What is Effective domain?

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 = R ∪ { ± ∞ } . {\displaystyle =\mathbb {R} \cup \{\pm \infty \}.}

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 + ∞ , {\displaystyle +\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 + ∞ {\displaystyle +\infty } at a point specifically to exluded that point from even being considered as a potential solution. Points at which the function takes the value − ∞ {\displaystyle -\infty } 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 + ∞ {\displaystyle +\infty } at that point instead.

When a minimum point of a function f : X → {\displaystyle f:X\to } is to be found but f {\displaystyle f} 's domain X {\displaystyle X} is a proper subset of some vector space V , {\displaystyle V,} then it often technically useful to extend f {\displaystyle f} to all of V {\displaystyle V} by setting f := + ∞ {\displaystyle f:=+\infty } at every x ∈ V ∖ X . {\displaystyle x\in V\setminus X.} By definition, no point of V ∖ X {\displaystyle V\setminus X} belongs to the effective domain of f , {\displaystyle f,} which is consistent with the desire to find a minimum point of the original function f : X → {\displaystyle f:X\to } rather than of the newly defined extension to all of V . {\displaystyle V.}

If the problem is instead a maximization problem then the effective domain instead consists of all points in the function's domain at which it is not equal to − ∞ . {\displaystyle -\infty.}