- #1

soundofsilence

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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

[itex]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[/itex]

[itex]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[/itex]

[itex](\frac{1}{2}s_{1}+\frac{1}{4}s_{2}+\frac{1}{4})^{5}-s_{1}=0[/itex]

[itex](\frac{1}{3}s_{1}+\frac{1}{3}s_{2}+\frac{1}{3})^{2}-s_{2}=0[/itex]

[itex]s_{1},s_{2} \geq 0[/itex]

[itex]\mu_{1},\mu_{2} \geq 0[/itex]

[itex]\mu_{1}s_{1}=0[/itex]

[itex]\mu_{2}s_{2}=0[/itex]

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

minimise [itex]s_{1}^{2}+s_{2}^{2}[/itex]

such that

[itex](\frac{1}{2}s_{1}+\frac{1}{4}s_{2}+\frac{1}{4})^{5}-s_{1}=0[/itex]

[itex](\frac{1}{3}s_{1}+\frac{1}{3}s_{2}+\frac{1}{3})^{2}-s_{2}=0[/itex]

Many thanks,

Peter