The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. It belongs to the family of quasi-Newton methods, which aim to approximate the inverse Hessian matrix used in Newton's method for optimization. BFGS is widely used due to its efficiency and the fact that it does not require the computation of second derivatives.
Given an objective function \( f(\mathbf{x}) \), the goal is to minimize \( f(\mathbf{x}) \) with respect to the vector \( \mathbf{x} \). The BFGS method iteratively updates an estimate of the inverse Hessian matrix and a search direction to find the minimum of \( f(\mathbf{x}) \).