A geometric program (GP ) is an optimization problem of the form
minimize
f
0
(
x
)
subject to
f
i
(
x
)
≤
1
,
i
=
1
,
…
,
m
g
i
(
x
)
=
1
,
i
=
1
,
…
,
p
,
{\displaystyle {\begin{array}{ll}{\mbox{minimize}}&f_{0}(x)\\{\mbox{subject to}}&f_{i}(x)\leq 1,\quad i=1,\ldots ,m\\&g_{i}(x)=1,\quad i=1,\ldots ,p,\end{array}}}
where
f
0
,
…
,
f
m
{\displaystyle f_{0},\dots ,f_{m}}
are posynomials and
g
1
,
…
,
g
p
{\displaystyle g_{1},\dots ,g_{p}}
are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from
R
+
+
n
{\displaystyle \mathbb {R} _{++}^{n}}
to
R
{\displaystyle \mathbb {R} }
defined as
x
↦
c
x
1
a
1
x
2
a
2
⋯
x
n
a
n
{\displaystyle x\mapsto cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}}
where
c
>
0
{\displaystyle c>0\ }
and
a
i
∈
R
{\displaystyle a_{i}\in \mathbb {R} }
. A posynomial is any sum of monomials.[ 1] [ 2]
Geometric programming is
closely related to convex optimization : any GP can be made convex by means of a change of variables.[ 2] GPs have numerous applications, including component sizing in IC design,[ 3] [ 4] aircraft design,[ 5] maximum likelihood estimation for logistic regression in statistics , and parameter tuning of positive linear systems in control theory .[ 6]
^ Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming . John Wiley and Sons. p. 278. ISBN 0-471-22370-0 .
^ a b S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. A Tutorial on Geometric Programming . Retrieved 20 October 2019.
^ M. Hershenson, S. Boyd, and T. Lee. Optimal Design of a CMOS Op-amp via Geometric Programming . Retrieved 8 January 2019.
^ S. Boyd, S. J. Kim, D. Patil, and M. Horowitz. Digital Circuit Optimization via Geometric Programming . Retrieved 20 October 2019.
^ W. Hoburg and P. Abbeel. Geometric programming for aircraft design optimization . AIAA Journal 52.11 (2014): 2414-2426.
^ Ogura, Masaki; Kishida, Masako; Lam, James (2020). "Geometric Programming for Optimal Positive Linear Systems" . IEEE Transactions on Automatic Control . 65 (11): 4648–4663. arXiv :1904.12976 . doi :10.1109/TAC.2019.2960697 . ISSN 0018-9286 . S2CID 140222942 .