The Cauchy method, also known as the steepest descent method, is an optimization algorithm used to find the minimum value of a function. It iteratively moves in the direction of steepest descent, calculated using the gradient of the function at the current point, in order to reach the minimum.
The Cauchy method is a specific type of gradient method that uses the steepest descent direction at each iteration. Other gradient methods may use different descent directions, such as the conjugate gradient method which takes into account previous iterations.
The gradient method is used to find the minimum value of a function. It is often used in machine learning and optimization problems where the goal is to minimize a cost or error function.
The gradient method works by iteratively updating the current point based on the direction of steepest descent, calculated using the gradient of the function at that point. This process continues until a stopping criterion is met, such as reaching a certain number of iterations or a small enough change in the function value.
The Cauchy and gradient methods can be slow to converge and may get stuck in local minima. They also require the function to be differentiable. In high dimensional problems, they may also suffer from the "curse of dimensionality" where the number of iterations needed to converge increases exponentially with the number of variables.