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A geometric program (GP) is an optimization problem of the form
- Minimize subject to
- where are posynomials and are monomials.
In the context of geometric programming (unlike all other disciplines), a monomial is defined as a function defined as
where and .
GPs have numerous application, such as components sizing in IC design and parameter estimation via logistic regression in statistics. The maximum likelihood estimator in logistic regression is a GP.
Geometric programs are not (in general) convex optimization problems, but they can be transformed to convex problems by a change of variables and a transformation of the objective and constraint functions. In particular, defining , the monomial , where . Similarly, if is the posynomial
then , where and . After the change of variables, a posynomial becomes a sum of exponentials of affine functions.
Several software packages and libraries exist to assist with formulating and solving geometric programs.
- MOSEK is a commercial solver capable of solving geometric programs as well as other non-linear optimization problems.
- CVXOPT is an open-source solver for convex optimization problems.
- GPkit is a Python package for cleanly defining and manipulating geometric programming models. There are a number of example GP models written with this package here.
- Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming. John Wiley and Sons. p. 278. ISBN 0-471-22370-0.
- S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A Tutorial on Geometric Programming
- S. Boyd, S. J. Kim, D. Patil, and M. Horowitz Digital Circuit Optimization via Geometric Programming