In convex analysis

Convex analysis

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....

, Danskin's theorem is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 which provides information about the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s of a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 of the form


The theorem has applications in optimization

Optimization (mathematics)

In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

, where it sometimes is used to solve minimax

Minimax

Minimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case scenario. Alternatively, it can be thought of as maximizing the minimum gain...

 problems.

Statement

The theorem applies to the following situation. Suppose is a continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 of two arguments,
where is a compact set. Further assume that is convex

Convex function

In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

 in for every .

Under these conditions, Danskin's theorem provides conclusions regarding the differentiability of the function
To state these results, we define the set of maximizing points as

Danskin's theorem then provides the following results.

Convexity

is convex

Convex function

In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

.

Directional derivatives

The directional derivative

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

 of in the direction , denoted , is given by

where is the directional derivative of the function at in the direction .

Derivative

is differentiable at if consists of a single element . In this case, the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of (or the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

 of if is a vector) is given by

Subdifferential

If is differentiable with respect to for all , and if is continuous with respect to for all , then the subdifferential of is given by

where indicates the convex hull

Convex hull

In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....

operation.