kvasiNewtonpäivitysmenetelmiä

kvasiNewtonpäi, also known as quasi-Newton methods, are a class of iterative numerical optimization algorithms used to find local optima of functions. They are closely related to Newton's method but offer an advantage by not requiring the explicit computation of the Hessian matrix, which can be computationally expensive or analytically difficult to obtain for many problems.

Instead of calculating the exact Hessian at each iteration, quasi-Newton methods approximate it. This approximation is typically updated at each step using information from the gradients of the function at the current and previous points. This update process is designed to satisfy a secant equation, which is derived from the Taylor expansion of the gradient. The secant equation ensures that the approximate Hessian is consistent with the observed changes in the gradient.

Popular quasi-Newton update formulas include the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method and the Davidon–Fletcher–Powell (DFP) method. BFGS is generally considered one of the most effective and widely used quasi-Newton methods. These methods often exhibit faster convergence than gradient descent methods, especially for problems with a large number of variables, while being more robust and less computationally demanding than standard Newton's method. They are frequently employed in machine learning, scientific computing, and engineering for solving nonlinear optimization problems.