# Posynomial

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Posynomial
> Markdown URL: https://mediated.wiki/source/Posynomial.md
> Source: https://en.wikipedia.org/wiki/Posynomial
> Source revision: 1251651558
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

A '''posynomial''', also known as a '''posinomial''' in some literature, is a [function](/source/function_(mathematics)) of the form

: <math>f(x_1, x_2, \dots, x_n) = \sum_{k=1}^K c_k x_1^{a_{1k}} \cdots x_n^{a_{nk}}</math>

where all the coordinates <math>x_i</math> and  coefficients <math>c_k</math> are positive [real number](/source/real_number)s, and the exponents <math>a_{ik}</math> are real numbers.  Posynomials are closed under addition, multiplication, and nonnegative scaling.

For example,

: <math>f(x_1, x_2, x_3) = 2.7 x_1^2x_2^{-1/3}x_3^{0.7} + 2x_1^{-4}x_3^{2/5}</math>

is a posynomial.

Posynomials are not the same as [polynomial](/source/polynomial)s in several independent variables.  A polynomial's exponents must be non-negative integers, but its independent variables and coefficients can be arbitrary real numbers; on the other hand, a posynomial's exponents can be arbitrary real numbers, but its independent variables and coefficients must be positive real numbers.  This terminology was introduced by [Richard J. Duffin](/source/Richard_Duffin), Elmor L. Peterson, and [Clarence Zener](/source/Clarence_Zener) in their seminal book on [geometric programming](/source/geometric_programming).

Posynomials are a [special case](/source/special_case) of [signomial](/source/signomial)s, the latter not having the restriction that the <math>c_k</math> be positive.

==References==

*{{cite book
 | author     = Richard J. Duffin
 |author2=Elmor L. Peterson |author3=Clarence Zener
  | title      = Geometric Programming
 | publisher  = John Wiley and Sons
 | date       = 1967
 | pages      = 278
 | isbn       = 0-471-22370-0
}}
*{{cite book
 | author      = Stephen P Boyd
 |author2=Lieven Vandenberghe
 | title      = Convex optimization 
 | publisher  = Cambridge University Press
 | date       = 2004
 | isbn       = 0-521-83378-7 
 | url = https://web.stanford.edu/~boyd/cvxbook/
}}
*{{cite book
 | author      = Harvir Singh Kasana
 |author2=Krishna Dev Kumar
 | title      = Introductory Operations Research: Theory and Applications
 | url      = https://archive.org/details/springer_10.1007-978-3-662-08011-5
 | publisher  = Springer
 | date       = 2004
 | isbn       = 3-540-40138-5 
}}
* {{cite journal | last1 = Weinstock | first1 = D. |author2-link=Joseph Appelbaum | last2 = Appelbaum | first2 = J. | title = Optimal solar field design of stationary collectors | journal = Journal of Solar Energy Engineering | date = 2004 | volume = 126 | issue = 3| pages = 898–905 | doi = 10.1115/1.1756137 }}

==External links==
* S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, [https://web.archive.org/web/20070308160245/http://www.stanford.edu/~boyd/gp_tutorial.html A Tutorial on Geometric Programming]

Category:Functions and mappings

{{mathapplied-stub}}

---
Adapted from the Wikipedia article [Posynomial](https://en.wikipedia.org/wiki/Posynomial) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Posynomial?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.