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Wikipedia / Subderivative

In mathematics, the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative.

Definition

Rigorously, a subderivative. By the converse of the mean value theorem, the set for a convex function is a nonempty closed interval are the one-sided limitsThe interval of all subderivatives is called the subdifferential.

Examples

which is convex. Then, the subdifferential at the origin is the interval is the singleton set. This is similar to the sign function, instead including all possible subderivatives.

,

while

Properties

  • if and only if.

  • is a global minimum. This last property is a generalization of the fact that the derivative of a function differentiable at a local minimum is zero.

  • (where the addition operator denotes the Minkowski sum). This reads as "the subdifferential of a sum is the sum of the subdifferentials."

The subgradient

is a real-valued convex function defined on a convex open set in the Euclidean space in that space is called a subgradient one has that

where the dot denotes the dot product is called the subdifferential. The subdifferential is always a nonempty convex compact set.

on a convex set in a locally convex space in the dual space is called a subgradient,

. The subdifferential is always a convex closed set. It can be an empty set; consider for example an unbounded operator is continuous, the subdifferential is nonempty.

Relation to convex conjugacy

on a locally convex space, the subdifferential can be characterized using the convex conjugate. The Fenchel–Young inequality states that

; that is,

if and only if

Equivalently,

Geometrically, this says that the affine function

.

History

The subdifferential on convex functions was introduced by Jean Jacques Moreau and R. Tyrrell Rockafellar in the early 1960s. The generalized subdifferential for nonconvex functions was introduced by Francis H. Clarke and R. Tyrrell Rockafellar in the early 1980s.

See also

References

  • .