Extended function

An extended function and the closely related concept of a real function is a function whose value range is extended infinitely by the symbolic value . This makes it easier to deal with the function, since one can concentrate on the pre-image sets of interest, all other sets are assigned the function value infinite. This may mean that case distinctions can be dispensed with.

definition

Given is a function and a set on which the function has a certain property of interest. Then the function is called with ${\displaystyle f\colon V\to \mathbb {R} }$ ${\displaystyle M\subset V}$ ${\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}$

${\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&{\text{ falls }}x\in M\\{+\infty }&{\text{ sonst}}\end{cases}}}$

advanced function too . Typical properties of interest are, for example, monotony , convexity or well-definedness . The extended function is also completely defined. The amount ${\displaystyle f}$ ${\displaystyle V}$

${\displaystyle \operatorname {dom} ({\tilde {f}})=\{x\in V\mid {\tilde {f}}(x)<{+\infty }\}}$

is called the essential domain of . Is , that's the name of a real function . ${\displaystyle {\tilde {f}}}$ ${\displaystyle \operatorname {dom} ({\tilde {f}})\neq \emptyset }$ ${\displaystyle {\tilde {f}}}$

Examples

monotony

As an example, consider the function defined by . It is monotonically decreasing on the interval . In order to transfer this property to the whole , we set . Hence: ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle f(x)=-x^{2}}$ ${\displaystyle [0,\infty )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle M=[0,\infty )}$

${\displaystyle {\tilde {f}}(x)={\begin{cases}-x^{2}&{\text{ falls }}x\in [0,\infty )\\{+\infty }&{\text{ sonst}}\end{cases}}}$

The extended function is now according to the calculation rules with infinitely monotonically decreasing to whole . ${\displaystyle \mathbb {R} }$

convexity

If the set is convex , then the extended convex function is through ${\displaystyle f}$ ${\displaystyle C}$ ${\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}$

${\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&{\text{ falls }}x\in C\\{+\infty }&{\text{ sonst}}\end{cases}}}$

Are defined. With the calculation rules for infinity, this function is now convex on the whole and not just on the set . For example, the sine function is convex on the interval . Thus the extended function reads ${\displaystyle V}$ ${\displaystyle C}$ ${\displaystyle [-\pi ,0]}$

${\displaystyle {\tilde {f}}(x)={\begin{cases}\sin(x)&{\text{ falls }}x\in [-\pi ,0]\\{+\infty }&{\text{ sonst.}}\end{cases}}}$

This function is now convex all over . ${\displaystyle \mathbb {R} }$

Definition gaps

If you look at the function , it is not defined at this point . If one now sets , where is the domain, then the following applies: ${\displaystyle f(x)={\tfrac {1}{x^{2}}}}$ ${\displaystyle x=0}$ ${\displaystyle M=D_{f}}$ ${\displaystyle D_{f}}$

${\displaystyle {\tilde {f}}(x)={\begin{cases}{+\infty }&{\text{ falls }}x=0\\{\frac {1}{x^{2}}}&{\text{ sonst}}\end{cases}}}$

The extended function is now completely defined and operations can be carried out with the function without taking into account the definition gap. However, it must not be concluded from the extended function that this applies, since the value was only determined afterwards. ${\displaystyle \mathbb {R} }$ ${\displaystyle {\tfrac {1}{0^{2}}}={+\infty }}$ ${\displaystyle +\infty }$

use

Extended functions can be found in many areas of analysis, especially optimization . Here they offer the advantage that you can still sensibly minimize with extended definitions, but you do not get any formal problems with definition gaps or non-convex areas of the function.

literature

• Carl Geiger, Christian Kanzow: Theory and numerics of restricted optimization tasks . Springer-Verlag, Berlin Heidelberg New York 2002, ISBN 3-540-42790-2 .
• Stephen Boyd, Lieven Vandenberghe: Convex Optimization . Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 ( online ).