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 ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}.}${\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 ${\displaystyle +\infty .}${\displaystyle +\infty .}^[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 ${\displaystyle +\infty }${\displaystyle +\infty } 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 ${\displaystyle -\infty }${\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 ${\displaystyle +\infty }${\displaystyle +\infty } at that point instead.

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

Suppose ${\displaystyle f:X\to [-\infty ,\infty ]}${\displaystyle f:X\to [-\infty ,\infty ]} is a map valued in the extended real number line ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} whose domain, which is denoted by ${\displaystyle \operatorname {domain} f,}${\displaystyle \operatorname {domain} f,} is ${\displaystyle X}${\displaystyle X} (where ${\displaystyle X}${\displaystyle X} will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the effective domain of ${\displaystyle f}${\displaystyle f} is denoted by ${\displaystyle \operatorname {dom} f}${\displaystyle \operatorname {dom} f} and typically defined to be the set^{[1] [2] [3]} $${\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}}$${\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}} unless ${\displaystyle f}${\displaystyle f} is a concave function or the maximum (rather than the minimum) of ${\displaystyle f}${\displaystyle f} is being sought, in which case the effective domain of ${\displaystyle f}${\displaystyle f} is instead the set^[2] $${\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)>-\infty \}.}$$-\infty \}.}"/>

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

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

${\displaystyle \operatorname {dom} f=\pi _{X}\left(\operatorname {epi} f\right)=\left\{x\in X~:~{\text{ there exists }}y\in \mathbb {R} {\text{ such that }}(x,y)\in \operatorname {epi} f\right\}.}${\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 ${\displaystyle f}${\displaystyle f} is concave rather than convex), the effective domain is instead equal to the image under ${\displaystyle \pi _{X}}${\displaystyle \pi _{X}} of ${\displaystyle f}${\displaystyle f}'s hypograph.

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

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

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

• 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.