BFGS METHOD Meaning and
Definition
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The BFGS method is an optimization algorithm that stands for Broyden-Fletcher-Goldfarb-Shanno. It is used to solve unconstrained nonlinear optimization problems. The BFGS method belongs to the class of quasi-Newton methods, which approximate the Hessian matrix of the objective function (second derivatives) without explicitly computing it.
In the BFGS method, an initial Hessian matrix is estimated, typically using the identity matrix. Then, the algorithm iteratively updates this estimate to better approximate the true Hessian. It uses the gradient information of the objective function and the difference between the current and previous iterations to calculate an approximation of the inverse Hessian matrix. This updated Hessian estimate is then used to find the next search direction.
The BFGS method adjusts the search direction to move towards the minimum of the objective function. At each iteration, it updates the current solution based on the step size determined by the line search. The process continues until convergence is reached, typically when a predefined tolerance level is achieved or when the maximum number of iterations is reached.
One advantage of the BFGS method is that it avoids the expensive computation of the Hessian matrix. Additionally, it is a quasi-Newton method, which means it converges relatively quickly and can handle large-scale optimization problems.
Overall, the BFGS method is a powerful tool for solving unconstrained nonlinear optimization problems efficiently and accurately.
Etymology of BFGS METHOD
The term "BFGS method" is an acronym that derives from the surnames of the individuals who developed and refined the method. The letters BFGS represent the names of four mathematicians: Broyden, Fletcher, Goldfarb, and Shanno.
The BFGS method is an optimization algorithm initially proposed by these mathematicians in 1970 as an improvement over the previously developed Davidon-Fletcher-Powell (DFP) method. The BFGS method aims to find the minimum of a smooth, nonlinear function by iteratively adjusting the search direction based on the gradient and curvature of the function.
Over time, the BFGS method has become widely used in the field of numerical optimization due to its efficiency and effectiveness. The name "BFGS" serves as a tribute to the contributions of Broyden, Fletcher, Goldfarb, and Shanno to the development of this algorithm.
