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In
convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...
, a branch of mathematics, the effective domain is an extension of the
domain of a function In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . ...
defined for functions that take values in the
extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
\infty, \infty= \mathbb \cup \. In
convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...
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, 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 \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 In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
\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 Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...
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 In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
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

* * *


References

* {{mathanalysis-stub Convex analysis Functions and mappings