Subgradienttimenetelmä

Subgradienttimenetelmä is a term that appears to be a combination of "subgradient" and "method" or "algorithm" in Finnish. A subgradient method is an iterative optimization technique used to find the minimum of a convex function, particularly when the function is not differentiable. Unlike gradient descent, which requires the gradient to be defined at each point, subgradient methods can handle non-smooth convex functions by using a subgradient instead. A subgradient of a function at a point is a generalization of the gradient.

The core idea of a subgradient method is to move in the direction opposite to a chosen subgradient at each iteration. The update rule typically involves subtracting a step size multiplied by the subgradient from the current point to get the next point. The choice of step size is crucial for convergence and can be determined by various rules, such as diminishing step sizes. These methods are widely applied in areas like machine learning, control theory, and operations research, especially in problems involving non-differentiable objective functions, such as L1 regularization or problems with constraints that lead to non-smoothness. While guaranteed to converge to a minimum for convex functions under certain conditions, subgradient methods may converge at a slower rate compared to gradient-based methods when the function is smooth.