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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 & Management Optimization, 2013, 9 (1) : 227-241. doi: 10.3934/jimo.2013.9.227
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show all references

References:
[1]

Appl. Numer. Math., 44 (2003), 257-280. doi: 10.1016/S0168-9274(02)00170-8.  Google Scholar

[2]

Optim. Methods Software, 20 (2005), 1-22.  Google Scholar

[3]

Optim. Meth. Software, 17 (2002), 605-626.  Google Scholar

[4]

Prentice-Hall, Englewood Cliffs, NJ, 1983.  Google Scholar

[5]

Optim. Meth. Software, 5 (1995), 319-345. doi: 10.1080/10556789508805619.  Google Scholar

[6]

IEEE Service Center, Piscataway, New Jersey, 1996. Google Scholar

[7]

Computational Optimization and Applications, 34 (2006), 47-62.  Google Scholar

[8]

Computing, 74 (2005), 23-39. doi: 10.1007/s00607-004-0083-1.  Google Scholar

[9]

Vol. 33, Kluwer Academic Publishers, The Netherlands, 1999.  Google Scholar

[10]

Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer, New York, 1981.  Google Scholar

[11]

in "Complementarity: Applications, Algorithms and Extensions"(M. C. Ferris, O. L. Mangasarian and J. S. Pang eds.), Kluwer Academic, Dordrecht, 2001, pp 179-200.  Google Scholar

[12]

Journal of Computational and Applied Mathematics, 173 (2005), 321-343.  Google Scholar

[13]

SIAM, Philadelphia, 1995.  Google Scholar

[14]

Comput. Appl. Math., 16 (1997), 215-235.  Google Scholar

[15]

Quart. Appl. Math., 2 (1944), 164-166.  Google Scholar

[16]

SIAM J. Appl. Math., 11 (1963), 431-441.  Google Scholar

[17]

Appl. Math. Comput., 22 (1987), 333-361.  Google Scholar

[18]

ACM Trans. Math. Software, 16 (1990), 143-151. Google Scholar

[19]

SIAM J. Optim., 9 (1999), 729-754. doi: 10.1137/S1052623497318980.  Google Scholar

[20]

in "Lecture Notes in Mathematics 630: Numerical Analysis"(G. A. Watson ed.), Springer-Verlag, Berlin, 1978, pp. 105-116.  Google Scholar

[21]

Academic Press, New York, 1970.  Google Scholar

[22]

Journal of Optimization Theories and Applications, 120 (2004), 601-649.  Google Scholar

[23]

SIAM J. Optim., 11 (2001), 889-917.  Google Scholar

[24]

Math. Programming, 74 (1996), 159-195.  Google Scholar

[25]

John Wiley and Sons, New York, NY, 1996. Google Scholar

[26]

Computing (Supplement 15), (2001), 237-249. Google Scholar

[27]

Numerical Algebra, Control and Optimization, 1 (2011), 15-34.  Google Scholar

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