# Software to solve Nonlinear Systems (ineq and eq)

1. Jan 14, 2013

### soundofsilence

Hi everyone,

I've got an optimisation/computing question. I have a system of nonlinear equalities and inequalities, which I've written below for reference. It's the conditions for a minimiser of a Karush-Kuhn-Tucker problem. Would anyone be kind enough to explain how I could use software to compute the solution? I've never really used matlab or maple which I assume might be able to do that. A second best would be computing the solution to the original problem, which I've also posted.

Ideally I would like to compute solutions to these, which I hope are the KKT conditions for a minimiser

$2s_{1}-\mu_{1}+\lambda_{1}(\frac{5}{2}(\frac{1}{2}s_{1}+ \frac{1}{4}s_{2}+\frac{1}{4})^{4}-1)+\lambda_{2}(\frac{2}{3} (\frac{1}{3}s_{1}+\frac{1}{3}s_{2}+\frac{1}{3}))=0$

$2s_{2}-\mu_{2}+\lambda_{1}(\frac{5}{4}(\frac{1}{2}s_{1}+ \frac{1}{4}s_{2}+\frac{1}{4})^{4})+\lambda_{2} (\frac{2}{3}(\frac{1}{3}s_{1}+\frac{1}{3}s_{2}+ \frac{1}{3})-1)=0$

$(\frac{1}{2}s_{1}+\frac{1}{4}s_{2}+\frac{1}{4})^{5}-s_{1}=0$
$(\frac{1}{3}s_{1}+\frac{1}{3}s_{2}+\frac{1}{3})^{2}-s_{2}=0$

$s_{1},s_{2} \geq 0$
$\mu_{1},\mu_{2} \geq 0$

$\mu_{1}s_{1}=0$
$\mu_{2}s_{2}=0$

If that seems a bit unlikely or difficult a second best would be just computing the solution to the original problem which is:

minimise $s_{1}^{2}+s_{2}^{2}$
such that
$(\frac{1}{2}s_{1}+\frac{1}{4}s_{2}+\frac{1}{4})^{5}-s_{1}=0$
$(\frac{1}{3}s_{1}+\frac{1}{3}s_{2}+\frac{1}{3})^{2}-s_{2}=0$

Many thanks,

Peter

2. Jan 16, 2013

### soundofsilence

Anyone? Thoughts appreciated!