Geometric Programming

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Monday, 10 February 2025

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Geometric programming

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Geometric programming

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

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

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

GPs have numerous application, such as components sizing in IC design 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.