proper convex function

{{About|the concept in convex analysis|the concept of properness in topology|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 (in the domain) at which some given function $f$ is minimized is typically sought, where $f$ is valued in the extended real number line $[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}.$ {{sfn|Rockafellar|Wets|2009|pp=1-28}} Such a point, if it exists, is called a {{em|global minimum point}} of the function and its value at this point is called the {{em|global minimum}} ({{em|value}}) of the function. If the function takes $-\infty$ as a value then $-\infty$ is necessarily the global minimum value and the minimization problem can be answered; this is ultimately the reason why the definition of "{{em|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 {{em|proper}}. Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases.

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 "{{em|proper}}" is defined in an analogous (albeit technically 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 {{em|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 a minimum point of $f$ is being sought, then $f$ is called {{em|proper}} if

: $f(x) > -\infty$ {{space|4}} for {{em|every}} $x \in X$

and if there also exists {{em|some}} point $x_0 \in X$ such that

: $f\left( x_0 \right) < +\infty.$

That is, a function is {{em|proper}} if it never attains the value $-\infty$ and its effective domain is nonempty.{{cite book|last1=Aliprantis|first1=C.D.|last2=Border|first2=K.C.|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|page=254}}

This means that there exists some $x \in X$ at which $f(x) \in \mathbb{R}$ and $f$ is also {{em|never}} equal to $-\infty.$ Convex functions that are not proper are called {{em|improper}} convex functions.{{cite book|author=Rockafellar, R. Tyrrell|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6|page=24}}

A {{em|proper concave function}} is by definition, any function $g : X \to [-\infty, \infty]$ such that $f := -g$ is a proper convex function. Explicitly, if $g : X \to [-\infty, \infty]$ is a concave function or if a maximum point of $g$ is being sought, then $g$ is called {{em|proper}} if its domain is not empty, it {{em|never}} takes on the value $+\infty,$ and it is not identically equal to $-\infty.$

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 $x \in \mathbb{R}^n.$

The sum of two proper convex functions is convex, but not necessarily proper.{{Cite book|title=Convex Optimization|last=Boyd|first=Stephen|publisher=Cambridge University Press|year=2004|isbn=978-0-521-83378-3|location=Cambridge, UK|pages=79}} 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.{{citation|title=Theory of extremal problems|volume=6|series=Studies in Mathematics and its Applications|first1=Aleksandr Davidovich|last1=Ioffe|first2=Vladimir Mikhaĭlovich|last2=Tikhomirov|publisher=North-Holland|year=2009|isbn=9780080875279|page=168|url=https://books.google.com/books?id=iDRVxznSxUsC&pg=PA168}}.

Citations

References

{{Rockafellar Wets Variational Analysis 2009 Springer}}

Category:Convex analysis

Category:Types of functions

proper convex function

Definitions

Properties

See also

Citations

References