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 \cup \.

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/math> 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/math> 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/math> is a map valued in the extended real number line \infty, \infty= \mathbb \cup \ whose domain, which is denoted by $\operatorname f,$ 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 $\operatorname f$ and typically defined to be the set
$\operatorname f = \$
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
$\operatorname f = \.$

In convex analysis and variational analysis, $\operatorname f$ is usually assumed to be $\operatorname f = \$ unless clearly indicated otherwise.

Characterizations

Let $\pi_ : X \times \mathbb \to X$ denote the canonical projection onto $X,$ which is defined by $(x, r) \mapsto x.$
The effective domain of f : X \to \infty, \infty/math> is equal to the image of $f$'s epigraph $\operatorname f$ under the canonical projection $\pi_.$ That is
:$\operatorname f = \pi_\left( \operatorname f \right) = \left\.$
For a maximization problem (such as if the $f$ is concave rather than convex), the effective domain is instead equal to the image under $\pi_$ 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/math> 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 $x \in X.$

See also

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References

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Convex analysis
Functions and mappings