Subgradientové

Subgradientové refers to a generalization of the gradient for convex functions that are not necessarily differentiable. While the gradient at a point provides a unique direction of steepest ascent for a differentiable function, a non-differentiable convex function at a particular point may have a set of possible "slopes" or directions of increase. The subgradient is an element of this set.

Formally, for a convex function f: R^n -> R, a vector g is a subgradient of f at a point x if f(y) >= f(x) + g^T(y-x) for all y in R^n. The set of all such vectors g at a point x is called the subdifferential of f at x, denoted by ∂f(x). If f is differentiable at x, then the subdifferential ∂f(x) contains only the gradient ∇f(x).

The concept of subgradients is crucial in convex optimization, particularly for problems involving non-smooth convex functions. Subgradient methods are iterative algorithms used to find the minimum of such functions. They work by repeatedly updating the current estimate of the minimum by moving in a direction opposite to a chosen subgradient.

Subgradient methods are often simpler to implement than other optimization techniques for non-smooth problems because they do not require the calculation of second-order derivatives or solving complex subproblems at each iteration. However, their convergence properties can be slower compared to methods that utilize more information about the function. Despite this, subgradient methods are widely applicable in fields such as machine learning, signal processing, and operations research where non-smoothness is common.

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