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 $+\infty .$ ^[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 $+\infty$ 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 $-\infty$ (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 $+\infty$ 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 $x\in V\setminus X.$ ^[1] 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^[1]^[2]^[3] $\operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}$ 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^[2] $\operatorname {dom} f=\{x\in X~:~f(x)>-\infty \}.$

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\}.

^[4]

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 $x\in X.$ ^[4]

References

^ ^a ^b ^c ^d ^e Rockafellar & Wets 2009, pp. 1–28.
^ ^a ^b 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.
^ ^a ^b 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. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

[FOOTNOTERockafellarWets20091–28-1] Rockafellar & Wets 2009, pp. 1–28.

[AB-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.

[Rockafellar-4] Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.

[1]

[2]

[3]

[4]

v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics

Definition

Characterizations

Properties

See also

References