Proper convex function
Encyclopedia
In mathematical analysis
(in particular convex analysis
) and optimization
, a proper convex function is a convex function
f taking values in the extended real number line
such that
for at least one x and
for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains
. Convex functions that are not proper are called improper convex functions.
A proper concave function is any function g such that
is a proper convex function.
for every x.
The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets
and 
are convex set
s in the vector space
X, then the indicator function
s
and 
are proper convex functions, but 
is not convex (unless 
is convex), and is possibly identically equal to 
if 
(i.e. are complimenatry halfspaces).
The infimal convolution of two proper convex functions is convex but not necessarily proper convex.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
(in particular convex analysis
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....
) and optimization
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
, a proper convex function is 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...
f taking values in the extended real number line
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
such that
for at least one x and
for every x. That is, a convex function is proper if its effective domain is nonempty and it never attains
. Convex functions that are not proper are called improper convex functions.A proper concave function is any function g such that
is a proper convex function.Properties
For every proper convex function f on Rn there exist some b in Rn and β in R such thatfor every x.
The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets
and are convex setConvex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
s in the 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 the indicator function
Characteristic function (convex analysis)
In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership of a given element in that set...
s
and are proper convex functions, but is not convex (unless is convex), and is possibly identically equal to if (i.e. are complimenatry halfspaces).The infimal convolution of two proper convex functions is convex but not necessarily proper convex.