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 ${\displaystyle -\infty }$ and also is not identically equal to ${\displaystyle +\infty .}$

In convex analysis and variational analysis, a point (in the domain) at which some given function ${\displaystyle f}$ is minimized is typically sought, where ${\displaystyle f}$ is valued in the extended real number line ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}.}$^[1] Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the function. If the function takes ${\displaystyle -\infty }$ as a value then ${\displaystyle -\infty }$ is necessarily the global minimum value and the minimization problem can be answered; this is ultimately the reason why the definition of "proper" requires that the function never take ${\displaystyle -\infty }$ as a value. Assuming this, if the function's domain is empty or if the function is identically equal to ${\displaystyle +\infty }$ 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 proper. 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 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 ${\displaystyle g}$ is called proper if its negation ${\displaystyle -g,}$ which is a convex function, is proper in the sense defined above.