quasiNewtonMethoden

Quasi-Newton methods are a class of iterative optimization algorithms used to find local minima of functions. They are related to Newton's method, which uses the gradient and Hessian matrix of the function to determine the search direction. However, Newton's method requires the computation and inversion of the Hessian matrix at each iteration, which can be computationally expensive, especially for high-dimensional problems. Quasi-Newton methods overcome this by approximating the Hessian matrix or its inverse using information from previous iterations.

The core idea behind Quasi-Newton methods is to build up an approximation of the Hessian or its inverse iteratively. At each step, the algorithm updates this approximation based on the gradient difference and the change in position between the current and previous iteration. A common update formula is the Broyden–Fletcher–Goldfarb–Shanno (BFGS) formula, which is widely used due to its good performance. Other popular updates include the Davidon–Fletcher–Powell (DFP) formula.

The general step in a Quasi-Newton method involves calculating a search direction, which is typically the inverse of the approximate Hessian multiplied by the negative gradient. A line search is then performed along this direction to find an optimal step size. The approximation of the Hessian or its inverse is updated using the gradient and position information from the new point. The process repeats until a convergence criterion is met, such as the gradient being close to zero or the change in position being very small.

Compared to Newton's method, Quasi-Newton methods avoid explicit Hessian calculation and inversion, making them more efficient for many practical problems. They are generally considered more robust than gradient descent methods as they incorporate curvature information, leading to faster convergence. However, they are not guaranteed to converge if the initial guess is too far from the minimum or if the function has certain pathological properties.