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]\displaystyle{ [-\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]\displaystyle{ +\infty. }[/math][1] 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]\displaystyle{ +\infty }[/math] at a point specifically to exclude that point from even being considered as a potential solution (to the minimization problem).[1] Points at which the function takes the value [math]\displaystyle{ -\infty }[/math] (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem,[1] 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]\displaystyle{ +\infty }[/math] at that point instead.
When a minimum point (in [math]\displaystyle{ X }[/math]) of a function [math]\displaystyle{ f : X \to [-\infty, \infty] }[/math] is to be found but [math]\displaystyle{ f }[/math]'s domain [math]\displaystyle{ X }[/math] is a proper subset of some vector space [math]\displaystyle{ V, }[/math] then it often technically useful to extend [math]\displaystyle{ f }[/math] to all of [math]\displaystyle{ V }[/math] by setting [math]\displaystyle{ f(x) := +\infty }[/math] at every [math]\displaystyle{ x \in V \setminus X. }[/math][1] By definition, no point of [math]\displaystyle{ V \setminus X }[/math] belongs to the effective domain of [math]\displaystyle{ f, }[/math] which is consistent with the desire to find a minimum point of the original function [math]\displaystyle{ f : X \to [-\infty, \infty] }[/math] rather than of the newly defined extension to all of [math]\displaystyle{ 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]\displaystyle{ -\infty. }[/math]
Definition
Suppose [math]\displaystyle{ f : X \to [-\infty, \infty] }[/math] is a map valued in the extended real number line [math]\displaystyle{ [-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \} }[/math] whose domain, which is denoted by [math]\displaystyle{ \operatorname{domain} f, }[/math] is [math]\displaystyle{ X }[/math] (where [math]\displaystyle{ X }[/math] will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the effective domain of [math]\displaystyle{ f }[/math] is denoted by [math]\displaystyle{ \operatorname{dom} f }[/math] and typically defined to be the set[1][2][3] [math]\displaystyle{ \operatorname{dom} f = \{ x \in X ~:~ f(x) \lt +\infty \} }[/math] unless [math]\displaystyle{ f }[/math] is a concave function or the maximum (rather than the minimum) of [math]\displaystyle{ f }[/math] is being sought, in which case the effective domain of [math]\displaystyle{ f }[/math] is instead the set[2] [math]\displaystyle{ \operatorname{dom} f = \{ x \in X ~:~ f(x) \gt -\infty \}. }[/math]
In convex analysis and variational analysis, [math]\displaystyle{ \operatorname{dom} f }[/math] is usually assumed to be [math]\displaystyle{ \operatorname{dom} f = \{ x \in X ~:~ f(x) \lt +\infty \} }[/math] unless clearly indicated otherwise.
Characterizations
Let [math]\displaystyle{ \pi_{X} : X \times \mathbb{R} \to X }[/math] denote the canonical projection onto [math]\displaystyle{ X, }[/math] which is defined by [math]\displaystyle{ (x, r) \mapsto x. }[/math] The effective domain of [math]\displaystyle{ f : X \to [-\infty, \infty] }[/math] is equal to the image of [math]\displaystyle{ f }[/math]'s epigraph [math]\displaystyle{ \operatorname{epi} f }[/math] under the canonical projection [math]\displaystyle{ \pi_{X}. }[/math] That is
- [math]\displaystyle{ \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][4]
For a maximization problem (such as if the [math]\displaystyle{ f }[/math] is concave rather than convex), the effective domain is instead equal to the image under [math]\displaystyle{ \pi_{X} }[/math] of [math]\displaystyle{ f }[/math]'s hypograph.
Properties
If a function never takes the value [math]\displaystyle{ +\infty, }[/math] such as if the function is real-valued, then its domain and effective domain are equal.
A function [math]\displaystyle{ f : X \to [-\infty, \infty] }[/math] is a proper convex function if and only if [math]\displaystyle{ f }[/math] is convex, the effective domain of [math]\displaystyle{ f }[/math] is nonempty, and [math]\displaystyle{ f(x) \gt -\infty }[/math] for every [math]\displaystyle{ x \in X. }[/math][4]
See also
References
- ↑ 1.0 1.1 1.2 1.3 1.4 Rockafellar & Wets 2009, pp. 1-28.
- ↑ 2.0 2.1 Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 254. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
- ↑ Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. p. 400. ISBN 978-3-11-018346-7.
- ↑ 4.0 4.1 Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.
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