Proper convex function

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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.


For every proper convex function f on Rn there exist some b in Rn and β in R such that

f(x) \ge 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 convex sets in the vector space X, then the indicator functions I_A and I_B are proper convex functions, but I_A + I_B is not convex (unless A \cup B is convex), and is identically equal to +\infty if A \cap B = \emptyset.

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.[citation needed]


  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.