Signomial

In mathematics, the signomial is a function of one more variables. It is perhaps most easily thought of as an extension of multi-dimensional polynomials to include non-integer powers.

More formally, let $X$ be a vector of real, positive numbers.

:$X\; =\; (x\_1,\; x\_2,\; x\_3,\; dots,\; x\_n)^T$

Then a signomial function has the form

: $f(x\_1,\; x\_2,\; dots,\; x\_n)\; =\; sum\_\{i=1\}^M\; left(c\_i\; prod\_\{j=1\}^N\; x\_j^\{a\_\{ij\; ight)$

where the coefficients $c\_k$ and the exponents $a\_\{ij\}$ 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. If in addition $m\; =\; 1$, then thefunction f is a monomial. If all exponents $a\_\{ij\}$ are integer and positive, then thesignomial becomes a polynomial.

For example,

: $f(x\_1,\; x\_2,\; x\_3)\; =\; 2.7\; x\_1^2x\_2^\{-1/3\}x\_3^\{0.7\}\; -\; 2x\_1^\{-4\}x\_3^\{2/5\}$

is a signomial.

Signomials are typically used in optimization problems [C. Maranas and C. Floudas, "Global optimization in generalized geometric programming", pp. 351–370, 1997.] where they are used to represent values to be optimized and constraints on the variables. Although signomial constraints and objectives are harder to solve than those using posynomials (unlike posynomials they are not guaranteed to be globally convex), they often allow a better match to real-world objectives and constraints.

References

External links

* S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, [http://www.stanford.edu/~boyd/gp_tutorial.html A Tutorial on Geometric Programming]