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 [ − ∞ , ∞ ] = R ∪ { ± ∞ } . {\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 .} 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 } 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 − ∞ {\displaystyle -\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 + ∞ {\displaystyle +\infty } at that point instead. When a minimum point (in X {\displaystyle X} ) of a function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is to be found but f {\displaystyle f} 's domain X {\displaystyle X} is a proper subset of some vector space V , {\displaystyle V,} then it often technically useful to extend f {\displaystyle f} to all of V {\displaystyle V} by setting f ( x ) := + ∞ {\displaystyle f(x):=+\infty } at every x ∈ V ∖ X . {\displaystyle x\in V\setminus X.} By definition, no point of V ∖ X {\displaystyle V\setminus X} belongs to the effective domain of f , {\displaystyle f,} which is consistent with the desire to find a minimum point of the original function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} rather than of the newly defined extension to all of 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 .}
Definition Suppose f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is a map valued in the extended real number line [ − ∞ , ∞ ] = R ∪ { ± ∞ } {\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} whose domain, which is denoted by domain f , {\displaystyle \operatorname {domain} f,} is X {\displaystyle X} (where X {\displaystyle X} will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the effective domain of f {\displaystyle f} is denoted by dom f {\displaystyle \operatorname {dom} f} and typically defined to be the set
dom f = { x ∈ X : f ( x ) < + ∞ } {\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}}
unless f {\displaystyle f} is a concave function or the maximum (rather than the minimum) of f {\displaystyle f} is being sought, in which case the effective domain of f {\displaystyle f} is instead the set
dom f = { x ∈ X : f ( x ) > − ∞ } . {\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)>-\infty \}.}
In convex analysis and variational analysis, dom f {\displaystyle \operatorname {dom} f} is usually assumed to be dom f = { x ∈ X : f ( x ) < + ∞ } {\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}} unless clearly indicated otherwise.
Characterizations Let π X : X × R → X {\displaystyle \pi _{X}:X\times \mathbb {R} \to X} denote the canonical projection onto X , {\displaystyle X,} which is defined by ( x , r ) ↦ x . {\displaystyle (x,r)\mapsto x.} The effective domain of f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is equal to the image of f {\displaystyle f} 's epigraph epi f {\displaystyle \operatorname {epi} f} under the canonical projection π X . {\displaystyle \pi _{X}.} That is
dom f = π X ( epi f ) = { x ∈ X : there exists y ∈ R such that ( x , y ) ∈ epi f } . {\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\}.}
For a maximization problem (such as if the f {\displaystyle f} is concave rather than convex), the effective domain is instead equal to the image under π X {\displaystyle \pi _{X}} of f {\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 f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} is a proper convex function if and only if f {\displaystyle f} is convex, the effective domain of f {\displaystyle f} is nonempty, and f ( x ) > − ∞ {\displaystyle f(x)>-\infty } for every x ∈ X . {\displaystyle x\in X.}
See also Proper convex function – Concept in convex analysis 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.