Two Modified Levenberg-Marquart Method And Their Applications

It is more and more important to develop optimization algorithms with the increasing application of optimization problems in science,engineering,economics and industry.As one of the classical optimization algorithms,LevenbergMarquart(LM)method has been widely used in other fields,such as image problems and data processing.However,with the increase of problem complexity,the existing LM algorithm is inevitably inefficient in numerical calculation.Two modified LM methods are proposed in this paper.Under certain assumptions,the convergence of the modified LM method is studied in this paper.The following results were obtained:(1)A modified LM method based on the parameter ?k1 is proposed in this paper and applied to solving nonlinear equations.Under the local error condition and Lipcshitz continuity condition,the convergence of the new LM method is shown to be at least superlinearly with not require full rank of the Jacobian matrix.In addition,numerical experiments show that the LM method is effective,compared with other algorithms in references,under certain conditions,the modified LM method has some advantages.(2)A modified LM method based on the parameter ?k2 is proposed and applied to nonlinear least squares problems in this paper.Under the newly proposed local error condition and the Lipcshitz continuous condition.According to the different rank of Jacobian matrix,it divided into two cases:constant rank or decreasing rank,the convergence of the modified LM method is discussed,the order of convergence is related to the new local error condition,and the order of convergence is not less than 1.The effectiveness of the LM method is also demonstrated by numerical experiments.Compared with the data in the other literature,it is found that the LM method is more efficient when the initial point is far away from the solution.