Geometric programming: Difference between revisions

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A '''geometric program''' ('''GP''') is an [[optimization (mathematics)|optimization]] problem of the form
:<math>
: Minimize <math>\ f_0(x)\ </math> subject to
\begin{array}{ll}
:: <math>f_i(x) \leq 1, \quad i = 1,\dots,m</math>
\mbox{minimize} & f_0(x) \\
:: <math>h_i(x) = 1,\quad i = 1,\dots,p</math>
::\mbox{subject to} & <math>f_i(x) \leq 1, \quad i = 1, \dotsldots, m</math>\\
:where <math>f_0,\dots,f_m</math> are [[posynomials]] and <math>h_1,\dots,h_p</math> are [[monomials]]. <ref>S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. ''[http://www.stanford.edu/~boyd/gp_tutorial.html A Tutorial on Geometric Programming].'' Retrieved 8 January 2019.</ref>
::& <math>h_ig_i(x) = 1, \quad i = 1, \dotsldots, p</math>,
\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 allstandard other disciplinesmathematics), a monomial is a function from <math>h:\mathbb{R}_{++}^n</math> \to <math>\mathbb{R}</math> defined as
 
:<math>h(x) =\mapsto c x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n} </math>
In the context of geometric programming (unlike all other disciplines), a monomial is a function <math>h:\mathbb{R}_{++}^n \to \mathbb{R}</math> defined as
 
:where <math>f_0, c > 0 \dots,f_m </math> are [[posynomials]] and <math>h_1,a_i \dots,h_pin \mathbb{R} </math>. areA posynomial is any sum of [[monomials]]. <ref>S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. ''[http://www.stanford.edu/~boyd/gp_tutorial.html A Tutorial on Geometric Programming].'' Retrieved 8 January 2019.</ref>
:<math>h(x) = 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>.
 
GPs have numerous applications, such as component sizing in [[Integrated circuit|IC]] design<ref>M. Hershenson, S. Boyd, and T. Lee. ''[http://www.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. ''[http://www.stanford.edu/~boyd/gp_digital_ckt.html Digital Circuit Optimization via Geometric Programming].'' Retrieved 8 January 2019.</ref> and parameter estimation via [[logistic regression]] in [[statistics]]. The [[maximum likelihood]] estimator in logistic regression is a GP.