- #1
miggimig
- 3
- 0
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{equation}<br /> \begin{split}<br /> & \mathrm{maximize} \qquad \underbrace{\log_2\det \left( 1 + \frac{p_{1}}{ p_{2} + \lambda(p_{3}+p_{4}) + \sigma^2} \right)}_{R_{1}} \\<br /> &\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{3}}{p_{4} + \lambda p_{2}+\sigma^2} \right)}_{R_{3}}\\<br /> &\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{2}}{\lambda<br /> p_{4}+\sigma^2} \right)}_{R_{2}}<br /> + \underbrace{\log_2\det \left( 1 + \frac{p_{4}}{\sigma^2} \right)}_{R_{4}}<br /> \\<br /> & \mathrm{subject\ to}\qquad p_{1}+p_{2} \leq 1;\ p_{3}+p_{4} \leq 1;\ R_{1}+R_{2}+R_{3} \leq 1<br /> \end{split}<br /> \end{equation}
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
\begin{equation}<br /> \begin{split}<br /> & \mathrm{maximize} \qquad \underbrace{\log_2\det \left( 1 + \frac{p_{1}}{ p_{2} + \lambda(p_{3}+p_{4}) + \sigma^2} \right)}_{R_{1}} \\<br /> &\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{3}}{p_{4} + \lambda p_{2}+\sigma^2} \right)}_{R_{3}}\\<br /> &\qquad\qquad\qquad + \underbrace{\log_2\det \left( 1 + \frac{p_{2}}{\lambda<br /> p_{4}+\sigma^2} \right)}_{R_{2}}<br /> + \underbrace{\log_2\det \left( 1 + \frac{p_{4}}{\sigma^2} \right)}_{R_{4}}<br /> \\<br /> & \mathrm{subject\ to}\qquad p_{1}+p_{2} \leq 1;\ p_{3}+p_{4} \leq 1;\ R_{1}+R_{2}+R_{3} \leq 1<br /> \end{split}<br /> \end{equation}
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
