Proper convex function - Definition

In mathematics, a proper convex function is a convex function f taking values in the extended real number line such that

<math>f(x) < +\infty<math>

for at least one x and

<math>f(x) > -\infty<math>

for every x. 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

<math>

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

instead of f itself, thereby avoiding some minor technicalities that may otherwise arise. The effective domain of a convex 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) ≥ <x,b> − β for every x.

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

References

• Rockafellar, Ralph Tyrell, Convex Analysis, Princeton University Press (1996). ISBN 0691015864