Proper convex function

A convex function ${\displaystyle f:X\to [-\infty ,\infty ]}$ taking values in the extended real number line ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}$ is called proper if there exists some point ${\displaystyle x_{0}}$ in its domain such that

${\displaystyle f\left(x_{0}\right)<+\infty }$

and also

${\displaystyle f(x)>-\infty }$

for every ${\displaystyle x.}$ That is, a convex function is proper if its effective domain is nonempty and it never attains ${\displaystyle -\infty }$.[1] This means that there exists some ${\displaystyle x\in \operatorname {domain} f}$ at which ${\displaystyle f(x)\in \mathbb {R} }$ and ${\displaystyle f}$ is also never equal to ${\displaystyle -\infty .}$ Convex functions that are not proper are called improper convex functions.[2]

A proper concave function is any function ${\displaystyle g}$ such that ${\displaystyle f=-g}$ is a proper convex function.

For every proper convex function ${\displaystyle f:\mathbb {R} ^{n}\to [-\infty ,\infty ],}$ there exist some ${\displaystyle b\in \mathbb {R} ^{n}}$ and ${\displaystyle r\in \mathbb {R} }$ such that

${\displaystyle f(x)\geq x\cdot b-r}$

for every ${\displaystyle x\in X.}$

The sum of two proper convex functions is convex, but not necessarily proper.[3] For instance if the sets ${\displaystyle A\subset X}$ and ${\displaystyle B\subset X}$ are non-empty convex sets in the vector space ${\displaystyle X,}$ then the characteristic functions ${\displaystyle I_{A}}$ and ${\displaystyle I_{B}}$ are proper convex functions, but if ${\displaystyle A\cap B=\varnothing }$ then ${\displaystyle I_{A}+I_{B}}$ is identically equal to ${\displaystyle +\infty .}$

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.[4]

• Effective domain

• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.CS1 maint: date and year (link)