Subgradients

Subgradients are a generalization of gradients used in convex analysis to handle nondifferentiable points. For a real-valued function f, a vector g is called a subgradient of f at a point x if the inequality f(y) ≥ f(x) + g^T(y − x) holds for all y in the domain. The set of all subgradients at x is the subdifferential, denoted ∂f(x). This set-valued object captures all linear underestimators of f at x.

If f is differentiable at x, the subdifferential contains only the gradient: ∂f(x) = {∇f(x)}. If f is not differentiable at x, ∂f(x) can contain multiple vectors. A classic example is f(x) = |x| on the real line, where ∂f(0) = [−1, 1]. For convex functions that are proper and lower semicontinuous, ∂f(x) is nonempty at interior points of the domain, though it may be empty at boundary points.

Subgradients lead to several important properties. The subdifferential mapping x ↦ ∂f(x) is monotone: if g1 ∈ ∂f(x1) and g2 ∈ ∂f(x2), then (g1 − g2)^T(x1 − x2) ≥ 0. They also underpin many algorithms, most notably subgradient methods for minimizing nondifferentiable convex functions, and they appear in duality and proximal optimization frameworks.

Generalizations extend the concept to nonconvex settings. The Clarke subdifferential broadens the idea to locally Lipschitz functions by taking limits of gradients at nearby differentiable points, forming a useful tool in nonsmooth analysis.