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Proper convex function ：ウィキペディア英語版

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

. Convex functions that are not proper are called ''improper convex functions''.
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) \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 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.〔.〕

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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