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

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 $[-\mathrm{\infty },\mathrm{\infty }]=ℝ\cup \{\pm \mathrm{\infty }\}.$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 $-\mathrm{\infty }$ as a value then $-\mathrm{\infty }$ 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 $-\mathrm{\infty }$ as a value. Assuming this, if the function's domain is empty or if the function is identically equal to $+\mathrm{\infty }$ 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 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 $g$ is called Template:Em if its negation $-g,$ which is a convex function, is proper in the sense defined above.

Definitions

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

$f(x)>-\mathrm{\infty }$ Template:Space for Template:Em $x\in X$

and if there also exists Template:Em point $x_0\in X$ such that

$f\left(x_0\right)<+\mathrm{\infty }.$

That is, a function is Template:Em if it never attains the value $-\mathrm{\infty }$ and its effective domain is nonempty. This means that there exists some $x\in X$ at which $f(x)\in ℝ$ and $f$ is also Template:Em equal to $-\mathrm{\infty }.$ Convex functions that are not proper are called Template:Em convex functions.

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

Properties

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

$f(x)\ge x\cdot b-r$

for every $x\in {ℝ}^n.$

The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets $A\subset X$ and $B\subset X$ are non-empty convex sets in the vector space $X,$ then the characteristic functions $I_A$ and $I_B$ are proper convex functions, but if $A\cap B=\varnothing$ then $I_A+I_B$ is identically equal to $+\mathrm{\infty }.$

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

See also

• Template:Annotated link

References

• Template:Rockafellar Wets Variational Analysis 2009 Springer