Geometric programming

Geometric programming

A Geometric Program is an optimization problem of the form

minimize f_0(x) subject to

: f_i(x) leq 1, quad i = 1,dots,m

: h_i(x) = 1,quad i = 1,dots,p

where f_0,dots,f_m are posynomials and h_1,dots,h_p are monomials. It should be noted that in the context of geometric programming (unlike all other disciplines), a monomial is defined as a function f:mathbb{R}^n o mathbb{R} with mathrm{dom} f = mathbb{R}_{++}^n defined as

:f(x) = c x_1^{a_1} x_2^{a_2} cdots x_n^{a_n}

where c > 0 and a_i in mathbb{R} .

GPs have numerous application, such as circuit sizing and parameter estimation via logistic regression in statistics. The maximum likelihood estimator in logistic regression is a GP.

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, defining y_i = log{x_i}, the monomial f(x) = c x_1^{a_1} cdots x_n^{a_n} mapsto e^{a^T y +b}, where b = log{c}.Similarly, if f is the posynomial

f(x) = sum_{k=1}^K c_k x_1^{a_{1k cdots x_n^{a_{nk

then f(x) = sum_{k=1}^K e^{a_k^T y + b_k}, where a_k = (a_{1k},dots,a_{nk} ) and b_k = log{c_k} . After the change of variables, a posynomial becomes a sum of exponentials of affine functions.

External links

* S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, [http://www.stanford.edu/~boyd/gp_tutorial.html A Tutorial on Geometric Programming]

* S. Boyd, S. J. Kim, D. Patil, and M. Horowitz [http://www.stanford.edu/~boyd/gp_digital_ckt.html Digital Circuit Optimization via Geometric Programming]


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Geometric dimensioning and tolerancing — (GD T) is a system for defining and communicating engineering tolerances. It uses a symbolic language on engineering drawings and computer generated three dimensional solid models for explicitly describing nominal geometry and its allowable… …   Wikipedia

  • Geometric median — The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one… …   Wikipedia

  • Geometric Description Language — In computer aided design, Geometric Description Language (GDL) is the programming language of ArchiCAD library parts. GSM is the file format of these CAD objects. Area of usageThese objects are similar to blocks in AutoCAD, but unlike blocks,… …   Wikipedia

  • Linear programming — (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.… …   Wikipedia

  • List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • Mathematical optimization — For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… …   Wikipedia

  • Convex optimization — Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. Given a real vector space X together with a convex, real valued function defined on a convex subset of X, the problem is to find …   Wikipedia

  • Clarence Zener — Clarence Melvin Zener (December 1, 1905 – July 15, 1993) was the American physicist who first described the property concerning the breakdown of electrical insulators.[1] These findings were later exploited by Bell Labs in the development of the… …   Wikipedia

  • TOMLAB — Infobox Software name = TOMLAB reg; programming language = MATLAB (FORTRAN, C) developer = Tomlab Optimization Inc. latest release version = 6.1 latest release date = 11 June 2008 size = 23 MB (Windows) operating system = Windows 32/64 bit, Linux …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”