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In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line $\left[-\infty, \infty\right] = \mathbb \cup \.$ In convex analysis and variational analysis, a point at which some given Extended real number line, 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 $+\infty,$ where the effective domain 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 $f : X \to \left[-\infty, \infty\right]$ 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 $f\left(x\right) := +\infty$ 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 $f : X \to \left[-\infty, \infty\right]$ 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 $f : X \to \left[-\infty, \infty\right]$ is a map valued in the extended real number line $\left[-\infty, \infty\right] = \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, 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 $\left(x, r\right) \mapsto x.$ The effective domain of $f : X \to \left[-\infty, \infty\right]$ is equal to the Image (function), image of $f$'s Epigraph (mathematics), epigraph $\operatorname f$ under the canonical projection $\pi_.$ That is :$\operatorname f = \pi_\left\left( \operatorname f \right\right) = \left\.$ 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 (mathematics), hypograph.

# Properties

If a function takes the value $+\infty,$ such as if the function is Real number, real-valued, then its Domain of a function, domain and effective domain are equal. A function $f : X \to \left[-\infty, \infty\right]$ is a proper convex function if and only if $f$ is convex, the effective domain of $f$ is nonempty, and $f\left(x\right) > -\infty$ for every $x \in X.$