Geometric programming

Authors: Wenjun Zhu(wz274), Sam Olsen(sgo23) (SYSEN5800, FALL 2021)

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: $f_{i}(x)\leqslant 1$, $i$ = 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 $f_{0}(x,y)=x^{2}y^{3}+3x+2xy$

Subject to: $x^{5}+2y^{6}+1\leqslant 1$,

$x+2y\leqslant 1$,

$xy=1$,

$x>0$,

$y>0$

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 $f_{0}(x_{1},x_{2},x_{3})=16(x_{1})^{2}(x_{2})^{2}+3(x_{2})^{3}(x_{3})^{4}-\left({\frac {16}{x_{2}}}\right)$

Subject to: $x^{5}+2y^{6}+1\leqslant 1$,

$x+2y\leqslant 1$,

$xy=1$,

$x>0$,

$y>0$