# Maximization of the sum of mutual information terms

1. Aug 3, 2009

### miggimig

I would like to compute the power distribution that maximizes the sum data rate of a certain communication scheme. The expression follows from the sum of the rate of four messages which interfere with each other. A certain amount of power (here 1 Watt) is assigned to these messages, and there is a further constraint on the sum rate of the first three messages.

$\begin{split} & \mathrm{maximize} \qquad \underbrace{\log_2\det \left( 1 + \frac{p_{1}}{ p_{2} + \lambda(p_{3}+p_{4}) + \sigma^2} \right)}_{R_{1}} \\ &\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{3}}{p_{4} + \lambda p_{2}+\sigma^2} \right)}_{R_{3}}\\ &\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{2}}{\lambda p_{4}+\sigma^2} \right)}_{R_{2}} + \underbrace{\log_2\det \left( 1 + \frac{p_{4}}{\sigma^2} \right)}_{R_{4}} \\ & \mathrm{subject\ to}\qquad p_{1}+p_{2} \leq 1;\ p_{3}+p_{4} \leq 1;\ R_{1}+R_{2}+R_{3} \leq 1 \end{split}$

The problem is not convex. However, I still would like to find the optimal power assignment $p_1, p_2, p_3, p_4$. 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