Geometric Programming 101

What is a GP?

A Geometric Program (GP) is a special type of constrained non-linear optimization problem.

A GP is made up of special types of functions called monomials and posynomials. In the context of a 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 the decision variables, and \(a_{1..n}\) are real exponents.

Building on this, a posynomial is defined as a sum of monomials:

\[g(x) = \sum_{k=1}^K c_k x_1^{a_1k} x_2^{a_2k} ... x_n^{a_nk}\]

Using these definitions, a GP in Standard Form is written as:

\[\begin{split}\begin{array}[lll]\text{} \text{minimize} & f_0(x) & \\ \text{subject to} & f_i(x) = 1, & i = 1,....,m \\ & g_i(x) \leq 1, & i = 1,....,n \end{array}\end{split}\]

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.

Where can I learn more?

To learn more about GPs, refer to the following resources: