User:Rileyjmurray/sandbox

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.