# Geometric program

### From Glossary

A mathematical program of the form

where each is a posynomial (k=0,...,m). Conventionally, the monomials are indexed sequentially, so that the posynomials appear as:

For example, consider:

Then, (variables), (constraints), (monomials in objective), ( monomial in constraint 1), and ( monomials in constraint 2).

The *exponent matrix* is the matrix whose i-th row contains
the exponents of the variables in the i-th monomial. The example has 5 rows and 3 columns:

The *degree of difficulty* is the number of terms in all posynomials
() minus the number of independent variables (one less than
the rank of exponent matrix). In the above example, the rank of is 3, so the degree of
difficulty is 5-3-1 = 1. If the last constraint were not present, only the first three
rows comprise the exponent matrix, and its rank is 2. In that case, the degree of difficulty is 0.
(Using the <a href="second.php?page=duals.html#Geometric">geometric dual</a>, the solution reduces
to solving a system of linear equations, which is what motivates the terminology.)

Also see superconsistent.

Since its inception, the posynomial form has been extended to
signomials. In that case, the duality need not
be strong, as there can be a duality gap. The
*general*, or *signomial*, geometric program is of the form:

for , where and are posynomials ().