Gauss-Newton Method for Solving Unconstrained Optimization Problems

The Gauss-Newton method is a classical iterative algorithm for solving unconstrained optimization problems, particularly well-suited for nonlinear least squares problems. Building upon Newton's method, it improves computational efficiency by approximating complex second-derivative calculations.

The core idea involves constructing iterative steps using local linear approximations of the objective function. For nonlinear least squares problems, the Gauss-Newton method approximates the objective function as a quadratic form and determines the iteration direction by solving linear equation systems. Compared to Newton's method, it eliminates the need to compute the full Hessian matrix, instead using the product of Jacobian matrices for approximation, thereby reducing computational complexity. In code implementation, this typically involves calculating the Jacobian matrix J and solving the normal equations JᵀJ δ = -Jᵀr, where r represents the residual vector.

In practical applications, the Gauss-Newton method requires combination with line search or trust region strategies to ensure convergence. Its advantage lies in fast convergence when dealing with moderately-sized parameter problems, but it remains sensitive to initial value selection and may encounter matrix singularity issues in certain scenarios. As optimization problem scales increase, modern improved algorithms like the Levenberg-Marquardt method often demonstrate better robustness. Code implementations typically include regularization parameters to handle ill-conditioned matrices and adaptive step size control for stability.