gp101.rst

Geometric Programming 101

What is a GP?

A Geometric Program (GP) is a type of non-linear optimization problem whose objective and constraints have a particular form. The decision variables in a GP must have strictly positive values (that is, they can't be zero).

GP objectives and inequalities are formed out of monomials and posynomials. In the context of GP, a monomial is defined as:

f(x) = cx₁^a₁x₂^a₂...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:

$$\begin{aligned} \begin{array}{lll}\text{} \text{minimize} & g_0(x) & \\\ \text{subject to} & f_i(x) = 1, & i = 1,....,m \\\ & g_i(x) \leq 1, & i = 1,....,n \end{array} \end{aligned}$$

Boyd et. al. give the following example of a GP in standard form:

$$\begin{aligned} \begin{array}{llll}\text{} \text{minimize} & x^{-1}y^{-1/2}z^{-1} + 2.3xz + 4xyz \\\ \text{subject to} & (1/3)x^{-2}y^{-2} + (4/3)y^{1/2}z^{-1} \leq 1 \\\ & x + 2y + 3z \leq 1 \\\ & (1/2)xy = 1 \end{array} \end{aligned}$$

Why are GPs special?

Geometric programs have several powerful properties:

1. Unlike most non-linear optimization problems, large GPs can be solved extremely quickly.
2. If there exists an optimal solution to a GP, it is guaranteed to be globally optimal.
3. 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.

What are Signomials / Signomial Programs?

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, and while they cannot be solved as quickly or to global optima, because they build on the structure of a GP they can be solved more quickly than a generic nonlinear program. More information on Signomial Programs can be found under Advanced Commands.

Where can I learn more?

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