Proper convex function

From formulasearchengine
Jump to navigation Jump to search

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 .[1] Convex functions that are not proper are called improper convex functions.[2]

A proper concave function is any function g such that is a proper convex function.


For every proper convex function f on Rn there exist some b in Rn and β in R such that

for every x.

The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets and are non-empty convex sets in the vector space X, then the indicator functions and are proper convex functions, but if then is identically equal to .

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.[3]


  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}.