Effective domain

In convex analysis, a branch of mathematics, the '''effective domain''' extends of the domain of a function defined for functions that take values in the extended real number line <math>[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}.</math>

In convex analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a global minimum point. The effective domain of this function is defined to be the set of all points in this function's domain at which its value is not equal to <math>+\infty.</math>{{sfn|Rockafellar|Wets|2009|pp=1-28}} It is defined this way because it is only these points that have even a remote chance of being a global minimum point. Indeed, it is common practice in these fields to set a function equal to <math>+\infty</math> at a point specifically to {{em|exclude}} that point from even being considered as a potential solution (to the minimization problem).{{sfn|Rockafellar|Wets|2009|pp=1-28}} Points at which the function takes the value <math>-\infty</math> (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem,{{sfn|Rockafellar|Wets|2009|pp=1-28}} with the reasoning being that if such a point was not acceptable as a solution then the function would have already been set to <math>+\infty</math> at that point instead.

When a minimum point (in <math>X</math>) of a function <math>f : X \to [-\infty, \infty]</math> is to be found but <math>f</math>'s domain <math>X</math> is a proper subset of some vector space <math>V,</math> then it often technically useful to extend <math>f</math> to all of <math>V</math> by setting <math>f(x) := +\infty</math> at every <math>x \in V \setminus X.</math>{{sfn|Rockafellar|Wets|2009|pp=1-28}} By definition, no point of <math>V \setminus X</math> belongs to the effective domain of <math>f,</math> which is consistent with the desire to find a minimum point of the original function <math>f : X \to [-\infty, \infty]</math> rather than of the newly defined extension to all of <math>V.</math>

If the problem is instead a maximization problem (which would be clearly indicated) then the effective domain instead consists of all points in the function's domain at which it is not equal to <math>-\infty.</math>

==Definition==

Suppose <math>f : X \to [-\infty, \infty]</math> is a map valued in the extended real number line <math>[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}</math> whose domain, which is denoted by <math>\operatorname{domain} f,</math> is <math>X</math> (where <math>X</math> will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the {{em|'''effective domain'''}} of <math>f</math> is denoted by <math>\operatorname{dom} f</math> and typically defined to be the set{{sfn|Rockafellar|Wets|2009|pp=1-28}}<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><ref>{{cite book|first1=Hans|last1=Föllmer|first2=Alexander|last2=Schied|title=Stochastic finance: an introduction in discrete time|publisher=Walter de Gruyter|year=2004|edition=2|isbn=978-3-11-018346-7|page=400}}</ref> <math display=block>\operatorname{dom} f = \{ x \in X ~:~ f(x) < +\infty \}</math> unless <math>f</math> is a concave function or the maximum (rather than the minimum) of <math>f</math> is being sought, in which case the {{em|'''effective domain'''}} of <math>f</math> is instead the set<ref name="AB" /> <math display=block>\operatorname{dom} f = \{ x \in X ~:~ f(x) > -\infty \}.</math>

In convex analysis and variational analysis, <math>\operatorname{dom} f</math> is usually assumed to be <math>\operatorname{dom} f = \{ x \in X ~:~ f(x) < +\infty \}</math> unless clearly indicated otherwise.

==Characterizations==

Let <math>\pi_{X} : X \times \mathbb{R} \to X</math> denote the canonical projection onto <math>X,</math> which is defined by <math>(x, r) \mapsto x.</math> The effective domain of <math>f : X \to [-\infty, \infty]</math> is equal to the image of <math>f</math>'s epigraph <math>\operatorname{epi} f</math> under the canonical projection <math>\pi_{X}.</math> That is :<math>\operatorname{dom} f = \pi_{X}\left( \operatorname{epi} f \right) = \left\{ x \in X ~:~ \text{ there exists } y \in \mathbb{R} \text{ such that } (x, y) \in \operatorname{epi} f \right\}.</math><ref name="Rockafellar">{{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=23}}</ref> For a maximization problem (such as if the <math>f</math> is concave rather than convex), the effective domain is instead equal to the image under <math>\pi_{X}</math> of <math>f</math>'s hypograph.

==Properties==

If a function {{em|never}} takes the value <math>+\infty,</math> such as if the function is real-valued, then its domain and effective domain are equal.

A function <math>f : X \to [-\infty, \infty]</math> is a proper convex function if and only if <math>f</math> is convex, the effective domain of <math>f</math> is nonempty, and <math>f(x) > -\infty</math> for every <math>x \in X.</math><ref name="Rockafellar" />

==See also==

* {{annotated link|Proper convex function}} * {{annotated link|Epigraph (mathematics)}} * {{annotated link|Hypograph (mathematics)}}

==References==

* {{Rockafellar Wets Variational Analysis 2009 Springer}}

Category:Convex analysis Category:Functions and mappings