In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, in particular the subfields of
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.
Convex sets
A subset C \subseteq X of ...
and
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, a proper convex function is an
extended real
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
-valued
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
with a
non-empty domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
, that never takes on the value
and also is not identically equal to
In
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.
Convex sets
A subset C \subseteq X of ...
and
variational analysis In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimizatio ...
, a point (in the domain) at which some given function
is minimized is typically sought, where
is valued in the
extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
Such a point, if it exists, is called a of the function and its value at this point is called the () of the function. If the function takes
as a value then
is necessarily the global minimum value and the minimization problem can be answered; this is ultimately the reason why the definition of "" requires that the function never take
as a value. Assuming this, if the function's domain is empty or if the function is identically equal to
then the minimization problem once again has an immediate answer. Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called . Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases.
If the problem is instead a maximization problem (which would be clearly indicated, such as by the function being
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset ...
rather than convex) then the definition of "" is defined in an analogous (albeit technically different) manner but with the same goal: to exclude cases where the maximization problem can be answered immediately. Specifically, a concave function
is called if its
negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...
which is a convex function, is proper in the sense defined above.
Definitions
Suppose that
extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
If
is a
convex function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
or if a minimum point of
is being sought, then
is called if
:
for
and if there also exists point
in its domain such that
:
That is, a function is if its
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 \infty, \infty= \mathbb \cup \.
In convex analysis and variation ...
is nonempty and it never attains the value
.
This means that there exists some
at which
and
is also equal to
Convex functions that are not proper are called convex functions.
A is by definition, any function
Properties
For every proper convex function
f : \mathbb^n \to \infty, \infty there exist some
b \in \mathbb^n and
r \in \mathbb such that
:
f(x) \geq x \cdot b - r
for every
x \in X.
The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets
A \subset X and
B \subset X are non-empty
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s in the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
X, then the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
s
I_A and
I_B are proper convex functions, but if
A \cap B = \varnothing then
I_A + I_B is identically equal to
+\infty.
The
infimal convolution of two proper convex functions is convex but not necessarily proper convex.
[.]
See also
*
Citations
References
*
{{Convex analysis and variational analysis
Convex analysis
Types of functions