Geometric programming

A Geometric Program is an optimization problem of the form

minimize $f_{0}(x)$ subject to

$f_{i}(x)\leq 1$, $i\in \left\{1,\dots ,m\right\}$

$h_{i}(x)=1$, $i\in \left\{1,\dots ,p\right\}.$

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

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

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

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.