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 $[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}.$

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 $+\infty.$ {{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 $+\infty$ 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 $-\infty$ (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 $+\infty$ at that point instead.

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

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 $-\infty.$

Definition

Suppose $f : X \to [-\infty, \infty]$ is a map valued in the extended real number line $[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}$ whose domain, which is denoted by $\operatorname{domain} f,$ is $X$ (where $X$ will be assumed to be a subset of some vector space whenever this assumption is necessary).

Then the {{em|effective domain}} of $f$ is denoted by $\operatorname{dom} f$ and typically defined to be the set{{sfn|Rockafellar|Wets|2009|pp=1-28}}{{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}}{{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}}

$\operatorname{dom} f = \{ x \in X ~:~ f(x) < +\infty \}$

unless $f$ is a concave function or the maximum (rather than the minimum) of $f$ is being sought, in which case the {{em|effective domain}} of $f$ is instead the set

$\operatorname{dom} f = \{ x \in X ~:~ f(x) > -\infty \}.$

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

Characterizations

Let $\pi_{X} : X \times \mathbb{R} \to X$ denote the canonical projection onto $X,$ which is defined by $(x, r) \mapsto x.$

The effective domain of $f : X \to [-\infty, \infty]$ is equal to the image of $f$ 's epigraph $\operatorname{epi} f$ under the canonical projection $\pi_{X}.$ That is

: $\operatorname{dom} f = \pi_{X}\left( \operatorname{epi} f \right) = \left\{ x \in X ~:~ \text{ there exists } y \in \mathbb{R} \text{ such that } (x, y) \in \operatorname{epi} f \right\}.$ {{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=23}}

For a maximization problem (such as if the $f$ is concave rather than convex), the effective domain is instead equal to the image under $\pi_{X}$ of $f$ 's hypograph.

Properties

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

A function $f : X \to [-\infty, \infty]$ is a proper convex function if and only if $f$ is convex, the effective domain of $f$ is nonempty, and $f(x) > -\infty$ for every $x \in X.$

References

{{Rockafellar Wets Variational Analysis 2009 Springer}}

Category:Convex analysis

Category:Functions and mappings

Effective domain

Definition

Characterizations

Properties

See also

References