Levenberg-Marquardt Optimization Method (LM Method) MATLAB Source Code Implementation

This article presents MATLAB source code implementation of the Levenberg-Marquardt optimization method (commonly referred to as the LM method). The LM algorithm serves as an efficient nonlinear optimization technique particularly effective for solving nonlinear least squares problems. In MATLAB, this algorithm can be applied to various practical scenarios including curve fitting, data regression, parameter estimation, and system identification tasks.

When implementing LM optimization, several factors require careful consideration such as initial parameter selection, convergence criteria specification, and damping parameter adjustment. The implementation typically involves defining the objective function using function handles, calculating Jacobian matrices either analytically or numerically, and setting appropriate termination conditions. MATLAB's built-in functions like lsqnonlin or custom implementations can utilize trust-region reflective algorithms with adaptive damping factors to balance between Gauss-Newton and gradient descent methods.

Key implementation aspects include proper handling of the Hessian matrix approximation through (J'J + λI) formulation, where J represents the Jacobian matrix and λ denotes the damping parameter. The algorithm automatically adjusts λ during iteration - decreasing it for successful steps (approaching Gauss-Newton) and increasing it when steps require more gradient descent characteristics. Practical implementation should include safeguards against ill-conditioned matrices and incorporate regularization techniques when necessary.

Through MATLAB's Levenberg-Marquardt optimization, engineers and researchers can effectively solve complex real-world problems with robust convergence properties. The provided code examples demonstrate proper function structuring, parameter tuning techniques, and performance monitoring through residual analysis. This comprehensive guide aims to help readers master both theoretical foundations and practical implementation skills for this powerful optimization methodology.