Proper convex function

In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value and also is not identically equal to
In convex analysis and variational analysis, a point at which some given function is minimized is typically sought, where is valued in the extended real number line 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 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 then the definition of "" is defined in an analogous 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 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 ' if
and if there also exists point such that
That is, a function is if it never attains the value and its effective domain is nonempty.
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 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 if its domain is not empty, it 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. 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 infimal convolution of two proper convex functions is convex but not necessarily proper convex.