Dual methods for nonconvex spectrum optimization of multicarrier systems

Abstract

Abstract-The design and optimization of multicarrier communications systems often involve a maximization of the total throughput subject to system resource constraints. The optimization problem is numerically difficult to solve when the problem does not have a convexity structure. This paper makes progress toward solving optimization problems of this type by showing that under a certain condition called the time-sharing condition, the duality gap of the optimization problem is always zero, regardless of the convexity of the objective function. Further, we show that the time-sharing condition is satisfied for practical multiuser spectrum optimization problems in multicarrier systems in the limit as the number of carriers goes to infinity. This result leads to efficient numerical algorithms that solve the nonconvex problem in the dual domain. We show that the recently proposed optimal spectrum balancing algorithm for digital subscriber lines can be interpreted as a dual algorithm. This new interpretation gives rise to more efficient dual update methods. It also suggests ways in which the dual objective may be evaluated approximately, further improving the numerical efficiency of the algorithm. We propose a low-complexity iterative spectrum balancing algorithm based on these ideas, and show that the new algorithm achieves near-optimal performance in many practical situations.Index Terms-Digital subscriber lines (DSLs), discrete multitone (DMT), duality theory, dynamic spectrum management (DSM), iterative spectrum balancing (ISB), nonconvex optimization, optimal spectrum balancing (OSB), orthogonal frequency-division multiplex (OFDM).

Sum-Throughput Maximization in Downlink OFDMA-based Cellular Cognitive Radio Networks

“…In this section, we propose a solution for (3) based on the results obtained in [25]- [27]. The purpose is to provide a solution for the problem (3) in the dual domain.…”

“…It is necessary to point out that even ) ( g is convex, however it may not be differentiable and does not have gradient. Thus, a search method based on sub-gradient approach [27] is performed as follows. (11) and (12) .…”

A New Approach for Sub-Channels Assignment in Downlink Fourth Generation Mobile Systems

“…However, this approach may introduce significant loss in optimality because of its relaxation of the data rate constraints and its high complexity in implementation. To cope with these inconveniences, authors in (Yu and Lui, 2006) make progress toward numerical solution of non-convex optimization problems for multicarrier systems based on the Lagrangian dual of these non-convex problems. Motivated by results proved in (Yu and Lui, 2006), authors Seong et al (2006) resolve OFDMA downlink resource allocation problems in the dual domain by using Lagrange dual decomposition.…”

“…In recent years, many dynamic resource allocation schemes for the OFDMA downlink systems have been developed to close the optimal solution of optimizing problems that aim to achieve the highest throughput with the minimum transmit power either with the users' data rates as the constraint and the total transmit power as the objective function or with the constraint on the power and the total throughput of the system as the objective, referred respectively as Margin Adaptive (MA) (Cheong et al, 1999;Liejune et al, 2011;Yu, and Lui, 2006;Seong et al, 2006) and Rate Adaptive (RA) (Jang and Lee, 2003) optimization problems. There is also a third type that aims to achieve the highest possible throughput by maintaining fairness among users (Rhee and Cioffi, 2000;Anas et al, 20011;Marabissi et al, 2008;Papoutsis et al, 2010).…”

Weighted Sum Rate Maximization in OFDM Based Cooperative Cognitive Radios: A Joint Optimization Approach

“…The gap is known as the duality gap. Recently, it is declared that the duality gap will be zero if the optimum answer of the optimization problem is a concave function of the constraints [20], and therefore the solution of the dual problem is the optimum solution to the original problem. Also it is shown in [21] and used in [14]-same approach as our problem in an OFDM downlink resource allocation problem-that the optimum value of the problem tends to be concave over constraints if the number of subcarriers (M) is large enough.…”