On the Power Allocation Problem in the Gaussian Interference Channel with Proportional Rate Constraints

This paper takes an analytical approach to solving the optimization problem of finding the power allocation that maximizes the sum-rate of the Gaussian interference channel with any linear power (interference) constraint and proportional rate constraints. It is proved that the sum-rate of the Gaussian interference channel restricted to proportional rate constraints does not have a critical point and the maximum sum-rate subject to said constraints occurs at the boundary of the domain formed by the plane representing the linear power constraint. This is accomplished by using analytic geometry in higher dimensions to show that the curve of intersection of the sum-rate and the proportional rate constraints is always increasing, and intersects the boundary plane representing the linear power constraint at a unique point. A polynomial time (in the number of users) centralized algorithm that finds this point of optimal power allocation is proposed. This is a significant improvement over existing algorithms for related power allocation problems which have exponential time complexity in the number of users. Two distributed algorithms with linear and constant complexities are also presented. Simulation results supporting the analysis and demonstrating the performances of the algorithms are presented.