Duality-Gap Bounds for Multi-Carrier Systems and Their Application to Periodic Scheduling

We investigate a novel cross-layer optimization problem for jointly performing dynamic spectrum management (DSM) and periodic rate-scheduling in time. The large number of carriers used in digital subscriber lines (DSL) makes DSM a large-scale optimization problem for which dual optimization is a commonly used method. The duality-gap which potentially accompanies the dual optimization for non-convex problems is typically assumed to be small enough to be neglected. Also, previous theoretical results show a vanishing duality-gap as the number of subcarriers approaches infinity. We will bound the potential performance improvements that can be achieved by the additional rate-scheduling procedure. This bound is found to depend on the duality-gap in the physical layer DSM problem. Furthermore, we will derive bounds on the duality-gap of the two most important optimization problems in DSL, namely the maximization of the weighted sum-rate and the minimization of the weighted sum-power. These bounds are derived for a finite number of subcarriers and are also applicable to the respective problems in orthogonal frequency division multiplex (OFDM) systems.