Utility Optimization Based on MSE for Parallel Broadcast Channels: The Square Root Law

Abstract

In order to design efficient online resource allocation algorithms, convexity of the underlying optimization problem is an important prerequisite. This paper covers two resource allocation problems: the sum-power constrained utility maximization and the sum-power minimization for minimum utility requirements for parallel broadcast channels. We derive a new class of utility functions for which both optimization problems can be transformed into convex representations and give necessary and sufficient conditions for the optimum solution of the original non-convex problems with regard to power. We thereby extend the known log-convexity class for which convex representations can be found and, by introducing the square-root criteria, present a straight forward test to check whether arbitrary utility functions belong to our class. For the new class of utility functions we present simple algorithms which operate in the non-convex domain, prove convergence to the global optimum and evaluate their performance by simulations. Besides, the paper reveals some insights on the general structure of the mean square error region and thereby disproves a former result.