A geometric program (GP) is an optimization problem of the form
where are posynomials and are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from to defined as
where and . 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 and taking the log of the objective and constraint functions, the functions , i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions , 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.
A geometric program (GP) is an optimization problem of the form
where are posynomials and are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from to defined as
where and . 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 and taking the log of the objective and constraint functions, the functions , i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions , 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.