The Levenberg-Marquardt algorithm=Implementation and theory
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THE LEVE_~BERG-MARQUARDT ALGORITHM:
IMPLEMENTATION AND THEORY
Jorge J. Mor~
i. Introduction
Let F: R n ÷ R m be continuously differentiable, and consider the nonlinear least
squares problem of finding a local minimizer of
I f (x) = 1 IIF x)IL2
(1.1) ~(x) = 7 7
i=l
Levenberg [1944] and Marquardt [1963] proposed a very elegant algorithm for the
numerical solution of (i.i). However, most implementations are either not robust,
or do not have a solid theoretical justification. In this work we discuss a robust
and efficient implementation of a version of the Levenberg-Marqnardt algorithm, and
show that it has strong convergence properties. In addition to robustness, the main
features of this implementation are the proper use of implicitly scaled variables,
and the choice of the Levenberg-Marquardt parameter via a scheme due to Hebden
[1973]. Numerical results illustrating the behavior of this implementation are also
presented.
Notation. in all cases If" II refers to the £2 vector norm or to the induced operator
norm. The Jacobian matrix of F evaluated at x is denoted by F' (x), but if we have a
of vectors {Xk} , then Jk and fk are used instead of F'(x k) and
F(x k)
sequence
respectively.
2. Derivation
The easiest way to derive the Levenberg-Marquardt algorithm is by a lineariza-
tion argument. If, given x ~ R n, we could minimize
~(P) = IIF( x+p) II
as a function of p, then x+p would be the desired solution. Since P is usually a
nonlinear function of p, we linearize F(x+p) and obtain the linear least squares
problem
~(p) = IIF(x) + F'(x)pl I •
Of course, this linearization is not valid for all values of p, and thus we con-
sider the constrained linear least squares problem
Work performed under the auspices of the U.S. Energy Research and Development
Administration
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