Geometric programming

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

Minimize $\ f_{0}(x)\$ subject to

$f_{i}(x)\leq 1,\quad i=1,\dots ,m$

$h_{i}(x)=1,\quad i=1,\dots ,p$

where $f_{0},\dots ,f_{m}$ are posynomials and $h_{1},\dots ,h_{p}$ are monomials.

In the context of geometric programming (unlike all other disciplines), a monomial is a function $h:\mathbb {R} _{++}^{n}\to \mathbb {R}$ defined as

$h(x)=cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$

where $c>0\$ and $a_{i}\in \mathbb {R}$.

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, defining $y_{i}=\log(x_{i})$, the monomial $f(x)=cx_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\mapsto e^{a^{T}y+b}$, where $b=\log(c)$. Similarly, if $f$ is the posynomial

$f(x)=\sum _{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{nk}}$

then $f(x)=\sum _{k=1}^{K}e^{a_{k}^{T}y+b_{k}}$, where $a_{k}=(a_{1k},\dots ,a_{nk})$ and $b_{k}=\log(c_{k})$. After the change of variables, a posynomial becomes a sum of exponentials of affine functions.

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.