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 $+\infty .$ 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 exclude that point from even being considered as a potential solution (to the minimization problem). 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, 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 $x\in V\setminus X.$ 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 effective domain of $f$ is denoted by $\operatorname {dom} f$ and typically defined to be the set $$\operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}$$ unless $f$ is a concave function or the maximum (rather than the minimum) of $f$ is being sought, in which case the 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

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

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 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 $x\in X.$

See also

• Proper convex function
• Epigraph (mathematics) – Region above a graph
• Hypograph (mathematics) – Region underneath a graph

References

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