Dynamic spectrum management: Complexity and duality

Zhi Quan Luo, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

608 Scopus citations


Consider a communication system whereby multiple users share a common frequency band and must choose their transmit power spectral densities dynamically in response to physical channel conditions. Due to co-channel interference, the achievable data rate of each user depends on not only the power spectral density of its own, but also those of others in the system. Given any channel condition and assuming Gaussian signaling, we consider the problem to jointly determine all users' power spectral densities so as to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. For the discretized version of this nonconvex problem, we characterize its computational complexity by establishing the NP-hardness under various practical settings, and identify subclasses of the problem that are solvable in polynomial time. Moreover, we consider the Lagrangian dual relaxation of this nonconvex problem. Using the Lyapunov theorem in functional analysis, we rigorously prove a result first discovered by Yu and Lui (2006) that there is a zero duality gap for the continuous (Lebesgue integral) formulation. Moreover, we show that the duality gap for the discrete formulation vanishes asymptotically as the size of discretization decreases to zero.

Original languageEnglish (US)
Pages (from-to)57-73
Number of pages17
JournalIEEE Journal on Selected Topics in Signal Processing
Issue number1
StatePublished - Feb 2008

Bibliographical note

Funding Information:
Manuscript received April 15, 2007; revised October 21, 2007. This work was supported in part by Hong Kong RGC Earmarked Grants CUHK418505 and CUHK418406, in part by the NSF under Grant DMS-0610037, and in part by the USDOD ARMY under Grant W911NF-05-1-0567. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ananthram Swami.


  • Complexity
  • Duality
  • Spectrum management
  • Sum-rate maximization


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