Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points.

Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.

In simple terms, a convex function graph is shaped like a cup

A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.

as a nonnegative real number) and an exponential function

Convex functions play an important role in many areas of mathematics.

They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties.

For instance, a strictly convex function on an open set has no more than one minimum.

Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations.

In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable.

be a convex subset of a real vector space and let

is called convex if and only if any of the following equivalent conditions hold: The second statement characterizing convex functions that are valued in the real line

is also the statement used to define convex functions that are valued in the extended real number line

is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of

The second statement can also be modified to get the definition of strict convexity, where the latter is obtained by replacing

is a function that the straight line between any pair of points on the curve

except for the intersection points between the straight line and the curve.

[3][4][5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph

is twice continuously differentiable and the domain is the real line, then we can characterize it as follows: For example, let

be strictly convex, and suppose there is a sequence of points

It is not necessary for a function to be differentiable in order to be strongly convex.

A third definition[14] for a strongly convex function, with parameter

is twice continuously differentiable, then it is strongly convex with parameter

Assuming still that the function is twice continuously differentiable, one can show that the lower bound of

by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.

The proof of this statement follows from the extreme value theorem, which states that a continuous function on a compact set has a maximum and minimum.

Like strictly convex functions, strongly convex functions have unique minima on compact sets.

This is a generalization of the concept of strongly convex function; by taking

It is worth noting that some authors require the modulus

to be an increasing function,[17] but this condition is not required by all authors.