Hessianközelítéseket

Hessianközelítéseket refers to approximations of the Hessian matrix, which is a square matrix of second-order partial derivatives of a scalar-valued function. In optimization and numerical analysis, calculating the exact Hessian can be computationally expensive or even impossible. Therefore, various methods have been developed to approximate it.

One common approach is the finite difference approximation. This method uses the function's values at nearby points to estimate the second-order derivatives. For example, a central difference approximation for a second partial derivative can be computed using the function's value at a point and at points slightly perturbed in the direction of the variable.

Another class of Hessian approximations includes quasi-Newton methods. These methods build up an approximation of the Hessian iteratively. Instead of directly calculating second derivatives, they use gradient information from successive iterations to update the Hessian approximation. Popular quasi-Newton methods like BFGS (Broyden–Fletcher–Goldfarb–Shanno) and DFP (Davidon–Fletcher–Powell) maintain a matrix that approximates the inverse of the Hessian or the Hessian itself. These methods are widely used because they often converge faster than gradient descent methods without requiring explicit Hessian computation.

The choice of Hessian approximation depends on the specific problem, the availability of gradient information, and the desired accuracy and computational efficiency. These approximations are crucial in algorithms for solving nonlinear optimization problems, where they help guide the search towards a minimum or maximum of the objective function.