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]

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, vol. 6, North-Holland, p. 168, ISBN 9780080875279.

[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, vol. 6, North-Holland, p. 168, ISBN 9780080875279.

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