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

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 [1] 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 at a point specifically to exclude that point from even being considered as a potential solution (to the minimization problem).[1] Points at which the function takes the value (if any) belong to the effective domain because such points are considered acceptable solutions to the minimization problem,[1] with the reasoning being that if such a point was not acceptable as a solution then the function would have already been set to at that point instead.

When a minimum point (in ) of a function is to be found but 's domain is a proper subset of some vector space then it often technically useful to extend to all of by setting at every [1] By definition, no point of belongs to the effective domain of which is consistent with the desire to find a minimum point of the original function rather than of the newly defined extension to all of

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

Definition

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Suppose   is a map valued in the extended real number line   whose domain, which is denoted by   is   (where   will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the effective domain of   is denoted by   and typically defined to be the set[1][2][3]   unless   is a concave function or the maximum (rather than the minimum) of   is being sought, in which case the effective domain of   is instead the set[2]  

In convex analysis and variational analysis,   is usually assumed to be   unless clearly indicated otherwise.

Characterizations

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Let   denote the canonical projection onto   which is defined by   The effective domain of   is equal to the image of  's epigraph   under the canonical projection   That is

 [4]

For a maximization problem (such as if the   is concave rather than convex), the effective domain is instead equal to the image under   of  's hypograph.

Properties

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If a function never takes the value   such as if the function is real-valued, then its domain and effective domain are equal.

A function   is a proper convex function if and only if   is convex, the effective domain of   is nonempty, and   for every  [4]

See also

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References

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  1. ^ a b c d e Rockafellar & Wets 2009, pp. 1–28.
  2. ^ a b Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 254. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
  3. ^ Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. p. 400. ISBN 978-3-11-018346-7.
  4. ^ a b Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.