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 $-\infty$ .[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.

Properties

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

f(x)\geq x\cdot b-\beta

for every x.

The sum of two proper convex functions is convex, but not necessarily proper.[3] For instance if the sets $A\subset X$ and $B\subset X$ are non-empty convex sets in the vector space X, then the characteristic 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.[4]

gollark: ```Get out of our way type system! We're going to reinterpret these bits or die trying! Even though this book is all about doing things that are unsafe, I really can't emphasize that you should deeply think about finding Another Way than the operations covered in this section. This is really, truly, the most horribly unsafe thing you can do in Rust. The railguards here are dental floss.```

gollark: _goes off to find rustonomicon_

gollark: https://pastebin.com/S2WeZawLI managed to cause an interesting compile error...

gollark: GAH! Even the inline assembly is too safe!

gollark: OH REALLY?

References

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.
Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 24. ISBN 978-0-691-01586-6.
Boyd, Stephen (2004). Convex Optimization. Cambridge, UK: Cambridge University Press. p. 79. ISBN 978-0-521-83378-3.
Ioffe, Aleksandr Davidovich; Tikhomirov, Vladimir Mikhaĭlovich (2009), Theory of extremal problems, Studies in Mathematics and its Applications, 6, North-Holland, p. 168, ISBN 9780080875279.

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[AB-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.