A Shamanskii-like Levenberg-Marquardt method for nonlinear equations

Authors:
Jinyan Fan
Affiliations:
Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China 200240
Venue:
Computational Optimization and Applications
Year:
2013

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Testing Unconstrained Optimization Software

ACM Transactions on Mathematical Software (TOMS)
On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption

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Abstract

In this paper, we propose a Shamanskii-like Levenberg-Marquardt method for nonlinear equations. At every iteration, not only a LM step but also m驴1 approximate LM steps are computed, where m is a positive integer. Under the local error bound condition which is weaker than nonsingularity, we show the Shamanskii-like LM method converges with Q-order m+1. The trust region technique is also introduced to guarantee the global convergence of the method. Since the Jacobian evaluation and matrix factorization are done after every m computations of the step, the overall cost of the Shamanskii-like LM method is usually much less than that of the general LM method (the m=1 case).