Effective domain

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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 [−∞,∞]=ℝ∪{±∞}. 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 X) of a function f:X→[−∞,∞] 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):=+∞ at every x∈V∖X.[1] By definition, no point of V∖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→[−∞,∞] 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 −∞.

Definition

Suppose f:X→[−∞,∞] is a map valued in the extended real number line [−∞,∞]=ℝ∪{±∞} whose domain, which is denoted by domain⁡f, is X (where X will be assumed to be a subset of some vector space whenever this assumption is necessary). Then the effective domain of f is denoted by dom⁡f and typically defined to be the set[1][2][3] dom⁡f={x∈X:f(x)<+∞} unless f is a concave function or the maximum (rather than the minimum) of f is being sought, in which case the effective domain of f is instead the set[2] dom⁡f={x∈X:f(x)>−∞}.

In convex analysis and variational analysis, dom⁡f is usually assumed to be dom⁡f={x∈X:f(x)<+∞} unless clearly indicated otherwise.

Characterizations

Let πX:X×ℝ→X denote the canonical projection onto X, which is defined by (x,r)↦x. The effective domain of f:X→[−∞,∞] is equal to the image of f's epigraph epi⁡f under the canonical projection πX. That is

dom⁡f=πX(epi⁡f)={x∈X: there exists y∈ℝ such that (x,y)∈epi⁡f}.[4]

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

Properties

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 f:X→[−∞,∞] is a proper convex function if and only if f is convex, the effective domain of f is nonempty, and f(x)>−∞ for every x∈X.[4]

See also

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Rockafellar & Wets 2009, pp. 1–28.
  2. ↑ 2.0 2.1 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. ↑ 4.0 4.1 Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6. 



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