Proper convex function

In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value ${\displaystyle -\infty }$ and also is not identically equal to ${\displaystyle +\infty .}$

Definitions

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.

Properties

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]

See also

• Effective domain

Citations

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.
3. Boyd, Stephen (2004). Convex Optimization. Cambridge, UK: Cambridge University Press. p. 79. ISBN 978-0-521-83378-3.
4. Ioffe, Aleksandr Davidovich; Tikhomirov, Vladimir Mikhaĭlovich (2009), Theory of extremal problems, Studies in Mathematics and its Applications, 6, North-Holland, p. 168, ISBN 9780080875279.

References

• 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)