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

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 ${\displaystyle -\infty }$. Convex functions that are not proper are called improper convex functions.

A proper concave function is any function g such that ${\displaystyle f=-g}$ is a proper convex function.

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 ${\displaystyle A\subset X}$ and ${\displaystyle B\subset X}$ are non-empty convex sets in the vector space X, then the indicator functions ${\displaystyle I_{A}}$ and ${\displaystyle I_{B}}$ are proper convex functions, but if ${\displaystyle A\cap B=\emptyset }$ then ${\displaystyle I_{A}+I_{B}}$ is identically equal to ${\displaystyle +\infty }$.

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