Quasi-Newton Optimization Algorithm

This article introduces a classic mathematical optimization method - the Quasi-Newton algorithm. As an unconstrained optimization technique, Quasi-Newton methods aim to find the minimum value of a function. Widely applied in optimization problems, their fundamental principle involves approximating the inverse Hessian matrix using first and second derivative information to minimize the objective function. The algorithm's implementation typically follows an iterative approach where at each step, the inverse Hessian approximation is updated using gradient differences and parameter changes. Key advantages include fast convergence rates, reduced computational complexity, and the elimination of direct Hessian matrix inversion, making it highly practical for implementation. Common variants like the BFGS (Broyden-Fletcher-Goldfarb-Shanno) and DFP (Davidon-Fletcher-Powell) methods employ different matrix update formulas - BFGS uses a rank-two update that maintains positive definiteness, while DFP applies a complementary updating scheme. These variations allow practitioners to select the most suitable approach based on specific application requirements, with BFGS being particularly popular due to its numerical stability and superior performance in many scientific computing scenarios.