Muller automaton

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

Minimize $f_0(x)$ subject to

$f_i(x)\le 1,i=1,\dots ,m$

$h_i(x)=1,i=1,\dots ,p$

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 $f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ with $\mathrm{d}\mathrm{o}\mathrm{m}f={\mathbb{ℝ}}_{++}^n$ defined as

$f(x)=c{x}_1^{a_1}{x}_2^{a_2}\cdots x_n^{a_n}$

where $c>0$ and $a_i\in \mathbb{ℝ}$.

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

See also

• Signomial

Footnotes

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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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