Robust Monotonic Optimization Framework for Multicell MISO Systems

Robust Monotonic Optimization Framework for Multicell MISO SystemsEmil Bjrnson1, Gan Zheng2, Mats Bengtsson1, Bjrn Ottersten1,2

1 Signal Processing Lab., ACCESS Linnaeus Centre, KTH Royal Institute of Technology, Sweden2 Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg

Published in IEEE Transactions on Signal Processing, vol. 60, no. 5, pp. 2508-2523, May 2012

1Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013IntroductionDownlink Coordinated Multicell SystemMany multi-antenna transmitters/BSsMany single-antenna receivers

Sharing a Frequency BandAll signals reach everyone!Limiting factor: co-user interference

Multi-Antenna TransmissionSpatially directed signalsKnown as: Beamforming/precoding

2Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Problem FormulationTypical Problem Formulations:

Bad News: NP-hard problem (treating co-user interference as noise)High complexity: Approximations are required in practiceCommon approach: Propose an approx. and compare with old approxs.Can we solve it optimally for benchmarking?maximize System Utility Precoding for all users

subject toPower ConstraintsWeighted sum of user performance,Proportional fairness,etc.Limited total power,Limited power per transmitter,Limited power per antenna

Z.-Q. Luo and S. Zhang, Dynamic spectrum management: Complexity and duality, IEEE Journal of Sel. Topics in Signal Processing, 2008.(1)3Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Contribution & TimelinessMain ContributionPropose an algorithm to solve Problem (1) optimallyTimely: Several concurrent works

Differences from Concurrent WorksUse state-of-the-art branch-reduce-and-bound (BRB) algorithmHandle robustness to channel uncertaintyArbitrary multicell scenarios and performance measuresW. Utschick and J. Brehmer, Monotonic optimization framework for coordinated beamforming in multicell networks, IEEE Trans. on Signal Processing, vol. 60, no. 4, pp. 18991909, 2012.L. Liu, R. Zhang, and K. Chua, Achieving global optimality for weighted sum rate maximization in the K-user Gaussian interference channel with multiple antennas, IEEE Trans. on Wireless Communications, vol. 11, no. 5, pp. 19331945, 2012.BRB Algorithm: H. Tuy, F. Al-Khayyal, and P. Thach, Monotonic optimization: Branch and cut methods,, Springer, 2005.4Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013System Model5Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Many Different Multicell Scenarios

Ideal Joint TransmissionCoordinated Beamforming(Interference channel)Underlay Cognitive Radio6

Multi-Tier CoordinationBjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Multicell System Model

7Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Robustness to Uncertain Channels

8Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013User Performance

9

Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Robust Performance Region

Weighting matrix(Positive semi-definite)Limit(Positive scalar)

2-UserPerformanceRegion

10Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Problem FormulationFind Optimal Solution to Detailed Version of (1):

For monotonic increasing system utility function :Sum performance:Proportional fairness:Max-min fairness:

Equivalent to Search in Performance Region:11

(2)Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Special Case:Fairness-Profile Optimization12Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Fairness-Profile OptimizationConsider Special Case of (2):

Called: Fairness-profile optimizationGeneralization of max-min fairness

Simple Geometric InterpretationCan we search on the line?Region is unknown

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Bjrnson et al.: Robust Monotonic Optimization Framework22 August 201314Fairness-Profile Optimization (2)How to Check if a Point on the Line is Feasible?

Proof: Based on S-lemma in robust optimization Theorem 1A point is in the region if and only if the following convex feasibility problem is feasible:

Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Fairness-Profile Optimization (3)Simple Line-Search: BisectionLine-search: Linear convergenceSub-problem: Feasibility checkWorks for any number of user

Bisection AlgorithmFind start intervalCheck feasibility of midpoint using Theorem 1If feasible: Remove lower halfElse: Remove upper halfIterate

15SummaryFairness-profile problem solvable in polynomial time!Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013BRB Algorithm16Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Computing Optimal Strategy: BRB AlgorithmSolve (2) for Any System Utility FunctionSystematic search in performance regionImprove lower/upper bounds on optimum:

Branch-Reduce-and-Bound (BRB) AlgorithmCover performance region with a boxDivide the box into two sub-boxesRemove parts with no solutions inSearch for solutions to improve boundsContinue with sub-box with largest value

End when bounds are tight enough:

Accuracy17Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Computing Optimal Strategy: Example

18Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Numerical Examples19Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Example 1: ConvergenceConvergence of Lower/Upper BoundsCompared with Polyblock algorithm (proposed only for perfect CSI)Scenario: 2 BSs, 3 antennas/BS, 2 users, perfect channel knowledgePlot relative error in lower/upper bounds (sum rate optimization)20

ObservationsBRB algorithm has faster convergenceLower bound converges rather quickly

Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Example 2: Benchmarking21

Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Conclusion22Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013ConclusionMaximize System Utility in Coordinated Multicell SystemsNP-hard problem in general: Only suboptimal solutions in practiceHow can we truly evaluate a suboptimal solution?

Robust Monotonic Optimization FrameworkSolves a wide range of system utility maximizationsHandles channel uncertainty and any monotone performance measuresSubproblem: Fairness-profile optimization (FPO) = polynomial timeBRB algorithm: Solves finite number of FPO problemsGeneralization: Problems where feasibility of a point is checked easily

Do you want to test it?Download Matlab code from the book Optimal Resource Allocation in Coordinated Multi-Cell Systems by E. Bjrnson & E. JorswieckBased on CVX package by Steven Boyd et al.23Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Thank you for your attention!Questions?24Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Backup Slides25Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Generic Multicell Setup

Many examples:Interference channelArbitrary overlapping cooperation clusters Global joint transmissionUnderlay cognitive radio, etc.

Dynamic Cooperation Clusters

Inner Circle : Serve users with dataOuter Circle : Suppress interferenceOutside Circles:Negligible impact modeled as noise

2626Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Dynamic Cooperation Clusters

27Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Dynamic Cooperation Clusters (2)

28Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Power Constraints: Examples

29Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Performance Region: ShapesCan the region have any shape?

No! Can prove that:Compact setNormal set

Upper corner in region, everything inside region30Bjrnson et al.: Robust Monotonic Optimization Framework22 August 2013Performance Region: Shapes (2)Some Possible Shapes

User-Coupling

Weak: ConvexStrong: Concave31