Effective domain

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 $[-\mathrm{\infty },\mathrm{\infty }]=ℝ\cup \{\pm \mathrm{\infty }\}.$

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 $+\mathrm{\infty }.$ 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 $+\mathrm{\infty }$ at a point specifically to exclude that point from even being considered as a potential solution (to the minimization problem). Points at which the function takes the value $-\mathrm{\infty }$ (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 $+\mathrm{\infty }$ at that point instead.

When a minimum point (in $X$) of a function $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ 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 $f(x):=+\mathrm{\infty }$ 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 $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ 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 $-\mathrm{\infty }.$

Definition

Suppose $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ is a map valued in the extended real number line $[-\mathrm{\infty },\mathrm{\infty }]=ℝ\cup \{\pm \mathrm{\infty }\}$ whose domain, which is denoted by $\mathrm{domain}f,$ 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 $\mathrm{dom}f$ and typically defined to be the set $$\mathrm{dom}f=\{x\in X:f(x)<+\mathrm{\infty }\}$$ 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 $$\mathrm{dom}f=\{x\in X:f(x)>-\mathrm{\infty }\}.$$

In convex analysis and variational analysis, $\mathrm{dom}f$ is usually assumed to be $\mathrm{dom}f=\{x\in X:f(x)<+\mathrm{\infty }\}$ unless clearly indicated otherwise.

Characterizations

Let ${\pi }_X:X\times ℝ\to X$ denote the canonical projection onto $X,$ which is defined by $(x,r)\mapsto x.$ The effective domain of $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ is equal to the image of $f$'s epigraph $\mathrm{epi}f$ under the canonical projection ${\pi }_X.$ That is

$\mathrm{dom}f={\pi }_X(\mathrm{epi}f)=\{x\in X:\text{ there exists }y\in ℝ\text{ such that }(x,y)\in \mathrm{epi}f\}.$

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

Properties

If a function never takes the value $+\mathrm{\infty },$ such as if the function is real-valued, then its domain and effective domain are equal.

A function $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ is a proper convex function if and only if $f$ is convex, the effective domain of $f$ is nonempty, and $f(x)>-\mathrm{\infty }$ for every $x\in X.$

See also

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References

• Template:Rockafellar Wets Variational Analysis 2009 Springer