Quasi-Newton Iteration Method - A Relatively Modern Optimization Technique

The Quasi-Newton iteration method is a relatively modern numerical optimization technique that addresses the drawbacks of Newton's method, which requires derivative computation and matrix inversion. By introducing a Quasi-Newton matrix to approximate the Hessian matrix, this approach eliminates the need for explicit derivative calculations and matrix inversions. This method can optimize nonlinear functions and solve both unconstrained and constrained optimization problems. In practical implementation, algorithms like BFGS (Broyden-Fletcher-Goldfarb-Shanno) or DFP (Davidon-Fletcher-Powell) update the approximation matrix iteratively using gradient information. Furthermore, it finds applications in machine learning for parameter optimization, such as enhancing the backpropagation algorithm in neural networks. With its efficient matrix update formulas that typically maintain positive definiteness, the Quasi-Newton method has widespread applications in numerical optimization and machine learning domains.