Effective domain

In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}.}$

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 ${\displaystyle +\infty ,}$[1] where the effective domain 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 ${\displaystyle +\infty }$ at a point specifically to exluded that point from even being considered as a potential solution (to the minimization problem).[1] Points at which the function takes the value ${\displaystyle -\infty }$ (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 ${\displaystyle +\infty }$ at that point instead.

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

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 ${\displaystyle -\infty .}$

Definition

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

${\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}}$[1][2][3]

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

${\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)>-\infty \}.}$[2]

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

Characterizations

Let ${\displaystyle \pi _{X}:X\times \mathbb {R} \to X}$ denote the canonical projection onto ${\displaystyle X,}$ which is defined by ${\displaystyle (x,r)\mapsto x.}$ The effective domain of ${\displaystyle f:X\to [-\infty ,\infty ]}$ is equal to the image of ${\displaystyle f}$'s epigraph ${\displaystyle \operatorname {epi} f}$ under the canonical projection ${\displaystyle \pi _{X}.}$ That is

${\displaystyle \operatorname {dom} f=\pi _{X}\left(\operatorname {epi} f\right)=\left\{x\in X~:~{\text{ there exists }}y\in \mathbb {R} {\text{ such that }}(x,y)\in \operatorname {epi} f\right\}.}$[4]

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

Properties

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

A function ${\displaystyle f:X\to [-\infty ,\infty ]}$ is a proper convex function if and only if ${\displaystyle f}$ is convex, the effective domain of ${\displaystyle f}$ is nonempty, and ${\displaystyle f(x)>-\infty }$ for every ${\displaystyle x\in X.}$[4]

See also

• Proper convex function
• Epigraph (mathematics)
• Hypograph (mathematics)

References

1. Rockafellar & Wets 2009, pp. 1-28. sfn error: no target: CITEREFRockafellarWets2009 (help)
2. 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.
3. 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. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.

• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.CS1 maint: date and year (link)