Subgradients and supporting hyperplanes

For a proper convex f on $\mathbb R^d$ and a finite-domain point x, a vector g is a subgradient when its affine prediction at x is no larger than f(z) for every z. The collection is the subdifferential. Outside the domain we define it to be empty. A differentiable convex function has subdifferential consisting only of its gradient, because directional derivatives force equality of every component. For |x|, the slope is −1 on the negative side, +1 on the positive side, and any value in [−1,1] at zero.

Subgradients are not arbitrary local trend estimates. A line that sits below a graph near x but crosses above it elsewhere fails the definition. For convex functions, local support around an interior point extends globally, but that conclusion uses convexity. Adding the two subgradient inequalities at x and y also shows monotonicity: the difference of supporting slopes has nonnegative inner product with x−y. In one dimension this generalizes the familiar statement that the derivative of a convex function is nondecreasing, now allowing vertical intervals of slopes at corners.

The condition zero belongs to the subdifferential is equivalent to x being a global minimizer, directly by setting g=0 in the definition. For a convex indicator, its subdifferential is the normal cone: vectors whose inner product with every feasible displacement is nonpositive. At an interior point of a full-dimensional convex set, that cone contains only zero. At a boundary it can contain outward-pointing vectors. A constrained optimum balances the loss gradient against a normal; the gradient alone generally does not vanish at a boundary solution.

When h is convex and differentiable everywhere and C is nonempty closed convex, the sum rule gives the subdifferential of h plus the indicator as gradient h plus the normal cone. More generally, sums of proper convex functions need a domain qualification for equality of subdifferential sums; meeting relative interiors is a standard finite-dimensional sufficient condition. For a maximum of finitely many affine functions, the subdifferential at a point is the convex hull of the active slopes. Inactive pieces do not touch the graph and cannot contribute to its supporting slope set there.

In finite dimensions a proper convex function has at least one subgradient at every point in the relative interior of its effective domain. At relative-boundary points this can fail even for a closed convex function. Take f(x)=−√x for x≥0 and infinity otherwise. A finite subgradient g at zero would require −√z≥gz for every positive z, hence g≤−1/√z for arbitrarily small z. No finite g satisfies that requirement. The epigraph has no supporting hyperplane with the nonvertical orientation required by a finite slope there.

This example prevents an unsafe recipe: one cannot always pick a subgradient at any finite point and run an algorithm. The evaluation oracle itself needs a domain guarantee. It also clarifies the role of the relative interior in calculus rules and duality qualifications. Finally, subgradients are exact mathematical lower bounds; inexact numerical slopes require an explicit tolerance and a modified inequality. Treating a noisy estimate as a certified support can produce a false lower bound or an incorrect stopping decision, even when the underlying function really is convex.