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 $-\infty$.^[1] Convex functions that are not proper are called improper convex functions.^[2]

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

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

for every x.

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

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.^[3]

References

1. Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 254. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
2. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 24. ISBN 978-0-691-01586-6.
3. Ioffe, Aleksandr Davidovich; Tikhomirov, Vladimir Mikhaĭlovich (2009), Theory of extremal problems, Studies in Mathematics and its Applications, 6, North-Holland, p. 168, ISBN 9780080875279.