# Signomial

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A '''signomial''' is an algebraic [function](/source/function_(mathematics)) of one or more independent variables.  It is perhaps most easily thought of as an algebraic extension of multivariable [polynomial](/source/polynomial)s—an extension that permits exponents to be arbitrary real numbers (rather than just non-negative integers) while requiring the independent variables to be strictly positive (so that division by zero and other inappropriate algebraic operations are not encountered).

Formally, a signomial is a function with domain <math>\mathbb{R}_{>0}^n</math> which takes values

: <math>f(x_1, x_2, \dots, x_n) = \sum_{i=1}^M \left(c_i \prod_{j=1}^n x_j^{a_{ij}}\right)</math>

where the coefficients <math>c_i</math> and the exponents <math>a_{ij}</math> are real numbers.  Signomials are [closed](/source/Closure_(mathematics)) under addition, subtraction, multiplication, and scaling.

If we restrict all <math>c_i</math> to be positive, then the function f is a [posynomial](/source/posynomial). Consequently, each signomial is either a posynomial, the negative of a posynomial, or the difference of two posynomials.  If, in addition, all exponents <math>a_{ij}</math> are  non-negative integers, then the signomial becomes a [polynomial](/source/polynomial) whose domain is the positive [orthant](/source/orthant).

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 signomial. 

The term "signomial" was introduced by Richard J. Duffin and Elmor L. Peterson in their seminal joint work on general algebraic optimization—published in the late 1960s and early 1970s.  A recent introductory exposition involves [optimization problem](/source/optimization_problem)s.<ref>C. Maranas and C. Floudas, ''Global optimization in generalized geometric programming'', pp. 351–370, 1997.</ref>  [Nonlinear optimization](/source/Nonlinear_optimization) problems with [constraints](/source/constrained_optimization) and/or [objectives](/source/objective_function) defined by signomials are harder to solve than those defined by only posynomials, because (unlike posynomials) signomials cannot necessarily be made [convex](/source/convex_function) by applying a logarithmic change of variables. Nevertheless, signomial optimization problems often provide a much more accurate mathematical representation of real-world nonlinear optimization problems.

== See also ==

* [Posynomial](/source/Posynomial)
* [Geometric programming](/source/Geometric_programming)

==References==
{{reflist}}

==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
Category:Mathematical optimization

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