Geometric programming

A geometric program (GP) is an optimization problem of the form

${\begin{array}{ll}{\mbox{minimize}}&f_{0}(x)\\{\mbox{subject to}}&f_{i}(x)\leq 1,\quad i=1,\ldots ,m\\&g_{i}(x)=1,\quad i=1,\ldots ,p,\end{array}}$

where $f_{0},\dots ,f_{m}$ are posynomials and $g_{1},\dots ,g_{p}$ are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from $\mathbb {R} _{++}^{n}$ to $\mathbb {R}$ defined as

$x\mapsto cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$

where $c>0\$ and $a_{i}\in \mathbb {R}$. A posynomial is any sum of monomials.

GPs have numerous applications, such as component sizing in IC design and parameter estimation via logistic regression in statistics. The maximum likelihood estimator in logistic regression is a GP.

Convex form

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, after performing the change of variables $y_{i}=\log(x_{i})$ and taking the log of the objective and constraint functions, the functions $f_{i}$, i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions $g_{i}$, i.e., the monomials, become affine. Hence, this transformation transforms every GP into an equivalent convex program.

Software

Several software packages 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.
• GGPLAB is a MATLAB toolbox for specifying and solving geometric programs (GPs) and generalized geometric programs (GGPs).
• CVXPY is a Python-embedded modeling language for specifying and solving convex optimization problems, including GPs, GGPs, and log-log convex programs.

See also

• Signomial
• Clarence Zener

Further Reading

• Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming. John Wiley and Sons. p. 278. ISBN 0-471-22370-0.