Levenberg-Marquardt Optimization Iteration Algorithm

The Levenberg-Marquardt optimization algorithm is a widely used method for solving nonlinear least squares problems. This algorithm introduces a damping parameter that dynamically balances between gradient descent and Gauss-Newton methods, significantly improving convergence efficiency. The core implementation involves starting with Gauss-Newton iterations and automatically switching to gradient descent when the loss function shows insufficient improvement. In practical code implementation, key components include: 1. A damping parameter (λ) adjustment mechanism that increases λ for gradient-dominant steps and decreases it for Gauss-Newton steps 2. Jacobian matrix computation using numerical differentiation or analytical derivatives 3. A trust region strategy to control step sizes and ensure convergence The algorithm's workflow typically follows: - Initialize parameters and compute initial residuals - Iteratively solve (JᵀJ + λI)δ = -Jᵀr where J is the Jacobian matrix - Accept or reject steps based on actual vs. predicted error reduction - Update λ using gain ratio comparisons Beyond scientific computing applications, this algorithm is extensively employed in machine learning for parameter optimization and computer vision for bundle adjustment problems. Its adaptive nature makes it particularly effective for ill-conditioned problems where pure Gauss-Newton methods might diverge.