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A vector that defines an affine global lower bound to a convex function at a point.
A subgradient of a convex function at a point (see domain) is a vector such that
Subgradients generalize derivatives: when is differentiable at (see derivative), the unique subgradient is . The set of all subgradients at is the subdifferential , and each subgradient induces a supporting hyperplane to the epigraph of .
Let be a vector space and let be an extended-real-valued function.
The function is convex if its epigraph (see epigraph) is a convex set in .
Context. This geometric definition is equivalent to analytic inequalities such as Jensen's inequality; see equivalent characterizations of convexity.
Let be a vector space and let be an extended-real-valued function (see extended reals).
The function is proper if .
Context. The epigraph turns function properties into geometric properties of sets; convexity of is defined by convexity of (see convex functions via epigraphs).
A subdifferential of a convex function at a point in its effective domain is the set
Any is called a subgradient of at .
Geometrically, encodes a supporting hyperplane to the epigraph of at , with normal determined by (compare hyperplane). If is differentiable at (in the sense of the derivative / differentiable map), then .
A supporting hyperplane of a convex function at a point (see domain) is an affine function
such that for all .
Equivalently, the graph is a hyperplane in that supports the epigraph of (a convex set when is convex). Such supporting hyperplanes are in one-to-one correspondence with subgradients: supports at exactly when (see subdifferential).