geometric programming — Jemoka Knowledge Base

A geometric program:

\begin{align} \min_{x}\quad & f_{0}\left(x\right) \\ \textrm{s.t.} \quad & f_{i}\left(x\right) \leq 1, i= 1\dots m \\ & h_{i}\left(x\right) = 1, i = 1 \dots p \end{align}

where f_{i} is posynomial, and h_{i} monomial. Notice that taking a log of this thing transforms the monomial into an affine function in \log \left(x\right), and into a logsumexp for posynomials. This also implies that solving for optimal \log \left(x\right), which is same as solving for x for positive x, is convex problem. monomial function \begin{equation} f\left(x\right) = cx_{1}^{a_{1}} x_{2}^{a_{2}} \dots x_{n}^{a_{n}} \end{equation} with c > 0, exponent a_{i} is any real. This term is not used in this way when not in the context of geometric programming. \text{dom}\left(f\right) = \mathbb{R}_{++}^{n}. posynomial function \begin{equation} f\left(x\right) = \sum_{k=1}^{k} c_{k}x_{1}^{a_{1k}} x_{2}^{a_{2k}} \dots x_{n}^{a_{nk}} \end{equation} \text{dom}\left(f\right) = \mathbb{R}_{++}^{n}.