scholarly journals A Modified Levenberg-Marquardt Method for Nonsmooth Equations with Finitely Many Maximum Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Shou-qiang Du ◽  
Yan Gao
Keyword(s):  
Variational Inequality ◽  
Nonlinear Programming ◽  
Present Method ◽  
Complementarity Problems ◽  
Nonsmooth Equations ◽  
Systems Of Equations ◽  
Marquardt Method ◽  
Nonsmooth Systems ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt

For solving nonsmooth systems of equations, the Levenberg-Marquardt method and its variants are of particular importance because of their locally fast convergent rates. Finitely many maximum functions systems are very useful in the study of nonlinear complementarity problems, variational inequality problems, Karush-Kuhn-Tucker systems of nonlinear programming problems, and many problems in mechanics and engineering. In this paper, we present a modified Levenberg-Marquardt method for nonsmooth equations with finitely many maximum functions. Under mild assumptions, the present method is shown to be convergent Q-linearly. Some numerical results comparing the proposed method with classical reformulations indicate that the modified Levenberg-Marquardt algorithm works quite well in practice.

scholarly journals A New Modified Efficient Levenberg–Marquardt Method for Solving Systems of Nonlinear Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Zhenxiang Wu ◽  
Tong Zhou ◽  
Lei Li ◽  
Liang Chen ◽  
Yanfang Ma
Keyword(s):  
Error Bound ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Systems Of Nonlinear Equations ◽  
Local Error ◽  
Local Error Bound ◽  
Marquardt Method ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Local Error Bound Condition

For systems of nonlinear equations, a modified efficient Levenberg–Marquardt method with new LM parameters was developed by Amini et al. (2018). The convergence of the method was proved under the local error bound condition. In order to enhance this method, using nonmonotone technique, we propose a new Levenberg–Marquardt parameter in this paper. The convergence of the new Levenberg–Marquardt method is shown to be at least superlinear, and numerical experiments show that the new Levenberg–Marquardt algorithm can solve systems of nonlinear equations effectively.

Generalized Tensor Complementarity Problems Over a Polyhedral Cone

2020 ◽  
Vol 37 (04) ◽  
pp. 2040006
Author(s):  
Liyun Ling ◽  
Chen Ling ◽  
Hongjin He
Keyword(s):  
Error Bound ◽  
Complementarity Problems ◽  
Polyhedral Cone ◽  
System Of Nonlinear Equations ◽  
Local Error ◽  
Local Error Bound ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Tensor Complementarity ◽  
Tensor Complementarity Problems

This paper addresses a class of generalized tensor complementarity problems (GTCPs) over a polyhedral cone. As a new generalization of the well-studied tensor complementarity problems (TCPs) in the literature, we first show the nonemptiness of the solution set of GTCPs when the involved tensor is cone ER. Then, we study bounds of solutions, and in addition to deriving a Hölderian local error bound of the problem under consideration. Finally, we reformulate GTCPs over a polyhedral cone as a system of nonlinear equations, which is helpful to employ the Levenberg–Marquardt algorithm for finding a solution of the problem. Some preliminary numerical results show that such an algorithm is efficient for GTCPs.

Sign in / Sign up

Export Citation Format

Share Document