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 defined as a function $f:\mathbb{R}_{++}^n \to \mathbb{R}$ defined as

$f(x) = c x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n}$

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

GPs have numerous application, such as components sizing in IC design[1] 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 $f(x) = c x_1^{a_1} \cdots x_n^{a_n} \mapsto e^{a^T y +b}$, 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.