A Geometric Program (GP) is a type of non-linear optimization problem whose objective and constraints have a particular form.
The decision variables must be strictly positive (non-zero, non-negative) quantities. This is a good fit for engineering design equations (which are often constructed to have only positive quantities), but any model with variables of unknown sign (such as forces and velocities without a predefined direction) may be difficult to express in a GP. Such models might be better expressed as :ref:`Signomials <signomials>`.
More precisely, GP objectives and inequalities are formed out of monomials and posynomials. In the context of GP, a monomial is defined as:
f(x) = c x_1^{a_1} x_2^{a_2} ... x_n^{a_n}
where c is a positive constant, x_{1..n} are decision variables, and a_{1..n} are real exponents. For example, taking x, y and z to be positive variables, the expressions
7x \qquad 4xy^2z \qquad \frac{2x}{y^2z^{0.3}} \qquad \sqrt{2xy}
are all monomials. Building on this, a posynomial is defined as a sum of monomials:
g(x) = \sum_{k=1}^K c_k x_1^{a_1k} x_2^{a_2k} ... x_n^{a_nk}
For example, the expressions
x^2 + 2xy + 1 \qquad 7xy + 0.4(yz)^{-1/3} \qquad 0.56 + \frac{x^{0.7}}{yz}
are all posynomials. Alternatively, monomials can be defined as the subset of posynomials having only one term. Using f_i to represent a monomial and g_i to represent a posynomial, a GP in standard form is written as:
\begin{array}{lll}\text{}
\text{minimize} & g_0(x) & \\
\text{subject to} & f_i(x) = 1, & i = 1,....,m \\
& g_i(x) \leq 1, & i = 1,....,n
\end{array}
Boyd et. al. give the following example of a GP in standard form:
\begin{array}{llll}\text{}
\text{minimize} & x^{-1}y^{-1/2}z^{-1} + 2.3xz + 4xyz \\
\text{subject to} & (1/3)x^{-2}y^{-2} + (4/3)y^{1/2}z^{-1} \leq 1 \\
& x + 2y + 3z \leq 1 \\
& (1/2)xy = 1
\end{array}
Geometric programs have several powerful properties:
- Unlike most non-linear optimization problems, large GPs can be solved extremely quickly.
- If there exists an optimal solution to a GP, it is guaranteed to be globally optimal.
- Modern GP solvers require no initial guesses or tuning of solver parameters.
These properties arise because GPs become convex optimization problems via a logarithmic transformation. In addition to their mathematical benefits, recent research has shown that many practical problems can be formulated as GPs or closely approximated as GPs.
When the coefficients in a posynomial are allowed to be negative (but the variables stay strictly positive), that is called a Signomial.
A Signomial Program has signomial constraints. While they cannot be solved as quickly or to global optima, because they build on the structure of a GP they can often be solved more quickly than a generic nonlinear program. More information can be found under :ref:`signomialprogramming`.
To learn more about GPs, refer to the following resources:
- A tutorial on geometric programming, by S. Boyd, S.J. Kim, L. Vandenberghe, and A. Hassibi.
- Convex optimization, by S. Boyd and L. Vandenberghe.
- Geometric Programming for Aircraft Design Optimization, Hoburg, Abbeel 2014