Posynomial

 ( Functions And Mappings ) 

A posynomial, also known as a posinomial in some literature, is a function of the form

$f(x\_1,\; x\_2,\; \backslash dots,\; x\_n)\; =\; \backslash sum\_\{k=1\}^K\; c\_k\; x\_1^\{a\_\{1k\}\}\; \backslash cdots\; x\_n^\{a\_\{nk\}\}$

where all the coordinates $x\_i$ and coefficients $c\_k$ are positive , and the exponents $a\_\{ik\}$ are real numbers. Posynomials are closed under addition, multiplication, and nonnegative scaling.

For example,

$f(x\_1,\; x\_2,\; x\_3)\; =\; 2.7\; x\_1^2x\_2^\{-1/3\}x\_3^\{0.7\}\; +\; 2x\_1^\{-4\}x\_3^\{2/5\}$

is a posynomial.

Posynomials are not the same as 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 Duffin, Elmor L. Peterson, and Clarence Zener in their seminal book on geometric programming.

Posynomials are a special case of , the latter not having the restriction that the $c\_k$ be positive.

• Richard J. Duffin (1967). 9780471223702, John Wiley and Sons. ISBN 9780471223702
• Stephen P Boyd (2025). 9780521833783, Cambridge University Press. . ISBN 9780521833783
• Harvir Singh Kasana (2025). 9783540401384, Springer. . ISBN 9783540401384

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External links

• S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A Tutorial on Geometric Programming

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