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2014-01-14T20:59:53Z

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In [[mathematical analysis]] (in particular [[convex analysis]]) and [[optimization (mathematics)|optimization]], 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''. That is, a convex function is ''proper'' if its [[effective domain]] is nonempty and it never attains <math>-\infty</math>.<ref name="AB">{{cite book|last1=Aliprantis|first1=C.D.|last2=Border|first2=K.C.|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|page=254}}</ref> Convex functions that are not proper are called ''improper convex functions''.<ref>{{cite book|author=[[Rockafellar, R. Tyrrell]]|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6|page=24}}</ref>

A ''proper concave function'' is any function ''g'' such that <math>f = -g</math> is a proper convex function.

== Properties ==

For every proper convex function ''f'' on '''R'''<sup>n</sup> there exist some ''b'' in '''R'''<sup>n</sup> and β in '''R''' such that

:<math>f(x) \ge x \cdot b - \beta</math>

for every ''x''.

The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets <math>A \subset X</math> and <math>B \subset X</math> are [[convex set]]s in the [[vector space]] ''X'', then the [[Characteristic function (convex analysis)|indicator function]]s <math>I_A</math> and <math>I_B</math> are proper convex functions, but <math>I_A + I_B</math> is not convex (unless <math>A \cup B</math> is convex), and is identically equal to <math>+\infty</math> if <math>A \cap B = \emptyset</math>.

The [[infimal convolute|infimal convolution]] of two proper convex functions is convex but not necessarily proper convex.{{Citation needed|date=October 2011}}

== References ==
{{Reflist}}

[[Category:Convex analysis]]
[[Category:Types of functions]]

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