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 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 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 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 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 $$x \in X.$$