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Maximization of the sum of mutual information terms

  1. Aug 3, 2009 #1
    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.

    [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
     
  2. jcsd
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