Geometric programming

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]

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, after performing the change of variables y i = log ⁡ ( x i ) {\displaystyle y_{i}=\log(x_{i})} and taking the log of the objective and constraint functions, the functions f i {\displaystyle f_{i}} , i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions g i {\displaystyle g_{i}} , i.e., the monomials, become affine. Hence, this transformation transforms every GP into an equivalent convex program.^[2] In fact, this log-log transformation can be used to convert a larger class of problems, known as log-log convex programming (LLCP), into an equivalent convex form.^[7]

Several software packages exist to assist with formulating and solving geometric programs.

• MOSEK is a commercial solver capable of solving geometric programs as well as other non-linear optimization problems.
• CVXOPT is an open-source solver for convex optimization problems.
• GPkit is a Python package for cleanly defining and manipulating geometric programming models. There are a number of example GP models written with this package here.
• GGPLAB is a MATLAB toolbox for specifying and solving geometric programs (GPs) and generalized geometric programs (GGPs).
• CVXPY is a Python-embedded modeling language for specifying and solving convex optimization problems, including GPs, GGPs, and LLCPs.^[7]

• Signomial
• Clarence Zener