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*P. C. Weeraddana **M. Codreanu, **M. Latva-Aho, ***A. Ephremides * KTH, Royal institute of Technology, Stockholm, Sweden ** CWC, University of Oulu, Finland ***University of Maryland, USA 2013.02.19 Multicell MISO Downlink Weighted Sum-Rate Maximization: A Distributed Approach
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Motivation Centralized RA requires gathering problem data at a central location -> huge overhead Large-scale communication networks -> large-scale problems Distributed solution methods are indeed desirable Many local subproblems -> small problems Coordination between subproblems -> light protocol 12.10.2015CWC | Centre For Wireless Communications2
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Motivation WSRMax: a central component of many NW control and optimization methods, e.g., Cross-layer control policies NUM for wireless networks MaxWeight link scheduling for wireless networks power and rate control policies for wireless networks achievable rate regions in wireless networks 12.10.2015CWC | Centre For Wireless Communications3
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Challenges WSRMax problem is nonconvex, in fact NP-hard At least a suboptimal solution is desirable Considering the most general wireless network (MANET) is indeed difficult A particular case is infrastructure based wireless networks Cellular networks Coordinating entities MS-BS, MS-MS, BS-BS Coordination between subproblems -> light protocol 12.10.2015CWC | Centre For Wireless Communications4
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Our contribution 10/12/2015CWC | Centre For Wireless Communications5 Distributed algorithm for WSRMax for MISO interfering BC channel; BS-BS coordination required Algorithm is based on primal decomposition methods and subgradient methods Split the problem into subproblems and a master problem local variables: Tx beamforming directions and power global variables: out-of-cell interference power Subproblems asynchronous (one for each BS) variables: Tx beamforming directions and power Master problem resolves out-of-cell interference (coupling) P. C. Weeraddana, M. Codreanu, M. Latva-aho, and A. Ephremides, IEEE Transactions on Signal Processing, February, 2013
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Key Idea 10/12/2015CWC | Centre For Wireless Communications6
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Key Idea 10/12/2015CWC | Centre For Wireless Communications7
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Key Idea 10/12/2015CWC | Centre For Wireless Communications8
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Key Idea 10/12/2015CWC | Centre For Wireless Communications9
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Key Idea 10/12/2015CWC | Centre For Wireless Communications10
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Key Idea 10/12/2015CWC | Centre For Wireless Communications11
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Key Idea 10/12/2015CWC | Centre For Wireless Communications12
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System model 10/12/2015CWC | Centre For Wireless Communications13 : number of BSs : set of BSs Tx regioninterference region : number of BS antennas : number of data streams : set of data streams : set of data streams of BS : transmitter node of d.s. : receiver node of d.s. 1 2 3 1 2 3 4 5 6 7 8 9 1011 12
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System model 10/12/2015CWC | Centre For Wireless Communications14 : information symbol;, : power signal vector transmitted by BS : beamforming vector;
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System model 10/12/2015CWC | Centre For Wireless Communications15 signal received at : channel; to intra-cell interference out-of-cell interference : cir. symm. complex Gaussian noise; variance
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System model 10/12/2015CWC | Centre For Wireless Communications16 received SINR of : out-of-cell interference power; th BS to intra-cell interference out-of-cell interference
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System model 10/12/2015CWC | Centre For Wireless Communications17 1 2 3 5 6 7 8 out-of-cell interference power, e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications18 1 2 3 5 6 7 8 out-of-cell interference power, e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications19 1 2 3 5 6 7 8 out-of-cell interference power, e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications20 1 2 3 5 6 7 8 out-of-cell interference power, e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications21 1 2 3 5 6 7 8 out-of-cell interference power, e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications22 received SINR of : out-of-cell interference power; th BS to intra-cell interference out-of-cell interference
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System model 10/12/2015CWC | Centre For Wireless Communications23 received SINR of intra-cell interference out-of-cell interference : set of out-of-cell interfering BSs that interferes : out-of-cell interference power (complicating variables)
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System model 10/12/2015CWC | Centre For Wireless Communications24 1 2 3 5 6 7 8 e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications25 1 2 3 1 2 3 4 5 6 7 8 9 1011 12 : set of d.s. that are subject to out-of-cell interference e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications26 1 2 3 1 2 3 4 5 6 7 8 9 1011 12 : set of d.s. that are subject to out-of-cell interference e.g.,
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System model 10/12/2015CWC | Centre For Wireless Communications27 1 2 3 66 9 12 : set of d.s. that are subject to out-of-cell interference e.g.,
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Problem formulation 10/12/2015CWC | Centre For Wireless Communications28 variables: and
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Primal decomposition 10/12/2015CWC | Centre For Wireless Communications29 variables: subproblems (for all ) : master problem: variables:
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Subproblem (BS optimization) 10/12/2015CWC | Centre For Wireless Communications30
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Subproblem 10/12/2015CWC | Centre For Wireless Communications31 variables: The problem above is NP-hard Suboptimal methods, approximations
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications32 The method is inspired from alternating convex optimization techniques Fix beamforming directions Approximate objective by an UB function resultant problem is a GP; variables Fix the resultant SINR values Find beamforming directions, that can preserve the SINR values with a power margin this can be cast as a SOCP Iterate until a stopping criterion is satisfied
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications33
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications34
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications35
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications36
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications37
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications38
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Subproblem: key idea 10/12/2015CWC | Centre For Wireless Communications39
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Master problem 10/12/2015CWC | Centre For Wireless Communications40
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Master problem 10/12/2015CWC | Centre For Wireless Communications41 variables: Recall: subproblems are NP-hard -> we cannot even compute the master objective value Suboptimal methods, approximations Problem is nonconvex -> subgradient method alone fails
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Master problem: key idea 10/12/2015CWC | Centre For Wireless Communications42 The method is inspired from sequential convex approximation (upper bound) techniques subgradient method is adopted to solve the resulting convex problems
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Integrate master problem & subproblem 10/12/2015CWC | Centre For Wireless Communications43 Increasingly important: convex approximations mentioned above are such that we can always rely on the results of BS optimizations to compute a subgradient for the subgradient method. thus, coordination of the BS optimizations
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10/12/2015CWC | Centre For Wireless Communications44 Integrate master problem & subproblem
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10/12/2015CWC | Centre For Wireless Communications45 out-of-cell interference: objective value computed by BS at ‘ F ’ : we can show that optimal sensitivity values of GP -> construct subgradient F subproblem convex
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10/12/2015CWC | Centre For Wireless Communications46 Integrate master problem & subproblem subgradient method:
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Integrate master problem & subproblem 10/12/2015CWC | Centre For Wireless Communications47 Algorithm 1 (subproblem) Algorithm 2 (overall problem)
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An example signaling frame structure 10/12/2015CWC | Centre For Wireless Communications48 Algorithm 2 (overall problem) note:in Alg.1 or subgradient method, ‘BS optimization GP’ is always carried out in our simulations: - fixed Alg.1 iterations ( ) - fixed subgrad iterations ( ) per switch
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications49 channel gains: : distance from to : small scale fading coefficients SNR operating point:
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications50 -> the degree of BS coordination note -> subgradient is not an ascent method out-of-cell interference is resolved -> objective value is increased smaller performs better compared to large light backhaul signaling
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10/12/2015CWC | Centre For Wireless Communications51 Numerical Examples B accuracy of the solution of an approximated master problem is irrelevant in the case of overall algorithm refining the approximation more often is more beneficial
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications52 same behavior smaller performs better compared to large
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications53 algorithm performs better than the centralized algorithm not surprising since both algorithms are suboptimal algorithms
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications54 algorithm performs better than the centralized algorithm not surprising since both algorithms are suboptimal algorithms
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications55 ; one subgradient iteration during BS coordination window 12% improvement within 5 BS coordination 99% of the centralized value within 5 BS coordination WMMSE performs better with no errors in user estimates WMMSE with even 1% error in signal covariance estimations at user perform poorly
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Numerical Examples 12.10.2015CWC | Centre For Wireless Communications56 ; one subgradient iteration during BS coordination window 24% improvement within 5 BS coordination 94% of the centralized value within 5 BS coordination WMMSE performs better with no errors in user estimates WMMSE with even 1% error in signal covariance estimations at user perform poorly
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Conclusions Problem: WSRMax for MISO interfering BC channel (NP-hard) Techniques: Primal decomposition Result: many subproblems (one for each BS) coordinating to find a suboptimal solution of the original problem Subproblem: alternating convex approximation techniques, GP, and SOCP Master problem: sequential convex approximation techniques and subgradient method Coordination: BS-BS (backhaul) signaling; favorable for practical implementation Substantial improvements with a small number of BS coordination -> favorable for practical implementation Numerical examples -> algorithm performance is significantly close to (suboptimal) centralized solution methods 12.10.2015CWC | Centre For Wireless Communications57
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