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. Convex functions that are not proper are called improper convex functions.

This definition takes account of the fact that the extended real number line does not constitute a field because, for example, the value of the expression ∞ − ∞ is left undefined.

It is always possible to consider the restriction of a proper convex function f to its effective domain

${\mbox{dom}}f=\left\{x:f(x)<\infty \right\}$

instead of f itself, thereby avoiding some minor technicalities that may otherwise arise. The effective domain of a conv|ex function is always a convex set.

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 convex but not necessarily proper convex. The infimal convolution of two proper convex functions is convex but not necessarily proper convex.

References

• Rockafellar, R. Tyrrell (1996) [1970]. Convex Analysis. Princeton University Press. ISBN 9780691015866.