wolfe conditions can control the trust-region radius. The new algorithm for which strong global convergence properties are proved is robust and efficient enough to solve systems of nonlinear equations.

MSC:

65H10 Numerical computation of solutions to systems of equations

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[1] Ahookhosh, M.; Amini, K.; Kimiaei, M., A globally convergent trust-region method for large-scale symmetric nonlinear systems, Numer. Funct. Anal. Optim., 36, 830-855 (2015) · Zbl 1330.65074
[2] Ahookhosh, M.; Esmaeili, H.; Kimiaei, M., An effective trust-region-based approach for symmetric nonlinear systems, Int. J. Comput. Math., 90, 3, 671-690 (2013) · Zbl 1273.90197
[3] Amini, K.; Esmaeili, H.; Kimiaei, M., A nonmonotone trust-region-approach with nonmonotone adaptive radius for solving nonlinear systems, Iran. J. Numer. Anal. Optim., 6, 1, 101-121 (2016) · Zbl 1343.65057
[4] Amini, K.; Shiker, MAK; Kimiaei, M., A line search trust-region algorithm with nonmonotone adaptive radius for a system of nonlinear equations, 4OR-Q J Oper, 14, 2, 132-152 (2016) · Zbl 1342.90178
[5] Bouaricha, A.; Schnabel, RB, Tensor methods for large sparse systems of nonlinear equations, Math. Program., 82, 377-400 (1998) · Zbl 0951.65046
[6] Broyden, CG, The convergence of an algorithm for solving sparse nonlinear systems, Math. Comput., 25, 114, 285-294 (1971) · Zbl 0227.65038
[7] Buhmiler, S.; Krejić, N.; Lužanin, Z., Practical quasi-Newton algorithms for singular nonlinear systems, Numer. Algorithm, 55, 481-502 (2010) · Zbl 1201.65076
[8] Conn, AR; Gould, NIM; Toint, PhL, Trust-Region Methods, Society for Industrial and Applied Mathematics (2000), Philadelphia: SIAM, Philadelphia · Zbl 0958.65071
[9] Dedieu, JP; Shub, M., Newton’s method for overdetermined systems of equations, Math. Comput., 69, 231, 1099-1115 (2000) · Zbl 0949.65061
[10] Dennis, JE, On the convergence of Broyden’s method for nonlinear systems of equations, Math. Comput., 25, 115, 559-567 (1971) · Zbl 0277.65030
[11] Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213 (2002) · Zbl 1049.90004
[12] Esmaeili, H.; Kimiaei, M., A new adaptive trust-region method for system of nonlinear equations, Appl. Math. Model., 38, 11-12, 3003-3015 (2014) · Zbl 1427.65080
[13] 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
[14] 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
[15] Fan, JY, An improved trust region algorithm for nonlinear equations, Comput. Optim. Appl., 48, 59-70 (2011) · Zbl 1230.90178
[16] Fasano, G.; Lampariello, F.; Sciandrone, M., truncated nonmonotone Gauss-Newton method for large-scale nonlinear least-squares problems, Comput. Optim. Appl., 34, 3, 343-358 (2006) · Zbl 1122.90094
[17] Gertz, EM, A quasi-Newton trust-region method, Math. Program. Ser. A, 100, 447-470 (2004) · Zbl 1068.90108
[18] Gertz, EM, Combination Trust-Region Line-Search Methods for Unconstrained Optimization (1999), San Diego: University of California, San Diego
[19] Gill, PhE; Murray, W., Algorithms for the solution of the nonlinear least-squares problem, SIAM J. Numer. Anal., 15, 5, 977-992 (1978) · Zbl 0401.65042
[20] Gill, P.E., Wright, M.H.: Department of Mathematics University of California San Diego. Course Notes for Numerical Nonlinear Optimization (2001)
[21] Griewank, A., The global convergence of Broyden-like methods with a suitable line search, J. Aust. Math. Soc. Ser. B, 28, 75-92 (1986) · Zbl 0596.65034
[22] Grippo, L.; Sciandrone, M., Nonmonotone derivative-free methods for nonlinear equations, Comput. Optim. Appl., 37, 297-328 (2007) · Zbl 1180.90310
[23] Gu, GZ; Li, DH; Qi, L.; Zhou, SZ, Descent directions of quasi-Newton methods for symmetric nonlinear equations, SIAM J. Numer. Anal., 40, 5, 1763-1774 (2003) · Zbl 1047.65032
[24] Kimiaei, M., A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints, Calcolo, 54, 3, 769-812 (2017) · Zbl 1373.90151
[25] Kimiaei, M.; Rahpeymaii, F., A new nonmonotone line-search trust-region approach for nonlinear systems, TOP, 27, 199-232 (2019) · Zbl 1416.65144
[26] Kimiaei, M., Nonmonotone self-adaptive Levenberg-Marquardt approach for solving systems of nonlinear equations, Numer. Funct. Anal. Optim., 39, 21, 47-66 (2018) · Zbl 1390.90517
[27] Kimiaei, M.; Esmaeili, H., A trust-region approach with novel filter adaptive radius for system of nonlinear equations, Numer. Algorithms, 73, 4, 999-1016 (2016) · Zbl 1358.90132
[28] Levenberg, K., A method for the solution of certain non-linear problems in least squares, Q. Appl. Math., 2, 164-166 (1944) · Zbl 0063.03501
[29] Li, Q.; Li, DH, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal., 31, 4, 1-11 (2011) · Zbl 1241.65047
[30] Li, D.; Fukushima, M., A global and superlinear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM J. Numer. Anal., 37, 152-172 (1999) · Zbl 0946.65031
[31] Marquardt, DW, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math., 11, 431-441 (1963) · Zbl 0112.10505
[32] Martinez, JM, A family of quasi-Newton methods for nonlinear equations with direct secant updates of matrix factorizations, SIAM J. Numer. Anal., 27, 4, 1034-1049 (1990) · Zbl 0702.65053
[33] Moré, JJ; Garbow, BS; Hillström, KE, Testing unconstrained optimization software, ACM Trans. Math. Softw., 7, 17-41 (1981) · Zbl 0454.65049
[34] Nocedal, J.; Wright, SJ, Numerical Optimization (2006), New York: Springer, New York · Zbl 1104.65059
[35] Nocedal, J., Yuan, Y.X.: Combining trust-region and line-search techniques, Optimization Technology Center mar OTC 98/04 (1998) · Zbl 0909.90243
[36] Ortega, JM; Rheinboldt, WC, Iterative Solution of Nonlinear Equations in Several Variables (1970), New York: Academic Press, New York · Zbl 0241.65046
[37] Powell, MJD; Rosen, JB; Mangasarian, OL; Ritter, K., A new algorithm for unconstrained optimization, Nonlinear Programming (1970), Cambridge: Academic Press, Cambridge · Zbl 0228.90043
[38] Powell, MJD; Mangasarian, OL; Meyer, RR; Robinson, SM, Convergence properties of a class of minimization algorithms, Nonlinear Programming 2, 1-27 (1975), Cambridge: Academic Press, Cambridge · Zbl 0321.90045
[39] Rahpeymaii, F.; Kimiaei, M.; Bagheri, A., A limited memory quasi-Newton trust-region method for box constrained optimization, J. Comput. Appl. Math., 303, 105-118 (2016) · Zbl 1381.90097
[40] Schnabel, RB; Frank, PD, Tensor methods for nonlinear equations, SIAM J. Numer. Anal., 21, 5, 815-843 (1984) · Zbl 0562.65029
[41] Thomas, SW, Sequential Estimation Techniques for Quasi-Newton Algorithms (1975), Ithaca: Cornell University, Ithaca
[42] Toint, PhL, Numerical solution of large sets of algebraic nonlinear equations, Math. Comput., 46, 173, 175-189 (1986) · Zbl 0614.65058
[43] Toint, PL; Duff, IS, Towards an efficient sparsity exploiting Newton method for minimization, Sparse Matrices and Their Uses, 57-87 (1982), Academic Press: New York, Academic Press
[44] Tong, XJ; Qi, L., On the convergence of a trust-region method for olving constrained nonlinear equations with degenerate solutions, J. Optim. Theory Appl., 123, 1, 187-211 (2004) · Zbl 1069.65055
[45] Yamashita, N.; Fukushima, M., On the rate of convergence of the Levenberg-Marquardt method, Computing, 15, 239-249 (2001) · Zbl 1001.65047
[46] Yuan, GL; Wei, ZX; Lu, XW, A BFGS trust-region method for nonlinear equations, Computing, 92, 4, 317-333 (2011) · Zbl 1241.65049
[47] Yuan, Y., Subspace methods for large scale nonlinear equations and nonlinear least squares, Optim. Eng., 10, 207-218 (2009) · Zbl 1171.65040
[48] Yuan, Y., Trust region algorithm for nonlinear equations, Information, 1, 7-21 (1998)
[49] Yuan, Y., Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numer. Algebra Control Optim., 1, 1, 15-34 (2011) · Zbl 1226.65045
[50] Zhang, J.; Wang, Y., A new trust region method for nonlinear equations, Math. Methods Oper. Res., 58, 283-298 (2003) · Zbl 1043.65072
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