Posynomial function

A posynomial function (also written as a posinomial function ) and the monomial function closely related to it are functions that are used in the formulation of geometric programs . They can be seen as generalizations of polynomial functions in several variables, since any real exponents are allowed.

definition

Be for as well as for . Then the function is called ${\displaystyle \mathbb {R} _{++}^{n}:=\{x\in \mathbb {R} ^{n}\,|\,x_{i}>0}$ ${\displaystyle i=1,\dots ,n\}}$ ${\displaystyle c_{k}>0}$ ${\displaystyle k=1,\dots ,N}$

${\displaystyle f\colon \mathbb {R} _{++}^{n}\to \mathbb {R} }$

${\displaystyle f(x_{1},\dots ,x_{n})=\sum _{k=1}^{N}c_{k}x_{1}^{a_{1,k}}\cdot \dots \cdot x_{n}^{a_{n,k}}}$

a posynomial function. Everyone is there . If the sum consists of only one sum term, one speaks of a monomial function. ${\displaystyle a_{i,j}\in \mathbb {R} }$

example

The function

${\displaystyle f(x_{1},x_{2})={\frac {{\sqrt {x_{1}}}x_{2}+x_{2}^{2{,}3}}{x_{1}x_{2}}}}$

is a posynomial function, it has the normal representation

${\displaystyle f(x_{1},x_{2})=x_{1}^{-0{,}5}+x_{1}^{-1}x_{2}^{1{,}3}}$

The function

${\displaystyle f(x_{1},x_{2},x_{3})={\frac {x_{1}^{17}x_{2}}{x_{3}^{\sqrt {2}}}}}$

is a monomial function, it has the normal representation

${\displaystyle f(x_{1},x_{2},x_{3})=x_{1}^{17}x_{2}x_{3}^{-{\sqrt {2}}}}$

properties

• Posynomial functions are closed under addition, multiplication and multiplication with positive scalars.
• Monomial functions are completed under multiplication, division and positive scaling.
• The posynomial functions thus in particular form a convex cone in the vector space of all functions , the monomial functions at least a (dotted) linear sub- cone . ${\displaystyle \mathbb {R} _{++}^{n}\to \mathbb {R} }$

literature

• Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 978-0-521-83378-3 . ( online )