Proper convex function

Article summary

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 -\infty and also is not identically equal to +\infty. In convex analysis and variational analysis, a point (in the domain) at which some given function f is minimized is typically sought, where f is valued in the extended real number line [-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}. 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 -\infty as a value then -\infty 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 -\infty as a value. Assuming this, if the function's domain is empty or if the function is identically equal to +\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 . Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases.…

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