IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 3, MARCH 2010 753 Ef ficient Power Allocation Schemes for Nonconvex Sum-Rate Maximization on Gaussian Cognitive MAC Sang-wook Han, Hoon Kim, Member, IEEE, Youngnam Han, Senior Member, IEEE, and John M. Cioffi, Fellow, IEEE Abstract—This letter considers a sum-rate-maximizing power allocation problem under the Gaussian cognitive multiple-access channel (MAC), where primary users and secondary users may communicate under mutual interference. Formulating the problem as a standard nonconvex quadratically constrained quadratic problem (QCQP) provides a simple method to find a solution using semidefinite relaxation (SDR). Simulation results show that the solution achieves almost the same performance of the exhaustive search, within polynomial time. Index Terms—Cognitive radio, sum-rate maximization, Gaus- sian cognitive MAC, semidefinite relaxation (SDR), nonconvex QCQP. I. I NTRODUCTION I NVESTIGATION of both spatial and temporal spectrum utilization reveals the fact that not all the spectrum is in use all the time. Cognitive radio technology inspired by the observation turns out to be a promising technique for the efficient use of this unused spectrum, potentially allowing a large amount of spectrum to become available for future high bandwidth applications [1]. Some works [2] - [4] make discussions on cognitive radio’s achievable rate from informa- tion theoretic point. In the seminal work [3], the achievable rate of a single cognitive radio user is provided under such constraints as (i) there is no interference to the primary user, and (ii) the primary encoder-decoder pair is oblivious to the presence of cognitive radio. In [4], they extend the result of [3] to the case with multiple cognitive radio users and characterize the cognitive radio’s achievable rate region for Gaussian multiple access channels (MACs). Maximization of the cognitive radio’s sum-rate on Gaussian MAC then raises Paper approved by B. Sikdar, the Editor for Wireless Packet Access and Cross-Layer Design of the IEEE Communications Society. Manuscript received November 3, 2008; revised June 1, 2009. S. Han is with the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver BC, Canada V6T 1Z4 (e- mail: swhan@ece.ubc.ca). H. Kim (corresponding author) is with the Department of Electron- ics Engineering, The University of Incheon, Incheon, Korea (e-mail: hoon@incheon.ac.kr). Y. Han is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejon, Korea (e-mail: yn- han@ee.kaist.ac.kr). J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: cioffi@stanford.edu). This research was supported by the MKE (Ministry of Knowledge Econ- omy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (IITA-2008-(C1090-0801-0036)). Parts of this work were done while the first author was with Korea Advanced Institute of Science and Technology (KAIST), Daejon, Korea. This paper was presented in part at the IEEE 68th Vehicular Technology Conference, Calgary, Alberta, Canada, September 2008. Digital Object Identifier 10.1109/TCOMM.2010.03.080151 the problem of the allocation of each cognitive user’s power ratio. This letter considers a power allocation problem to ap- proximate the maximization of the sum-rate of cognitive radio on the Gaussian MAC. The maximization formulation is formulated as a standard nonconvex quadratically con- strained quadratic problem (QCQP). In [4], they solve the same problem to result in a nonconvex quadratic optimization problem, whose solution is difficult to get. Thereby, they rather handle the nonconvexity through a heuristic scheme that uses Lagrangian multipliers, by which the solution can only be obtained through iterative numerical computations. When the complexity of the solution is an issue, the solution of a standard type of the nonconvex QCQP can be achieved more easily in polynomial time by applying semidefinite relaxation (SDR) [8]. Through the formulation by a standard nonconvex QCQP, this work uses a conceptually simple SDR to solve the relaxation of the power allocation problem for the nonconvex sum-rate maximization on Gaussian cognitive MAC, instead of using a heuristic algorithm in [4]. Our simplified, well- formulated optimization problem of the original problem is solved very easily compared to the heuristic, highly computa- tional approach to the original problem [4]. Performance and computational complexity of the solutions are made through Monte-Carlo simulations. Simulation results reveal that the SDR solution achieves almost the same performance as an exhaustive search, within polynomial time. Note that this SDR can be solved using any of the commercially available SDP solvers, such as CVX [9], based on an interior-point method. The rest of this letter is structured as follows. Section II presents a cognitive-radio MAC model. Section III investigates the formulation of sum-rate maximization on Gaussian cogni- tive MAC as a nonconvex QCQP. Section IV deals with the convex relaxation of the nonconvex QCQP. Simulation results appear in Section V. Section VI draws conclusions. II. COGNITIVE RADIO MULTIPLE ACCESS MODELING Figure 1 describes the system model [4]. This model has a primary mobile station (MS) communicating with a primary base station (BS) and multiple secondary MSs, who desire access to a secondary BS using primary frequency bands without license, where   is the path gain from the primary MS to the primary BS, and   ( =1, 2, .... ) is the path gain of the interference link from a secondary MS to the primary BS and ℎ  ( =1, 2, .... ) is the path gain from a secondary MS to the secondary BS and ℎ  is the interference link from 0090-6778/10$25.00 c ⃝ 2010 IEEE Authorized licensed use limited to: Jeppiaar Engineering College. Downloaded on May 01,2010 at 03:47:19 UTC from IEEE Xplore. Restrictions apply.