Proper convex function
In
In convex analysis and variational analysis, a point (in the domain) at which some given function is minimized is typically sought, where is valued in the extended real number line
If the problem is instead a maximization problem (which would be clearly indicated, such as by the function being concave rather than convex) then the definition of "proper" 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 proper if its negation which is a convex function, is proper in the sense defined above.
Definitions
Suppose that is a function taking values in the extended real number line If is a convex function or if a minimum point of is being sought, then is called proper if
- for every
and if there also exists some point such that
That is, a function is proper if it never attains the value and its effective domain is nonempty.[2] This means that there exists some at which and is also never equal to Convex functions that are not proper are called improper convex functions.[3]
A proper concave function is by definition, any function such that is a proper convex function. Explicitly, if is a concave function or if a maximum point of is being sought, then is called proper if its domain is not empty, it never takes on the value and it is not identically equal to
Properties
For every proper convex function there exist some and such that
for every
The sum of two proper convex functions is convex, but not necessarily proper.[4] For instance if the sets and are non-empty convex sets in the vector space then the characteristic functions and are proper convex functions, but if then is identically equal to
The
See also
Citations
- ^ Rockafellar & Wets 2009, pp. 1–28.
- ISBN 978-3-540-32696-0.
- ISBN 978-0-691-01586-6.
- ISBN 978-0-521-83378-3.
- ISBN 9780080875279.
References
- OCLC 883392544.