Effective domain explained

[ -inf ty, inf ty] = R \cup \{ \pm inf ty \}.

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

+inf ty. 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 +inf ty at a point specifically to that point from even being considered as a potential solution (to the minimization problem). Points at which the function takes the value -inf ty (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 +inf ty at that point instead.

When a minimum point (in

X ) of a function f : X \to [ -inf ty, inf ty] 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) : = +inf ty 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 [ -inf ty, inf ty] 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

-inf ty.

Definition

Suppose

f : X \to [ -inf ty, inf ty] is a map valued in the extended real number line [ -inf ty, inf ty] = R \cup \{ \pm inf ty \} 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 of f is denoted by \operatorname{dom} f and typically defined to be the set^{[1] [2]} $$\backslash operatorname\; f\; =\; \backslash$$unless f is a concave function or the maximum (rather than the minimum) of f is being sought, in which case the of f is instead the set$$\backslash operatorname\; f\; =\; \backslash .$$

In convex analysis and variational analysis,

\operatorname{dom} f is usually assumed to be \operatorname{dom} f = \{ x \in X ~:~ f(x) < +inf ty \} unless clearly indicated otherwise.

Characterizations

Let

\pi _{X} : X x R \to X denote the canonical projection onto X, which is defined by (x, r) \mapsto x. The effective domain of f : X \to [ -inf ty, inf ty] 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 ~:~ thereexists y \in R suchthat (x, y) \in \operatorname{epi} f \right\}. ^[3] 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 takes the value

+inf ty, such as if the function is real-valued, then its domain and effective domain are equal.

A function

f : X \to [ -inf ty, inf ty] is a proper convex function if and only if f is convex, the effective domain of f is nonempty, and f(x) > -inf ty for every x \in X.

Notes and References

1. Book: Aliprantis. C.D.. Border. K.C.. Infinite Dimensional Analysis: A Hitchhiker's Guide. 3. Springer. 2007. 978-3-540-32696-0. 10.1007/3-540-29587-9. 254.
2. Book: Hans. Föllmer. Alexander. Schied. Stochastic finance: an introduction in discrete time. Walter de Gruyter. 2004. 2. 978-3-11-018346-7. 400.
3. Book: Rockafellar, R. Tyrrell. Rockafellar, R. Tyrrell. Convex Analysis. Princeton University Press. Princeton, NJ. 1997. 1970. 978-0-691-01586-6. 23.