Posynomial

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

f ( x 1 , x 2 , … , x n ) = ∑ k = 1 K c k x 1 a 1 k ⋯ x n a n k {\displaystyle f(x_{1},x_{2},\dots ,x_{n})=\sum _{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{nk}}}

where all the coordinates x i {\displaystyle x_{i}} and coefficients c k {\displaystyle c_{k}} are positive real numbers, and the exponents a i k {\displaystyle 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 2 x 2 − 1 / 3 x 3 0.7 + 2 x 1 − 4 x 3 2 / 5 {\displaystyle f(x_{1},x_{2},x_{3})=2.7x_{1}^{2}x_{2}^{-1/3}x_{3}^{0.7}+2x_{1}^{-4}x_{3}^{2/5}}

is a posynomial.

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

Posynomials are a special case of signomials, the latter not having the restriction that the c k {\displaystyle c_{k}} be positive.

• Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming. John Wiley and Sons. p. 278. ISBN 0-471-22370-0.
• Stephen P Boyd; Lieven Vandenberghe (2004). Convex optimization. Cambridge University Press. ISBN 0-521-83378-7.
• Harvir Singh Kasana; Krishna Dev Kumar (2004). Introductory Operations Research: Theory and Applications. Springer. ISBN 3-540-40138-5.
• Weinstock, D.; Appelbaum, J. (2004). "Optimal solar field design of stationary collectors". Journal of Solar Energy Engineering. 126 (3): 898–905. doi:10.1115/1.1756137.

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