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A signomial is type of a mathematical function that generalizes polynomials to allow for arbitrary real exponents. There are two conventions for defining signomials that are equivalent to one another under a change of variables. Both of the two conventions parameterize a given signomial by an exponent matrix $A\in \mathbb {R} ^{m\times n}$ and a coefficient vector $c\in \mathbb {R} ^{m}$. The more common convention from a mathematical modeling perspective is to say that a signomial takes values

$f(x_{1},x_{2},\dots ,x_{n})=\sum _{i=1}^{m}c_{i}\prod _{j=1}^{n}x_{j}^{a_{ij}}$.

In this case the domain of a signomial is restricted to the set of elementwise positive vectors. The restriction to nonnegative inputs avoids ambiguities involving such as the value of ${\sqrt[{n}]{-1}}$ (the complex roots of unity), and further restriction to positive inputs is necessary to avoid division by zero. In pure mathematics and mathematical optimization it is common to define a signomial as

$g(y_{1},y_{2},\dots ,y_{n})=\sum _{i=1}^{m}c_{i}\exp \left(\sum _{j=1}^{n}a_{ij}y_{j}\right)$.

Using this exponential convention, the domain of a signomial is all of $n$-dimensional real space. These two conventions are equivalent to one another under the nonlinear change of variables $x_{i}=\exp y_{i}$.

History

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 problems.

Properties

Signomials are closed under addition, subtraction, multiplication, and scaling.

If we restrict all $c_{i}$ to be positive, then the function f is a posynomial. Consequently, each signomial is either a posynomial, the negative of a posynomial, or the difference of two posynomials. If, in addition, all exponents $a_{ij}$ are non-negative integers, then the signomial becomes a polynomial whose domain is the positive orthant.

Applications

Nonlinear optimization problems with constraints and/or objectives defined by signomials are harder to solve than those defined by only posynomials, because (unlike posynomials) signomials cannot necessarily be made convex 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.

External links

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

Category:Functions and mappings Category:Mathematical optimization

Advances in Geometric Programming, doi:10.1007/978-1-4615-8285-4