Geometric programming: Difference between revisions
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== Numerical Examples == | == Numerical Examples == | ||
=== | === Solve a Geometric Programming problem in standard form === | ||
'''Minimize''' <math>f_0(x,y) = x^2y^3 + 3x + 2xy </math> | '''Minimize''' <math>f_0(x,y) = x^2y^3 + 3x + 2xy </math> | ||
'''Subject to:''' <math> | '''Subject to:''' <math>x^5+2y^6+1\leqslant1 </math>, | ||
<math> | <math>x+2y\leqslant1 </math>, | ||
<math> | <math>xy = 1 </math>, | ||
<math>x > 0 </math>, | |||
<math>y > 0 </math> | |||
For the problem above, this is a geometric optimization problem in standard form. | |||
The solution is: ... | |||
=== Transform Nonconvex Optimization Problems to Convex Optimization problem === | |||
'''Reformulate the following non-convex MINLP to a convex one''' | |||
'''Minimize''' <math>f_0(x_1,x_2,x_3) = 16(x_1)^2(x_2)^2 + 3(x_2)^3(x_3)^4 - \left ( \frac{16}{x_2} \right ) </math> | |||
'''Subject to:''' <math>x^5+2y^6+1\leqslant1 </math>, | |||
<math>x+2y\leqslant1 </math>, | |||
<math>xy = 1 </math>, | |||
<math>x > 0 </math>, | |||
<math>y > 0 </math> | |||
Revision as of 16:04, 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
Subject to: , = 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
Solve a Geometric Programming problem in 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,y) = x^2y^3 + 3x + 2xy }
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 x^5+2y^6+1\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 x+2y\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 xy = 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 x > 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 y > 0 }
For the problem above, this is a geometric optimization problem in standard form.
The solution is: ...
Transform Nonconvex Optimization Problems to Convex Optimization problem
Reformulate the following non-convex MINLP to a convex one
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_1,x_2,x_3) = 16(x_1)^2(x_2)^2 + 3(x_2)^3(x_3)^4 - \left ( \frac{16}{x_2} \right ) }
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 x^5+2y^6+1\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 x+2y\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 xy = 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 x > 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 y > 0 }