# Maximization of the sum of mutual information terms

#### 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

#### fresh_42

Mentor
2018 Award
You could start and compute the function value at the vertices, take the maximal vertex, and see if they increase or decrease if you leave this point. Another idea is to use Lagrange multipliers.

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