Effective domain
Encyclopedia
In convex analysis
, a branch of mathematics, the effective domain is an extension of the domain of a function.
Given a vector space
X then a convex function
mapping to the extended reals,
, has an effective domain defined by
If the function is concave
, then the effective domain is
The effective domain is equivalent to the projection of the epigraph
of a function
onto X. That is
Note that if a convex function is mapping to the normal real number line given by
then the effective domain is the same as the normal definition of the domain.
A function
is a proper convex function
if and only if the effective domain of f is nonempty and
for every 
.
Convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory....
, a branch of mathematics, the effective domain is an extension of the domain of a function.
Given a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
X then a convex function
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
mapping to the extended reals,
, has an effective domain defined byIf the function is concave
Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
, then the effective domain is
The effective domain is equivalent to the projection of the epigraph
Epigraph
Epigraph may refer to:* an inscription, as studied in the archeological sub-discipline of epigraphy* Epigraph , a phrase, quotation, or poem that is set at the beginning of a document or component...
of a function
onto X. That isNote that if a convex function is mapping to the normal real number line given by
then the effective domain is the same as the normal definition of the domain.A function
is a proper convex functionProper convex function
In mathematical analysis and optimization, a proper convex function is a convex function f taking values in the extended real number line such thatf -\infty...
if and only if the effective domain of f is nonempty and
for every .