A vector that generalizes the gradient for a convex (possibly nondifferentiable) function at a point: a subgradient g at x satisfies f(y) ≥ f(x) + g·(y−x) for all y in the domain; the set of all such g is the subdifferential.

Principle

Subgradients define supporting hyperplanes to the epigraph of a convex function and provide first-order optimality conditions and descent directions when classical derivatives do not exist.

Demonstration

For f(t)=|t| at t=0 the subdifferential is the interval [−1,1]; any g in [−1,1] satisfies |y| ≥ 0 + g·(y−0) for all y, illustrating nondifferentiability handled by a set of valid subgradients.

Misapplication

Using subgradient-based algorithms without convexity assumptions or interpreting any selection of a one-sided slope at a nondifferentiable point as a valid subgradient for nonconvex functions leads to incorrect descent claims.

Consequence

When correctly applied to convex optimization, subgradients yield necessary and sufficient optimality criteria (0 in subdifferential) and support iterative methods (subgradient descent) that converge under proper step-size rules.

Reversal

The classical gradient is the unique subgradient at a point when the function is differentiable there; conversely, a singleton subdifferential implies differentiability of a convex function at that point.

Boundary

Primarily formulated for convex functions on convex domains; generalized notions (Clarke, limiting subgradients) extend to certain nonconvex settings but have different calculus rules and interpretations.

Semantic Tension

Tension exists between the convex subgradient and generalized gradients for nonconvex functions (Clarke, Mordukhovich): they agree on convex cases but differ in set structure and calculus properties in nonconvex analysis.

Synthesis

A subgradient is a supporting vector that replaces the gradient for convex nondifferentiable functions, forming a subdifferential set that furnishes optimality conditions and first-order methods for nonsmooth convex optimization.