Proper convex function

This article is about the concept in convex analysis. For the concept of properness in topology, see proper map.

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 $-\infty$ and also is not identically equal to $+\infty .$

In convex analysis and variational analysis, a point at which some given function $f,$ valued in the extended real number line $[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \},$ is minimized is typically sought, where such a point (if it exists) is called a global minimum point. If the function takes $-\infty$ as a value then it is necessarily a global minimum value and the minimization problem can be answered; this is why the definition of "proper" requires that the function never take $-\infty$ as a value. Assuming this, if the function's domain is empty or if the function is identically equal to $+\infty$ then the minimization problem once again has an immediate answer. Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper.

If the problem is instead a maximization problem (which would be clearly indicated, such as by the function being concave rather than convex) then the definition of "proper" is defined in an analogous, but different, manner but with the same goal: to exclude cases where the maximization problem can be answered immediately. Specifically, a concave function $g$ is called proper if its negation $-g,$ which is a convex function, is proper in the sense defined above.

Definitions

Suppose that $f:X\to [-\infty ,\infty ]$ is a function taking values in the extended real number line $[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}.$ If $f$ is a convex function or if the minimum of $f$ is being sought, then $f$ is called proper if there exists some point $x_{0}$ in its domain such that

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

and also

$f(x)>-\infty$

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

A proper concave function is by definition, any function $g:X\to [-\infty ,\infty ]$ such that $f:=-g$ is a proper convex function.

Properties

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

$f(x)\geq x\cdot b-r$

for every $x\in X.$

The sum of two proper convex functions is convex, but not necessarily proper. 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 $A\cap B=\varnothing$ 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.

See also

• Effective domain

References

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