HOME

TheInfoList




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 [-\infty, \infty] = \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 [-\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 \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 (x, r) \mapsto x. The effective domain of f : X \to [-\infty, \infty] 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( \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 (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 [-\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.


See also

* * *


References

* {{mathanalysis-stub Convex analysis Functions and mappings