Gauss-Newton Method for Solving Unconstrained Optimization Problems

In mathematics, there are various fundamental methods for solving unconstrained optimization problems. One such approach is the Gauss-Newton method, which finds extensive applications in computer science, machine learning, and artificial intelligence domains. The Gauss-Newton method is an iterative algorithm designed for efficiently solving nonlinear least-squares problems. It builds upon Newton's method concept but employs an approximation of the inverse Hessian matrix instead of computing the exact inverse required by standard Newton's method. This approximation significantly reduces computational complexity and improves efficiency. From an implementation perspective, the Gauss-Newton algorithm typically involves initializing parameters, computing the Jacobian matrix of residuals at each iteration, and updating parameters using the formula: θ_new = θ_old - (JᵀJ)⁻¹Jᵀr, where J represents the Jacobian matrix and r denotes the residual vector. Key functions in programming implementations often include Jacobian calculation using automatic differentiation or finite differences, matrix inversion using Cholesky decomposition for stability, and convergence checking through residual norms. In practical applications, the Gauss-Newton method demonstrates superior performance compared to gradient descent when handling nonlinear problems, particularly excelling in scenarios involving small residuals and well-behaved Jacobian matrices. However, developers should implement safeguards against ill-conditioned matrices by incorporating regularization techniques like Levenberg-Marquardt modifications when (JᵀJ) becomes near-singular.