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January  2013, 9(1): 227-241. doi: 10.3934/jimo.2013.9.227

On the Levenberg-Marquardt methods for convex constrained nonlinear equations

Jinyan Fan 1,

1. 

Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

Received  January 2012 Revised  July 2012 Published  December 2012

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.
Keywords: Convex constrained nonlinear equations, quadratic convergence., local error bound, constrained Levenberg-Marquardt method, projected Levenberg-Marquardt method.
Mathematics Subject Classification: 90C30, 65K0.
Citation: Jinyan Fan. On the Levenberg-Marquardt methods for convex constrained nonlinear equations. Journal of Industrial and Management Optimization, 2013, 9 (1) : 227-241. doi: 10.3934/jimo.2013.9.227
References:
[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.

[2]

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

[3]

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.

[4]

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

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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

[11]

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.

[12]

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.

[13]

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

[14]

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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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.

[20]

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.

[21]

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

[22]

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.

[23]

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.

[24]

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.

[25]

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

[26]

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

[27]

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

show all references

References:
[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.

[2]

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

[3]

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.

[4]

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

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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

[11]

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.

[12]

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.

[13]

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

[14]

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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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.

[20]

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.

[21]

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

[22]

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.

[23]

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.

[24]

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.

[25]

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

[26]

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

[27]

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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