×

zbMATH — the first resource for mathematics

An inexact derivative-free Levenberg-Marquardt method for linear inequality constrained nonlinear systems under local error bound conditions. (English) Zbl 1410.49037
Summary: In this paper, a derivative-free affine scaling inexact Levenberg-Marquardt method with interior backtracking line search technique is considered for solving linear inequality constrained nonlinear systems. The proposed algorithm is designed to take advantage of the problem structure by building polynomial interpolation models for each function of nonlinear systems subject to the linear inequality constraints on variables. Each iterate switches to backtracking step generated by affine scaling inexact Levenberg-Marquardt method and satisfies strict interior point feasibility by line search backtracking technique. Under local error bounded assumption, the method is superlinear and quadratic convergent on \(F(x)\). The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

MSC:
49M37 Numerical methods based on nonlinear programming
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C56 Derivative-free methods and methods using generalized derivatives
Software:
WEDGE; levmar
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bellavia, S.; Morini, B., Subspace trust-region methods for large bound-constrained nonlinear equations, SIAM J. Numer. Anal., 44, 1535-1555, (2006) · Zbl 1128.65033
[2] Behling, R.; Fischer, A., A unified local convergence analysis of inexact constrained Levenberg-Marquardt methods, Optim. Lett., 6, 927-940, (2012) · Zbl 1279.90159
[3] Conn, A. R.; Scheinberg, K.; Vicente, L. N., Geometry of interpolation sets in derivative free optimization, Math. Program., 111, 141-172, (2008) · Zbl 1163.90022
[4] Conn, A. R.; Scheinberg, K.; Toint, P. L., On the convergence of derivative-free methods for unconstrained optimization, (Powell, M. J.D.; Buhmann, M. D.; Iserles, A., (1997), Cambridge University Press Cambridge, UK), 83-108 · Zbl 1042.90617
[5] Dan, H.; Yamashita, N.; Fukushima, M., Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions, Technical Report 2001-001, (January 2001), Department of Applied Mathematics and Physics, Kyoto University
[6] Dikin, I. I., Iterative solution of problems of linear and quadratic programming, Soviet Math. Dokl., 8, 18-35, (1967) · Zbl 0189.19504
[7] Elster, C.; Neumaier, A., A grid algorithm for bound constrained optimization of noisy functions, IMA J. Numer. Anal., 15, 585-608, (1995) · Zbl 0831.65063
[8] Elhaway, M. E., Optimal Power Flow: Solution Techniques, Requirements and Challenges, (1996), IEEE Service Center Piscataway, New Jersey
[9] Fischer, A., Local behavior of an iterative framework for generalized equations with nonisolated solutions, Math. Prog., 94, 91-124, (2002) · Zbl 1023.90067
[10] Fischer, A.; Shukla, P. K.; Wang, M., On the inexactness level of robust Levenberg-Marquardt methods, Optimization, 59, 273-287, (2010) · Zbl 1196.65097
[11] (Floudas, C. A.; Pardalos, P. M., Encyclopedia of Optimization, (2008), Springer Berlin) · Zbl 1156.90001
[12] Ferris, M. C.; Pang, J. S., Engineering and economic applications of complementarity problems, SIAM Rev., 39, 669-713, (1997) · Zbl 0891.90158
[13] Fan, J.; Yuan, Y., On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing., 74, 23-39, (2005) · Zbl 1076.65047
[14] Glad, T.; Goldstein, A., Optimization of functions whose values are subject to small errors, BIT, 17, 160-169, (1977) · Zbl 0365.65044
[15] Hock, W.; Schittkowski, K., Test examples for nonlinear programming codes, Lect. Notes Econ. Math. Syst., 187, 1-177, (1981)
[16] Kanzow, C.; Yamashita, N.; Fukushima, M., Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, J. Comput. Appl. Math., 172, 375-397, (2004) · Zbl 1064.65037
[17] Levenberg, K., A method for the solution of certain non-linear problems in least squares, Q. Appl. Math., 2, 164-168, (1944) · Zbl 0063.03501
[18] Marazzi, M.; Nocedal, J., Wedge trust region methods for derivative free optimization, Math. Program., 91, 289-305, (2002) · Zbl 1049.90134
[19] Macconi, M.; Morini, B.; Porcelli, M., Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities, Appl. Numer. Math., 59, 859-876, (2009) · Zbl 1165.65030
[20] Marquardt, D. W., An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math., 11, 431-441, (1963) · Zbl 0112.10505
[21] Martinez, J. M.; Qi, L., Inexact Newton methods for solving nonsmooth equations, J. Comput. Appl. Math., 60, 127-145, (1995) · Zbl 0833.65045
[22] Powell, M. J.D., Least Frobenius norm updating of quadratic models that satisfy interpolation conditions, Math. Program. Ser. B, 100, 183-215, (2004) · Zbl 1146.90526
[23] Stewart, G. M.; Sun, J. G., Matrix Perturbation Theory, (1990), Academic Press San Diego
[24] Wang, P.; Zhu, D., An affine scaling derivative-free trust-region method for solving nonlinear systems subject to linear inequality constraints, Int. J. Comput. Math., 98, 1660-1687, (2015) · Zbl 1319.49051
[25] Wang, T.; Monteiro, R. D.C.; Pang, K. J.S., An interior point potential reduction method for constrained equations, Math. Program., 74, 159-195, (1996) · Zbl 0855.90128
[26] Wood, A. J.; Wollenberg, B. F., Power Generation, Operation and Control, (1996), John Wiley and Sons New York, NY
[27] Yamashita, N.; Fukushima, M., On the rate of convergence of the Levenberg-Marquardt method, Computing (Suppl), 15, 227-238, (2001) · Zbl 1001.65047
[28] Zhang, H.; Conn, A. R.; Scheinberg, K.; Optim, J., A derivative-free algorithm for least-squares minimization, SIAM, 20, 3555-3576, (2010) · Zbl 1213.65091
[29] Zhang, H.; Conn, A. R., On the local convergence of a derivative-free algorithm for least-squares minimization, Comput. Optim., 51, 481-507, (2012) · Zbl 1268.90043
[30] Zhang, J. L., On the convergence properties of the Levenberg-Marquardt method, Optimization, 52, 739-756, (2003) · Zbl 1059.90132
[31] Zhu, D., Affine scaling interior Levenberg-Marquardt method for bound-constrained semismooth equations under local error bound conditions, J. Comput. Appl. Math., 219, 198-215, (2008) · Zbl 1151.65047
[32] Zhu, D., An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems, J. Comput. Appl. Math., 184, 343-361, (2005) · Zbl 1087.65047
[33] Zhu, D., Anew affine scaling interior point algorithm for nonlinear optimization subject to linear equality and inequality constraints, J. Comput. Appl. Math., 161, 1-25, (2003) · Zbl 1050.65064
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.