Geometric programming

A Geometric Program is an optimization problem of the form

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

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

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

where ${\displaystyle f_{0},\dots ,f_{m}}$ are posynomials and ${\displaystyle h_{1},\dots ,h_{p}}$ 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 ${\displaystyle y_{i}=\log {x_{i}}}$ , the monomial ${\displaystyle f(x)=cx_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\mapsto e^{a^{T}y+b}}$ , where ${\displaystyle b=log{c}}$ . Similarly, if ${\displaystyle f}$  is the posynomial

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

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