# Maximizing sum-rate with constraints

I have an SNR measure, which is a ratio of two linear functions, and I need to maximize the sum rate of a cellular system given by

$$R = \sum_{i=1}^{N_{1}}\sum_{j=1}^{N_{2}}\mathrm{log}_{2}\left(1 + \frac{f(x_{i,j})}{g(x_{i,j})}\right)$$

subject to the constraints

$$0 \leq x_{i,j} \leq 1$$ $$\sum_{j=1}^{N_{2}}x_{i,j} = 1 \quad i = 1, \dots, N_{1}$$

where, $$f$$ and $$g$$ are linear functions in $$x_{i,j}$$

Can anyone suggest a way (rough algorithm) to solve such a problem? I am unable to figure out how to account for the second constraint.

• Try a Lagrange Multiplier approach. It's what we in information theory usually do to show things like the maximum of entropy being the equidistribution, and entropy just happens to be a sum of functions of x and the logarithms of x... Apr 5, 2020 at 23:39
• Thanks for suggesting it. In that case, should I bother about the inequality? It looks like the second constraint guarantees $x_{i,j} \leq 1$. Apr 6, 2020 at 8:00
• you're right about that Apr 6, 2020 at 8:10