{{Short description|Optimization problem}} A '''geometric program''' ('''GP''') is an optimization problem of the form :<math> \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} </math> where <math>f_0,\dots,f_m</math> are posynomials and <math>g_1,\dots,g_p</math> are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from <math>\mathbb{R}_{++}^n</math> to <math>\mathbb{R}</math> defined as
:<math>x \mapsto c x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n} </math>
where <math> c > 0 \ </math> and <math>a_i \in \mathbb{R} </math>. A posynomial is any sum of monomials.<ref name="duffin">{{cite book | author = Richard J. Duffin |author2=Elmor L. Peterson |author3=Clarence Zener | title = Geometric Programming | publisher = John Wiley and Sons | year = 1967 | pages = 278 | isbn = 0-471-22370-0 }}</ref><ref name="tutorial">S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. ''[https://web.stanford.edu/~boyd/papers/gp_tutorial.html A Tutorial on Geometric Programming].'' Retrieved 20 October 2019.</ref>
Geometric programming is closely related to convex optimization: any GP can be made convex by means of a change of variables.<ref name="tutorial"/> GPs have numerous applications, including component sizing in IC design,<ref>M. Hershenson, S. Boyd, and T. Lee. ''[https://web.stanford.edu/~boyd/papers/opamp.html Optimal Design of a CMOS Op-amp via Geometric Programming].'' Retrieved 8 January 2019.</ref><ref>S. Boyd, S. J. Kim, D. Patil, and M. Horowitz. ''[https://web.stanford.edu/~boyd/papers/gp_digital_ckt.html Digital Circuit Optimization via Geometric Programming].'' Retrieved 20 October 2019.</ref> aircraft design,<ref>W. Hoburg and P. Abbeel. ''[https://people.eecs.berkeley.edu/~pabbeel/papers/2014-AIAA-GP-aircraft-design.pdf Geometric programming for aircraft design optimization].'' AIAA Journal 52.11 (2014): 2414-2426.</ref> maximum likelihood estimation for logistic regression in statistics, and parameter tuning of positive linear systems in control theory.<ref>{{Cite journal|last1=Ogura|first1=Masaki|last2=Kishida|first2=Masako|last3=Lam|first3=James|date=2020|title=Geometric Programming for Optimal Positive Linear Systems|journal=IEEE Transactions on Automatic Control|volume=65|issue=11|pages=4648–4663|doi=10.1109/TAC.2019.2960697|issn=0018-9286|arxiv=1904.12976|bibcode=2020ITAC...65.4648O |s2cid=140222942 }}</ref>
==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, after performing the change of variables <math>y_i = \log(x_i)</math> and taking the log of the objective and constraint functions, the functions <math>f_i</math>, i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions <math>g_i</math>, i.e., the monomials, become affine. Hence, this transformation transforms every GP into an equivalent convex program.<ref name="tutorial"/> 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.<ref name="dgp">A. Agrawal, S. Diamond, and S. Boyd. ''[https://arxiv.org/abs/1812.04074 Disciplined Geometric Programming.]'' Retrieved 8 January 2019.</ref>
==Software== Several software packages exist to assist with formulating and solving geometric programs. * [https://www.mosek.com/ MOSEK] is a commercial solver capable of solving geometric programs as well as other non-linear optimization problems. * [http://cvxopt.org/ CVXOPT] is an open-source solver for convex optimization problems. * [https://github.com/convexengineering/gpkit 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 [https://github.com/convexengineering/gplibrary here]. *[https://web.stanford.edu/~boyd/ggplab/ GGPLAB] is a MATLAB toolbox for specifying and solving geometric programs (GPs) and generalized geometric programs (GGPs). * [https://www.cvxpy.org/tutorial/dgp/index.html CVXPY] is a Python-embedded modeling language for specifying and solving convex optimization problems, including GPs, GGPs, and LLCPs.<ref name="dgp"/>
==See also== *Signomial *Clarence Zener
==References== {{reflist}}
Category:Convex optimization