Geometric programming

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

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

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

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

where ${\displaystyle c>0\ }$ and ${\displaystyle a_{i}\in \mathbb {R} }$. A posynomial is any sum of monomials.^[1][2]

Geometric programming is closely related to convex optimization: any GP can be made convex by means of a change of variables.^[2] GPs have numerous applications, including component sizing in IC design,^[3][4] aircraft design,^[5] maximum likelihood estimation for logistic regression in statistics, and parameter tuning of positive linear systems in control theory.^[6]