Duality Gap Estimation and Polynomial Time Approximation for Optimal Spectrum Management

Abstract

Consider a communication system whereby multiple users share a common frequency band and must choose their transmit power spectra jointly in response to physical channel conditions including the effects of interference. The goal of the users is to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. A popular approach to solve the discretized version of this nonconvex problem is by Lagrangian dual relaxation. Unfortunately the discretized spectrum management problem is NP-hard and its Lagrangian dual is in general not equivalent to the primal formulation due to a positive duality gap. In this paper, we use a convexity result of Lyapunov to estimate the size of duality gap for the discretized spectrum management problem and show that the duality gap vanishes asymptotically at the rate O(1/ √ N ), where N is the size of the uniform discretization of the shared spectrum. If the channels are frequency flat, the duality gap estimate improves to O(1/N ). Moreover, when restricted to the FDMA spectrum sharing strategies, we show that the Lagrangian dual relaxation, combined with a linear programming scheme, can generate an -optimal solution for the continuous formulation of the spectrum management problem in polynomial time for any > 0.

Energy-Efficiency Optimization With Model Convexification for Wireless Ad Hoc Networks With Multi-Packet Reception Capability

“…Unfortunately, when an energy-efficiency optimization problem of a wireless ad hoc network, subject to both individual power constraints and coupling interference constraints, cannot meet the mathematical convexity, convex optimization techniques proposed in the prior works cannot be applied directly. Due to the existence of a positive duality gap, the Lagrangian dual formulation is in general not equivalent to its primal non-convex problem [21], and only local convergence is guaranteed by primal-dual approaches.…”

“…As shown in [18], SINR-based utility functions or their linear combinations are neither convex nor concave in the power feasible region due to the impact of the coupling interferences. The presence of non-convexity can make current convex optimization methods (like interior-point methods, sequential quadratic programming, and semidefinite cone programming [19], [20]) less effective or even infeasible in solving a global optimal and feasible point [21]. It remains an open issue to effectively tackle the nonlinear constrained energy-efficiency optimization problem in the presence of non-convex SINR-based utility functions.…”