Convex function

In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have

$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y).$

Convex function on an interval.

In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set. A function is also said to be strictly convex if

$f(tx+(1-t)y)<tf(x)+(1-t)f(y)\,$

for any t in (0,1).

Properties of convex functions

A convex function f defined on some convex open interval C is continuous on C and differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C.

A continuous function on an interval C is convex if and only if

$f\left({\frac {x+y}{2}}\right)\leq {\frac {f(x)+f(y)}{2}}.$

for all x and y in C.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval.

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: f(y) ≥ f(x) + f'(x) (y − x) for all x and y in the interval.

A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the opposite is not true, as shown by f(x) = x⁴.

More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.

If two functions f and g are convex, then so is any weighted combination a f + b g with non-negative coefficients a and b. Likewise, if f and g are convex, then the function max{f,g} is convex.

Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.

For a convex function f, the level sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets.

Convex functions respect Jensen's inequality.

Examples

• The second derivative of x² is 2; it follows that x² is a convex function of x.
• The absolute value function |x| is convex, even though it does not have a derivative at x = 0.
• The function f with domain [0,1] defined by f(0)=f(1)=1, f(x)=0 for 0<x<1 is convex; it is continuous on the open interval (0,1), but not continuous at 0 and 1.
• The function x³ has second derivative 6x; thus it is convex for x ≥ 0 and concave for x ≤ 0.
• Every linear transformation is convex but not strictly convex, since if f is linear, then $f(a+b)=f(a)+f(b)$. This implies that the identity map (i.e., f(x) = x) is convex but not strictly convex. The fact holds if we replace "convex" by "concave".
• An affine function is simultaneously convex and concave.

See also

• Logarithmically convex function
• Subderivative of a convex function