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 <math>-\infty</math> and also is not identically equal to <math>+\infty.</math>

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

Definitions

Suppose that <math>f : X \to [-\infty, \infty]</math> is a function taking values in the extended real number line <math>[-\infty, \infty] = \mathbb{R} \cup \{ \pm\infty \}.</math> If <math>f</math> is a convex function or if a minimum point of <math>f</math> is being sought, then <math>f</math> is called Template:Em if

<math>f(x) > -\infty</math> Template:Space for Template:Em <math>x \in X</math>

and if there also exists Template:Em point <math>x_0 \in X</math> such that

<math>f\left( x_0 \right) < +\infty.</math>

That is, a function is Template:Em if it never attains the value <math>-\infty</math> and its effective domain is nonempty.<ref name="AB">Template:Cite book</ref> This means that there exists some <math>x \in X</math> at which <math>f(x) \in \mathbb{R}</math> and <math>f</math> is also Template:Em equal to <math>-\infty.</math> Convex functions that are not proper are called Template:Em convex functions.<ref>Template:Cite book</ref>

A Template:Em is by definition, any function <math>g : X \to [-\infty, \infty]</math> such that <math>f := -g</math> is a proper convex function. Explicitly, if <math>g : X \to [-\infty, \infty]</math> is a concave function or if a maximum point of <math>g</math> is being sought, then <math>g</math> is called Template:Em if its domain is not empty, it Template:Em takes on the value <math>+\infty,</math> and it is not identically equal to <math>-\infty.</math>

Properties

For every proper convex function <math>f : \mathbb{R}^n \to [-\infty, \infty],</math> there exist some <math>b \in \mathbb{R}^n</math> and <math>r \in \mathbb{R}</math> such that

<math>f(x) \geq x \cdot b - r</math>

for every <math>x \in \mathbb{R}^n.</math>

The sum of two proper convex functions is convex, but not necessarily proper.<ref>Template:Cite book</ref> For instance if the sets <math>A \subset X</math> and <math>B \subset X</math> are non-empty convex sets in the vector space <math>X,</math> then the characteristic functions <math>I_A</math> and <math>I_B</math> are proper convex functions, but if <math>A \cap B = \varnothing</math> then <math>I_A + I_B</math> is identically equal to <math>+\infty.</math>

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.<ref>Template:Citation.</ref>

See also

• Template:Annotated link

References

• Template:Rockafellar Wets Variational Analysis 2009 Springer