Convex (Concave) Functions

An n-dimensional region A is said to be convex if, whenever x = (x1, …, xn) and y = (y1, …, y“) both belong to A, their convex combination {kx + (1 - k)y}, also belongs to A. Geometrically this means that the straight line connecting any two points of a convex region belongs to the region. In the two-dimensional plane, the area enclosed by a triangle or a circle is a convex region. Similarly the region enclosed by a tetrahedron is convex in three-dimensional space.

Two well-known types of functions which are related and which occur frequently are monotone functions and convex functions.

A function f (x) is monotone increasing on an interval if, for any two values x1 and x2 in the interval, with x1 < x2, the relation f(x1) f(x2) holds. It is strictly increasing if only the inequality holds. It is monotone decreasing if holds. If the derivative of a function is nonnegative (nonpositive) in an interval, then the function is monotone increasing (decreasing) in the interval.

Convex functions occur in an important way in optimization theory. We shall give conditions for determining when a function is concave or convex. These conditions are related to the foregoing discussion of quadratic forma as applied to determining an extremum of an unconstraint function.

A function of a single variable f (x) is convex (also called concave upward) on an interval if, for any points x1, x2 on the interval, with x1 x2, it satisfies Jensen´s inequality

.

Geometrically, this says that the value of the function at a point which is the average of x1 and x2 is less than its average value at the two points.

The function f(x) is strictly convex if only the inequality holds. A twice-differentiable function f(x) on an opera interval (not necessarily including the end points of the interval) is convex if and only if on the interval.

This definition may be generalized to a function of several variables.