Proper convex function

{{Short description|Concept in convex analysis}} {{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 <math>-\infty</math> and also is not identically equal to <math>+\infty.</math>

In convex analysis and variational analysis, a point (in the domain) at which some given function <math>f</math> is minimized is typically sought, where <math>f</math> is valued in the extended real number line <math>[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}.</math>{{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 <math>-\infty</math> as a value then <math>-\infty</math> 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 <math>-\infty</math> as a value. Assuming this, if the function's domain is empty or if the function is identically equal to <math>+\infty</math> 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 <math>g</math> is called {{em|proper}} if its negation <math>-g,</math> which is a convex function, is proper in the sense defined above.

==Definitions==

Suppose that <math>f : X \to [-\infty, \infty]</math> is a function taking values in the extended real number line <math>[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}.</math> If <math>f</math> is a convex function or if a minimum point of <math>f</math> is being sought, then <math>f</math> is called '''{{em|proper}}''' if

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

and if there also exists {{em|some}} point <math>x_0 \in X</math> such that

:<math>f\left( x_0 \right) < +\infty.</math>

That is, a function is {{em|proper}} if it never attains the value <math>-\infty</math> and its effective domain is nonempty.<ref name="AB">{{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}}</ref> This means that there exists some <math>x \in X</math> at which <math>f(x) \in \mathbb{R}</math> and <math>f</math> is also {{em|never}} equal to <math>-\infty.</math> Convex functions that are not proper are called '''{{em|improper}}''' convex functions.<ref>{{cite book|author=Rockafellar, R. Tyrrell|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|orig-date=1970|isbn=978-0-691-01586-6|page=24}}</ref>

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

==Properties==

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

:<math>f(x) \geq x \cdot b - r</math>

for every <math>x \in \mathbb{R}^n.</math>

The sum of two proper convex functions is convex, but not necessarily proper.<ref>{{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}}</ref> For instance if the sets <math>A \subset X</math> and <math>B \subset X</math> are non-empty convex sets in the vector space <math>X,</math> then the characteristic functions <math>I_A</math> and <math>I_B</math> are proper convex functions, but if <math>A \cap B = \varnothing</math> then <math>I_A + I_B</math> is identically equal to <math>+\infty.</math>

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.<ref>{{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}}.</ref>

==See also==

* {{annotated link|Effective domain}}

==Citations==

==References==

* {{Rockafellar Wets Variational Analysis 2009 Springer}}

Category:Convex analysis Category:Types of functions