Proper convex function

In mathematical analysis (in particular convex analysis) and optimization, a proper convex function is a convex function f taking values in the extended real number line such that

$f(x) < +\infty$

for at least one x and

$f(x) > -\infty$

for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains $-\infty$. Convex functions that are not proper are called improper convex functions.

A proper concave function is any function g such that f = − g is a proper convex function.

Properties

For every proper convex function f on Rⁿ there exist some b in Rⁿ and β in R such that

$f(x) \ge x \cdot b - \beta$

for every x.^{[citation needed]}

The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets $A \subset X$ and $B \subset X$ are convex sets in the vector space X, then the indicator functions I_A and I_B are proper convex functions, but I_A + I_B is not convex (unless $A \cup B$ is convex), and is possibly identically equal to $+\infty$ if $A \cup B = X$ (i.e. are complimenatry halfspaces).

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.^{[citation needed]}