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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
∪
{
±
∞
}
.
{\displaystyle [-\infty ,\infty ]=\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 .}
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
+
∞
{\displaystyle +\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
−
∞
{\displaystyle -\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
+
∞
{\displaystyle +\infty }
at that point instead.
When a minimum point (in
X
{\displaystyle X}
) of a function
f
:
X
→
[
−
∞
,
∞
]
{\displaystyle f:X\to [-\infty ,\infty ]}
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
(
x
)
:=
+
∞
{\displaystyle f(x):=+\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 [-\infty ,\infty ]}
rather than of the newly defined extension to all of
V
.
{\displaystyle 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
−
∞
.
{\displaystyle -\infty .}
Definition
Suppose
f
:
X
→
[
−
∞
,
∞
]
{\displaystyle f:X\to [-\infty ,\infty ]}
is a map valued in the extended real number line
[
−
∞
,
∞
]
=
R
∪
{
±
∞
}
{\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}
whose domain, which is denoted by
domain
f
,
{\displaystyle \operatorname {domain} f,}
is
X
{\displaystyle X}
(where
X
{\displaystyle X}
will be assumed to be a subset of some vector space whenever this assumption is necessary).
Then the effective domain of
f
{\displaystyle f}
is denoted by
dom
f
{\displaystyle \operatorname {dom} f}
and typically defined to be the set
dom
f
=
{
x
∈
X
:
f
(
x
)
<
+
∞
}
{\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}}
unless
f
{\displaystyle f}
is a concave function or the maximum (rather than the minimum) of
f
{\displaystyle f}
is being sought, in which case the effective domain of
f
{\displaystyle f}
is instead the set
dom
f
=
{
x
∈
X
:
f
(
x
)
>
−
∞
}
.
{\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)>-\infty \}.}
In convex analysis and variational analysis,
dom
f
{\displaystyle \operatorname {dom} f}
is usually assumed to be
dom
f
=
{
x
∈
X
:
f
(
x
)
<
+
∞
}
{\displaystyle \operatorname {dom} f=\{x\in X~:~f(x)<+\infty \}}
unless clearly indicated otherwise.
Characterizations
Let
π
X
:
X
×
R
→
X
{\displaystyle \pi _{X}:X\times \mathbb {R} \to X}
denote the canonical projection onto
X
,
{\displaystyle X,}
which is defined by
(
x
,
r
)
↦
x
.
{\displaystyle (x,r)\mapsto x.}
The effective domain of
f
:
X
→
[
−
∞
,
∞
]
{\displaystyle f:X\to [-\infty ,\infty ]}
is equal to the image of
f
{\displaystyle f}
's epigraph
epi
f
{\displaystyle \operatorname {epi} f}
under the canonical projection
π
X
.
{\displaystyle \pi _{X}.}
That is
dom
f
=
π
X
(
epi
f
)
=
{
x
∈
X
:
there exists
y
∈
R
such that
(
x
,
y
)
∈
epi
f
}
.
{\displaystyle \operatorname {dom} f=\pi _{X}\left(\operatorname {epi} f\right)=\left\{x\in X~:~{\text{ there exists }}y\in \mathbb {R} {\text{ such that }}(x,y)\in \operatorname {epi} f\right\}.}
For a maximization problem (such as if the
f
{\displaystyle f}
is concave rather than convex), the effective domain is instead equal to the image under
π
X
{\displaystyle \pi _{X}}
of
f
{\displaystyle f}
's hypograph.
Properties
If a function never takes the value
+
∞
,
{\displaystyle +\infty ,}
such as if the function is real-valued, then its domain and effective domain are equal.
A function
f
:
X
→
[
−
∞
,
∞
]
{\displaystyle f:X\to [-\infty ,\infty ]}
is a proper convex function if and only if
f
{\displaystyle f}
is convex, the effective domain of
f
{\displaystyle f}
is nonempty, and
f
(
x
)
>
−
∞
{\displaystyle f(x)>-\infty }
for every
x
∈
X
.
{\displaystyle x\in X.}
See also
Proper convex function
Epigraph (mathematics) – Region above a graph
Hypograph (mathematics) – Region underneath a graph
References
Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.