Geometric programming: Difference between revisions
No edit summary |
No edit summary |
||
| Line 8: | Line 8: | ||
== Theory/Methodology == | == Theory/Methodology == | ||
=== Definition === | |||
The standard form of Geometric Programming optimization is to minimize the objective function which must be posynomial. The inequality constraints can only have the form of a posynomial less than or equal to one, and the equality constraints can only have the form of a monomial equal to one. | |||
==== Standard Form ==== | |||
'''Minimize''' <math>f_0(x)</math> | |||
'''Subject to:''' <math>f_i(x)\leqslant1 </math>, <math>i </math> = 1,...,m, | |||
<math>g_i(x) = 1 </math>, <math>i </math> = 1,...,p, | |||
<math>x_i > 0 </math>, <math>i </math> = 1,...,q, | |||
where <math>f_i(x)</math> are posynomial functions, <math>g_i(x)</math> are monomials, and <math>x_i </math> are the optimization variables. | |||
| Line 14: | Line 27: | ||
== Numerical Examples == | == Numerical Examples == | ||
=== To solve a standard form Geometric Programming problem === | |||
'''Minimize''' <math>f_0( </math> | |||
'''Subject to:''' <math>f_i(x)\leqslant1 </math>, <math>i </math> = 1,...,m, | |||
<math>g_i(x) = 1 </math>, <math>i </math> = 1,...,p, | |||
<math>x_i > 0 </math>, <math>i </math> = 1,...,q, | |||
Revision as of 16:42, 13 November 2021
Authors: Wenjun Zhu(wz274), Sam Olsen(sgo23) (SYSEN5800, FALL 2021)
Introduction
Theory/Methodology
Definition
The standard form of Geometric Programming optimization is to minimize the objective function which must be posynomial. The inequality constraints can only have the form of a posynomial less than or equal to one, and the equality constraints can only have the form of a monomial equal to one.
Standard Form
Minimize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_0(x)}
Subject to: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i(x)\leqslant1 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } = 1,...,m,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i(x) = 1 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } = 1,...,p,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i > 0 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } = 1,...,q,
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i(x)} are posynomial functions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i(x)} are monomials, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i } are the optimization variables.
Numerical Examples
To solve a standard form Geometric Programming problem
Minimize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_0( }
Subject to: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_i(x)\leqslant1 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } = 1,...,m,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i(x) = 1 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } = 1,...,p,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i > 0 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } = 1,...,q,