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
for at least one x and
for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains
A proper concave function is any function g such that
Properties
For every proper convex function f on Rn there exist some b in Rn and β in R such that
for every x.
The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets
The infimal convolution of two proper convex functions is convex but not necessarily proper convex.