Geometric programming

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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}}} {\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}} {\displaystyle f_{0},\dots ,f_{m}} are posynomials and g 1 , … , g p {\displaystyle g_{1},\dots ,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}} {\displaystyle \mathbb {R} _{++}^{n}} to R {\displaystyle \mathbb {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}}} {\displaystyle x\mapsto cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}}

where c > 0   {\displaystyle c>0\ } {\displaystyle c>0\ } and a i ∈ R {\displaystyle a_{i}\in \mathbb {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]

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 y i = log ⁡ ( x i ) {\displaystyle 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}} {\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}} {\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]

Software

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]

See also

References

  1. Richard J. Duffin; Elmor L. Peterson; Clarence Zener (1967). Geometric Programming. John Wiley and Sons. p. 278. ISBN 0-471-22370-0.
  2. S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. A Tutorial on Geometric Programming. Retrieved 20 October 2019.
  3. M. Hershenson, S. Boyd, and T. Lee. Optimal Design of a CMOS Op-amp via Geometric Programming. Retrieved 8 January 2019.
  4. S. Boyd, S. J. Kim, D. Patil, and M. Horowitz. Digital Circuit Optimization via Geometric Programming. Retrieved 20 October 2019.
  5. W. Hoburg and P. Abbeel. Geometric programming for aircraft design optimization. AIAA Journal 52.11 (2014): 2414-2426.
  6. 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. Bibcode:2020ITAC...65.4648O. doi:10.1109/TAC.2019.2960697. ISSN 0018-9286. S2CID 140222942.
  7. A. Agrawal, S. Diamond, and S. Boyd. Disciplined Geometric Programming. Retrieved 8 January 2019.