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On the Levenberg-Marquardt methods for convex constrained nonlinear equations

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  • In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.
    Mathematics Subject Classification: 90C30, 65K05.


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  • [1]

    S. Bellavia, M. Macconi and B. Morini, An affine scaling trust-region approach to bound-constrained nonlinear systems, Appl. Numer. Math., 44 (2003), 257-280.doi: 10.1016/S0168-9274(02)00170-8.


    S. Bellavia and B. Morini, An interior global method for nonlinear systems with simple bounds, Optim. Methods Software, 20 (2005), 1-22.


    H. Dan, N. Yamashita and M. Fukushima, Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions, Optim. Meth. Software, 17 (2002), 605-626.


    J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations," Prentice-Hall, Englewood Cliffs, NJ, 1983.


    S. P. Dirkse, M. C. Ferris, MCPLIB: a collection of nonlinear mixed complementarity problems, Optim. Meth. Software, 5 (1995), 319-345.doi: 10.1080/10556789508805619.


    M. E. El-Hawary, "Optimal Power Flow: Solutions Techniques, Requirements, and Challenges," IEEE Service Center, Piscataway, New Jersey, 1996.


    J. Y. Fan and J. Y. Pan, Convergence properties of a self-adaptive Levenberg-Marquardt algorithm under local error bound condition, Computational Optimization and Applications, 34 (2006), 47-62.


    J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23-39.doi: 10.1007/s00607-004-0083-1.


    C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer and C. A. Schweiger, "Handbook of Test Problems in Local and Global Optimization, Nonconvex Optimization and its Applications," Vol. 33, Kluwer Academic Publishers, The Netherlands, 1999.


    W. Hock and K. Schittkowski, "Test Examples for Nonlinear Programming Codes," Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer, New York, 1981.


    C. Kanzow, An active set-type Newton method for constrained nonlinear systems, in "Complementarity: Applications, Algorithms and Extensions"(M. C. Ferris, O. L. Mangasarian and J. S. Pang eds.), Kluwer Academic, Dordrecht, 2001, pp 179-200.


    C. Kanzow, N. Yamashita and M. Fukushima, Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, Journal of Computational and Applied Mathematics, 173 (2005), 321-343.


    C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," SIAM, Philadelphia, 1995.


    D. N. Kozakevich, J. M. Martinez and S. A. Santos, Solving nonlinear systems of equations with simple bounds, Comput. Appl. Math., 16 (1997), 215-235.


    K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-166.


    D. W. Marquardt, An algorithm for least-squares estimation of nonlinear inequalities, SIAM J. Appl. Math., 11 (1963), 431-441.


    K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361.


    K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Trans. Math. Software, 16 (1990), 143-151.


    R. D. C. Monteiro and J. S. Pang, A potential reduction Newton method for constrained equations, SIAM J. Optim., 9 (1999), 729-754.doi: 10.1137/S1052623497318980.


    J. J. Moré, The LM algorithm: implementation and theory, in "Lecture Notes in Mathematics 630: Numerical Analysis"(G. A. Watson ed.), Springer-Verlag, Berlin, 1978, pp. 105-116.


    J. M. Ortega and W. C. Rheinboldt, "Iterative solution of Nonlinear Equations in Several Variables," Academic Press, New York, 1970.


    L. Qi, X. J. Tong and D. H. Li, An active-set projected trust region algorithm for box constrained nonsmooth equations, Journal of Optimization Theories and Applications, 120 (2004), 601-649.


    M. Ulbrich, Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems, SIAM J. Optim., 11 (2001), 889-917.


    T. Wang, R. D. C. Monteiro and J. S. Pang, An interior point potential reduction method for constrained equations, Math. Programming, 74 (1996), 159-195.


    A. J. Wood and B. F. Wollenberg, "Power Generation, Operation, and Control," John Wiley and Sons, New York, NY, 1996.


    N. Yamashita and M. Fukushima, On the rate of convergence of the LM method, Computing (Supplement 15), (2001), 237-249.


    Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least sqaures, Numerical Algebra, Control and Optimization, 1 (2011), 15-34.

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