arg max

The function arg max ( argumentum maximi , dt. Argument of the maximum ) is a function used in analysis and optimization to calculate the point at which a function assumes its maximum . The function arg min is used in the same way .

definition

If the domain of a function is , then from is the point at which the function assumes its maximum, that is ${\displaystyle D}$ ${\displaystyle f}$ ${\displaystyle \arg \max }$ ${\displaystyle f}$ ${\displaystyle x_{\mathrm {max} }}$

${\displaystyle x_{\mathrm {max} }={\underset {x\in D}{\operatorname {arg\,max} }}\,f(x)\Leftrightarrow f(x_{\mathrm {max} })=\max _{x\in D}f(x).}$

So it's not about the value of the maximum itself, but about a value from the domain of definition. This value is not well-defined if the function assumes its maximum in several places or has no maximum.

example

The function has the maximum value that is assumed at the point . Therefore applies ${\displaystyle f(x)=x(10-x)}$ ${\displaystyle 25}$ ${\displaystyle x_{\mathrm {max} }=5}$

${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}(x(10-x))=5.}$

Alternative definition

In order to achieve well-definition, the following is alternatively explained as a set- valued function : ${\displaystyle \arg \max }$

${\displaystyle {\underset {x\in D}{\operatorname {arg\,max} }}\,f(x):=\{x\in D\ |\ f(x){\text{ maximal}}\}=\{x\in D\ |\ \forall y\in D\ f(y)\leq f(x)\}=f^{-1}\left(\max _{x\in D}f(x)\right)}$

Analogous to this

${\displaystyle {\underset {x\in D}{\operatorname {arg\,min} }}\,f(x):=\{x\in D\ |\ f(x){\text{ minimal}}\}=\{x\in D\ |\ \forall y\in D\ f(y)\geq f(x)\}=f^{-1}\left(\min _{x\in D}f(x)\right)}$

Are defined.

example

${\displaystyle {\underset {x\in [0,4\pi ]}{\operatorname {arg\,max} }}\,\cos(x)=\{0,2\pi ,4\pi \}.}$

literature

• Peter Gritzmann Fundamentals of Mathematical Optimization , Springer, 2013, ISBN 978-3-528-07290-2 , page 3.