Subgradin

Subgradin is a term sometimes used as a variant spelling for subgradient, a concept in convex analysis that generalizes the gradient to non-differentiable functions. It plays a central role in convex optimization by describing linear underestimators of a function at a given point.

Formally, let f: R^n → R be convex. A vector g is a subgradient of f at x if for all y in R^n, f(y) ≥ f(x) + g^T (y − x). The collection of all subgradients at x is called the subdifferential and is denoted ∂f(x). If f is differentiable at x, then ∂f(x) contains only the gradient ∇f(x). For convex functions, the subdifferential is a convex set and is nonempty at points in the interior of the domain.

Subgradient methods are iterative optimization procedures that use elements of the subdifferential. Starting from x0, one picks gk ∈ ∂f(xk) and updates xk+1 = xk − αk gk with a positive step size αk. These methods generalize gradient descent to non-differentiable convex problems and often require appropriate step-size rules for convergence.

Examples help illustrate the concept. For f(x) = |x| on the real line, ∂f(x) = {sign(x)} for x ≠ 0, and ∂f(0) = [−1, 1]. For norms and indicator functions of convex sets, subgradients relate to normal vectors to supporting hyperplanes and normal cones.

Extensions include Clarke’s subdifferential for certain nonconvex, locally Lipschitz functions, which broadens the idea of a generalized gradient. In practice, subgradin (subgradient) is a foundational tool in convex analysis, optimization, and related fields, where non-differentiability is common.