A signomial is an algebraic

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

of one or more independent variables. It is perhaps most easily thought of as an algebraic extension of multivariable

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

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

\mathbb_^n

which takes values :

f(x_1, x_2, \dots, x_n) = \sum_^M \left(c_i \prod_^n x_j^\right)

where the coefficients

c_i

and the exponents

a_

are real numbers. 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_

are non-negative integers, then the signomial becomes a

whose domain is the positive

orthant In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutu ...

. For example, :

f(x_1, x_2, x_3) = 2.7 x_1^2x_2^x_3^ - 2x_1^x_3^

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 In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...

s.C. Maranas and C. Floudas, ''Global optimization in generalized geometric programming'', pp. 351–370, 1997. Nonlinear optimization problems with

constraints Constraint may refer to: * Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies * Constraint (mathematics), a condition of an optimization problem that the solution m ...

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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

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.

References

{{reflist

External links

* S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi
A Tutorial on Geometric Programming
Functions and mappings Mathematical optimization

See also

References

External links