[itex] \begin{equation}

\begin{split}

& \mathrm{maximize} \qquad \underbrace{\log_2\det \left( 1 + \frac{p_{1}}{ p_{2} + \lambda(p_{3}+p_{4}) + \sigma^2} \right)}_{R_{1}} \\

&\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{3}}{p_{4} + \lambda p_{2}+\sigma^2} \right)}_{R_{3}}\\

&\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{2}}{\lambda

p_{4}+\sigma^2} \right)}_{R_{2}}

+ \underbrace{\log_2\det \left( 1 + \frac{p_{4}}{\sigma^2} \right)}_{R_{4}}

\\

& \mathrm{subject\ to}\qquad p_{1}+p_{2} \leq 1;\ p_{3}+p_{4} \leq 1;\ R_{1}+R_{2}+R_{3} \leq 1

\end{split}

\end{equation} [/itex]

The problem is not convex. However, I still would like to find the optimal power assignment [itex] p_1, p_2, p_3, p_4 [/itex]. I have little knowledge of optimization techniques, therefore I would be happy about any advice how to proceed.

My first idea is, to test if the function is quasi-convex, but I do not know how to proof this property.

I also would be very happy about concrete advices which technique or solver is suitable to find the solution.

If the second constraint is a big problem, I would also be happy to find the optimal power assignment for the relaxed problem without this constraint.

Thank you very much for your attention!

Michael