# Optimization with polynomial objective and constraints

Let us be given the following optimization problem.

\begin{aligned} & \text{minimize} && u \cdot x^2 + v \cdot y^2 + w \cdot z^2, \\ & \text{subject to} && x + y + z = 1, \\ & {} && 0 \leq x < y < z \leq 1,\\ & {} && u \cdot x +2 v \cdot y \leq 3, \\ & {} && 3 v \cdot y + 2 w \cdot z \leq 1, \\ & {} && 0 \leq u \leq 1, \\ & {} && 0 \leq v \leq 1, \\ & {} && 0 \leq w \leq 1. \end{aligned}

All $x,y,z,u,v,w$ are variables to be optimized. In general, we have a set of polynomial constraints and the objective function to be minimized is also a polynomial. This problem seems to be done by geometric programming. However, in geometric programming, only monomial equality constraints are allowed. But here, we have the equality constraint $x+y+z = 1$.

So what should be a possible way to solve this kind of problems?

• You could replace each $z$ by $1-x-y$ if you could figure the right restriction so that $0 \le x<y<z \le 1$ would be satisfied. Then there would no longer be equality constraints. – coffeemath May 24 '13 at 12:36