Geometric programming
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 $ f_0(x) $
Subject to: $ f_i(x)\leqslant1 $, $ i $ = 1,...,m,
$ g_i(x) = 1 $, $ i $ = 1,...,p,
$ x_i > 0 $, $ i $ = 1,...,q,
where $ f_i(x) $ are posynomial functions, $ g_i(x) $ are monomials, and $ x_i $ are the optimization variables.
Numerical Examples
Solve a Geometric Programming problem in standard form
Minimize $ f_0(x,y) = x^2y^3 + 3x + 2xy $
Subject to: $ x^5+2y^6+1\leqslant1 $,
$ x+2y\leqslant1 $,
$ xy = 1 $,
$ x > 0 $,
$ 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 $ f_0(x_1,x_2,x_3) = 16(x_1)^2(x_2)^2 + 3(x_2)^3(x_3)^4 $
Subject to: $ x^5+2y^6+1\leqslant1 $,
$ x+2y\leqslant1 $,
$ xy = 1 $,
$ x > 0 $,
$ y > 0 $