How to handle convex $x_1\log(1+x_2/x_1)$ in DCP ruleset?

1

The function f(x1,x2)=x1*log(1+x2/x1) is a concave function. However, it can not be used in the context of cvx as it does not appear in DCP rule set. It is a typical function in communication systems, where x1 is the bandwidth and x2 is the transmit power.

Does any one know how I can handle this function in CVX?

2

How to implement following function \sum\limits_i {\sum\limits_k {{y_i}\log (1 + \frac{{x_i^k}}{{{y_i}}})} }

3

Does anyone know if there is a standard name for this function? If so perhaps we should add it to CVX.

4

This comment may help. The Shannon capacity of a communication channel with gain H and noise power spectral density N_0 is defined as C = Wlog_2 (1+\frac{|H|^2 P}{ N_0W}).

5

use -rel_entr((x1,x1+x2) to define the function in cvx. Then it works

Edit by Mark L. Stone: I have taken the liberty of correcting a typo in your answer.