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 ${\displaystyle h:\mathbb {R} _{++}^{n}\to \mathbb {R} }$ defined as

${\displaystyle h(x)=cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}}$

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 $f(x)=cx_{1}^{{a_{1}}}\cdots x_{n}^{{a_{n}}}\mapsto e^{{a^{T}y+b}}$, 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.

See also

• Signomial

Footnotes

1. http://www.stanford.edu/~boyd/papers/opamp.html

References

• Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming. John Wiley and Sons. p. 278. ISBN 0-471-22370-0.

External links

• S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A Tutorial on Geometric Programming
• S. Boyd, S. J. Kim, D. Patil, and M. Horowitz Digital Circuit Optimization via Geometric Programming