Optimal Resource Allocation in Coordinated Multi-Cell Systems

Optimal Resource Allocation in CoordinatedMulti-Cell SystemsEmil Bjrnson1 and Eduard Jorswieck2

1 Signal Processing Lab., KTH Royal Institute of Technology, Sweden and Alcatel-Lucent Chair on Flexible Radio, Suplec, France2 Dresden University of Technology,Department of Electrical Engineering and Information Technology, Germany

8 September 2013Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20131

PIMRC 2013 Tutorial1Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20132Biography: Emil Bjrnson1983: Born in Malm, Sweden

2007: Master of Science inEngineering Mathematics,Lund University, Sweden

2011: PhD in Telecommunications,KTH, Stockholm, SwedenAdvisors: Bjrn Ottersten, Mats Bengtsson

2012: Recipient of International Postdoc Grant from Sweden. Work with Prof. Mrouane Debbah at Suplec on Optimization of Green Small-Cell Networks

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20133Biography: Eduard Jorswieck1975: Born in Berlin, Germany

2000: Dipl.-Ing. in Electrical Engineering and Computer Science, TU Berlin, Germany

2004: PhD in Electrical Engineering, TU Berlin, GermanyAdvisor: Holger Boche

2006: Post-Doc Fellowship and Assistant Professorship atKTH Stockholm, Sweden

2008: Full Professor and Head of Chair of Communications Theory at TU Dresden, Germany

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20134Book ReferenceTutorial is Based on a Recent Book:

270 pagesE-book for free (from our homepages)Printed book: Special price $35, use link:https://ecommerce.nowpublishers.com/shop/add_to_cart?id=1595Matlab code is available online

Check out: http://flexible-radio.com/emil-bjornsonOptimal Resource Allocation in Coordinated Multi-Cell SystemsResearch book by E. Bjrnson and E. JorswieckFoundations and Trends in Communications and Information Theory, Vol. 9, No. 2-3, pp. 113-381, 2013

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20135General OutlineBook Consists of 4 Chapters

IntroductionProblem formulation and general system model

Optimal Single-Objective Resource AllocationWhich problems are practically solvable?

Structure of Optimal Resource AllocationHow does to optimal solution look like?

Extensions and GeneralizationsApplications to 9 current research topicsE.g., channel uncertainty, distributed optimization,hardware impairments, cognitive radio, etc.Covered by Emil Bjrnson in first 90 minsCovered by Eduard Jorswieckin second 90 minsBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20136Outline: Part 1IntroductionMulti-cell structure, system model, performance measure

Problem FormulationResource allocation: Multi-objective optimization problem

Subjective Resource AllocationUtility functions, different computational complexity

Structure of Optimal BeamformingBeamforming parametrization and its applications

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20137SectionIntroductionBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20138IntroductionProblem Formulation (vaguely):Transfer information wirelessly to usersDivide radio resources among users (time, frequency, space)

Downlink Coordinated Multi-Cell SystemMany transmitting base stations (BSs)Many receiving users

Sharing a Frequency BandAll signals reach everyone!

Limiting FactorInter-user interference

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 20139Introduction: Multi-Antenna TransmissionTraditional Ways to Manage InterferenceAvoid and suppress in time and frequency domainResults in orthogonal access techniques: TDMA, OFDMA, etc.

Multi-Antenna TransmissionBeamforming: Spatially directed signalsAdaptive control of interferenceServe multiple users: Space-division multiple access (SDMA)

Main difference from classical resource allocation!Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201310Introduction: From Single-Cell to Multi-CellNave Multi-Cell ExtensionDivide BS into disjoint clustersSDMA within each clusterAvoid inter-cluster interferenceFractional frequency-reuse

Coordinated Multi-Cell TransmissionSDMA in multi-cell: All BSs collaborateFrequency-reuse one: Interference managed by beamformingMany names: co-processing, coordinated multi-point (CoMP), network MIMO, multi-cell processing

Almost as One Super-CellBut: Different data knowledge, channel knowledge, power constraints!

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201311Basic Multi-Cell Coordination StructureGeneral Multi-Cell CoordinationAdjacent base stations coordinate interferenceSome users served by multiple base stations

Dynamic Cooperation Clusters

Inner Circle : Serve users with dataOuter Circle : Suppress interferenceOutside Circles:Negligible impactImpractical to acquire informationDifficult to coordinate decisions

E. Bjrnson, N. Jaldn, M. Bengtsson, B. Ottersten, Optimality Properties, Distributed Strategies, and Measurement-Based Evaluation of Coordinated Multicell OFDMA Transmission, IEEE Trans. on Signal Processing, 2011.Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201312Example: Ideal Joint TransmissionAll Base Stations Serve All Users Jointly = One Super Cell

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201313Example: Wyner ModelAbstraction: User receives signals from own and neighboring base stations

One or Two Dimensional VersionsJoint Transmission or Coordination between Cells

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201314Example: Coordinated BeamformingOne Base Station Serves Each UserInterference Coordination Across Cells

Special Case

Interference channelBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201315Example: Soft-Cell CoordinationHeterogeneous DeploymentConventional macro BS overlaid by short-distance small BSsInterference coordination and joint transmission between layers

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201316Example: Cognitive RadioSecondary System Borrows Spectrum of Primary SystemUnderlay: Interference limits for primary users

Other Examples

Spectrum sharing between operators

Physical layer securityBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201317Resource Allocation: First DefinitionProblem Formulation (imprecise):Select beamforming to maximize system utilityMeans: Allocate power to users and in spatial dimensionsSatisfy: Physical, regulatory & economic constraints

Some Assumptions:Linear transmission and receptionPerfect synchronization (whenever needed)Flat-fading channels (e.g., using OFDM)

Perfect channel knowledgeIdeal transceiver hardwareCentralized optimization

Will be relaxed in Part 2Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201318Multi-Cell System Model

One System Model for Any Multi-Cell Scenario!

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201319Multi-Cell System Model: Dynamic Cooperation Clusters

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201320Multi-Cell System Model: Dynamic Cooperation Clusters (2)

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201321Multi-Cell System Model: Power ConstraintsWeighting matrix(Positive semi-definite)Limit(Positive scalar)

All at the same timeBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201322Multi-Cell System Model: Power Constraints (2)

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201323Introduction: How to Measure User Performance?Mean Square Error (MSE)Difference: transmitted and received signalEasy to AnalyzeFar from User Perspective?

Bit/Symbol Error Probability (BEP/SEP)Probability of error (for given data rate)Intuitive interpretationComplicated & ignores channel coding

Information RateBits per channel useMutual information: perfect and long codingAnyway closest to reality?

All improveswith SINR:

SignalInterference + NoiseBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201324Introduction: Generic Measure User Performance

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201325Section: IntroductionQuestions?Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201326SectionProblem FormulationBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201327Problem FormulationGeneral Formulation of Resource Allocation:

Multi-Objective Optimization ProblemGenerally impossible to maximize for all users!Must divide power and cause inter-user interference

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201328Performance Region

2-UserPerformanceRegionCare aboutuser 2Care aboutuser 1BalancebetweenusersPart of interest:Pareto boundaryPareto Boundary

Cannot improve for any user without degrading for other usersOther Names

Rate RegionCapacity RegionMSE Region, etc.Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201329Performance Region (2)Definitions of Pareto BoundaryStrong Pareto point: Improve for any user Degrade for other userWeak Pareto point: Cannot simultaneously improve for all users

Weak Definition is More ConvenientBoundary is compact and simply-connected

Optimality Condition 1

Sending one stream per user is sufficient (assumed earlier)Optimality Condition 2

At least one power constraint is active (=holds with equality)X. Shang, B. Chen, H. V. Poor, Multiuser MISO Interference Channels With Single-User Detection: Optimality of Beamforming and the Achievable Rate Region, IEEE Trans. on Information Theory, 2011.R. Mochaourab and E. Jorswieck, Optimal Beamforming in Interference Networks with Perfect Local Channel Information, IEEE Trans. on Signal Processing, 2011.Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201330Performance Region (3)Can the region have any shape?

No! Can prove that:Compact setNormal set

Upper corner in region, everything inside regionBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201331Performance Region (4)Some Possible Shapes

User-Coupling

Weak: ConvexStrong: ConcaveScheduling

Time-sharing for strongly coupled users

Select multiple pointsHard: Unknown regionUtopia point(Combine user points)

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201332Performance Region (5)Which Pareto Optimal Point to Choose?Tradeoff: Aggregate Performance vs. Fairness

PerformanceRegionUtilitarian point(Max sum performance)

Egalitarian point(Max fairness)

Single user point

Single user point

No Objective Answer

Utopia point outside of region

Only subjective answers exist!Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201333Section: Problem FormulationQuestions?Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201334SectionSubjective Resource AllocationBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201335Subjective ApproachSystem Designer Selects Utility FunctionDescribes subjective preferenceIncreasing and continuous function

Examples:

Sum performance:Proportional fairness:Harmonic mean:Max-min fairness:

Known as A Priori Approach

Select utility function before optimizationPut different weights to move betweenextremesBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201336Subjective Approach (2)

Symmetric region: Same pointAsymmetric region: Different points

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201337Subjective Approach (3)Utility Function gives Single-Objective Optimization Problem:

This is the Starting Point of Many ResearchersAlthough Selection of is Inherently SubjectiveAffects the Solvability

Pragmatic Approach

Try to Select Utility Function to Enable Efficient OptimizationBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201338Complexity of Single-Objective Optimization ProblemsClasses of Optimization ProblemsDifferent scaling with number of parameters and constraints

Main ClassesConvex: Polynomial time solutionMonotonic: Exponential time solutionArbitrary: More than exponential time

Approximations neededPractically solvable Hard to even approximateBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201339Complexity of Resource Allocation ProblemsWhat is a Convex Problem?Recall definitions:

Convex Function

For any two points on the graph of the function,the line between the points is above the graph

Examples:

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201340Complexity of Resource Allocation Problems (2)

Can be selected to be convexConvex powerconstraintsSINR constraints:Main complication!Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201341Classification of Resource Allocation ProblemsClassification of Three Important ProblemsThe Easy problemWeighted max-min fairnessWeighted sum performance

We will see: These have Different ComplexitiesDifficulty: Too many spatial degrees of freedom Convex problem only if search space is particularly limitedMonotonic problem in generalBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201342Complexity Example 1: The Easy Problem

Second order cone: ConvexM. Bengtsson, B. Ottersten, Optimal Downlink Beamforming Using Semidefinite Optimization, Proc. Allerton, 1999.A. Wiesel, Y. Eldar, and S. Shamai, Linear precoding via conic optimization for fixed MIMO receivers, IEEE Trans. on Signal Processing, 2006.W. Yu and T. Lan, Transmitter optimization for the multi-antenna downlink with per-antenna power constraints, IEEE Trans. on Signal Processing, 2007.E. Bjrnson, G. Zheng, M. Bengtsson, B. Ottersten, Robust Monotonic Optimization Framework for Multicell MISO Systems, IEEE Trans. on Signal Processing, 2012.

Total PowerConstraints

Per-AntennaConstraints

General Constraints

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201343Complexity Example 2: Max-Min Fairness

Solution is on this lineBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201344Complexity Example 2: Max-Min Fairness (2)Simple Line-Search: BisectionIteratively Solving Convex Problems (i.e., quasi-convex)

Find start intervalSolve the easy problem at midpointIf feasible: Remove lower halfElse: Remove upper halfIterate

Subproblem: Convex optimizationLine-search: Linear convergenceOne dimension (independ. #users)Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201345Complexity Example 2: Max-Min Fairness (3)Classification of Weighted Max-Min Fairness:Quasi-convex problem (belongs to convex class)Polynomial complexity in #users, #antennas, #constraintsMight be feasible complexity in practice

T.-L. Tung and K. Yao, Optimal downlink power-control design methodology for a mobile radio DS-CDMA system, in IEEE Workshop SIPS, 2002.M. Mohseni, R. Zhang, and J. Cioffi, Optimized transmission for fading multiple-access and broadcast channels with multiple antennas, IEEE Journal on Sel. Areas in Communications, 2006.A. Wiesel, Y. Eldar, and S. Shamai, Linear precoding via conic optimization for fixed MIMO receivers, IEEE Trans. on Signal Processing, 2006.E. Bjrnson, G. Zheng, M. Bengtsson, B. Ottersten, Robust Monotonic Optimization Framework for Multicell MISO Systems, IEEE Trans. on Signal Processing, 2012.

Early work

Main references

Channel uncertaintyBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201346Complexity Example 3: Weighted Sum Performance

Opt-value is unknown!Distance from origin is unknownLine Hyperplane (dim: #user 1)Harder than max-min fairnessNon-convex problemBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201347Complexity Example 3: Weighted Sum Performance (2)Classification of Weighted Sum Performance:Non-convex problemPower constraints: ConvexUtility: Monotonic increasing/decreasing in beamforming vectorsTherefore: Monotonic problem

Can There Be a Magic Algorithm?No, provably NP-hard (Non-deterministic Polynomial-time hard)Exponential complexity but in which parameters?(#users, #antennas, #constraints)

Z.-Q. Luo and S. Zhang, Dynamic spectrum management: Complexity and duality, IEEE Journal of Sel. Topics in Signal Processing, 2008.Y.-F. Liu, Y.-H. Dai, and Z.-Q. Luo, Coordinated beamforming for MISO interference channel: Complexity analysis and efficient algorithms, IEEE Trans. on Signal Processing, 2011.Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201348Complexity Example 3: Weighted Sum Performance (3)Are Monotonic Problems Impossible to Solve?No, not for small problems!Monotonic Optimization AlgorithmsImprove Lower/upper bounds on optimum:Continue untilSubproblem: Essentially weighted max-min fairness problem

H. Tuy, Monotonic optimization: Problems and solution approaches, SIAM Journal of Optimization, 2000.L. Qian, Y. Zhang, and J. Huang, MAPEL: Achieving global optimality for a non-convex wireless power control problem, IEEE Trans. on Wireless Commun., 2009.E. Jorswieck, E. Larsson, Monotonic Optimization Framework for the MISO Interference Channel, IEEE Trans. on Communications, 2010.W. Utschick and J. Brehmer, Monotonic optimization framework for coordinated beamforming in multicell networks, IEEE Trans. on Signal Processing, 2012.E. Bjrnson, G. Zheng, M. Bengtsson, B. Ottersten, Robust Monotonic Optimization Framework for Multicell MISO Systems, IEEE Trans. on Signal Processing, 2012.

Monotonicoptimization

Early works

PolyblockalgorithmBRBalgorithmBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201349Complexity Example 3: Weighted Sum Performance (4)

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201350Complexity Example 3: Weighted Sum Performance (5)Maximizing Sum Performance has High ComplexityOther ShortcomingsNot all Pareto Points are AttainableWeights have no Clear InterpretationNot Robust to Perturbations

What about other utility functions?Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201351Summary: Complexity of Resource Allocation ProblemsRecall: The SINR constraints are complicating factor

Three conditions that simplify:Fixed SINRs (easy problem)Allow no interference (zero-forcing)Multiplication Addition (change of variable, single antenna BSs)

SignalSINRInterferenceGeneralSum PerformanceNP-hardMax-Min FairnessQuasi-ConvexEasy ProblemConvexGeneralZero ForcingSingle AntennaSum PerformanceNP-hardConvexNP-hardMax-Min FairnessQuasi-ConvexQuasi-ConvexQuasi-ConvexEasy ProblemConvexConvexLinearProportional FairnessNP-hardConvexConvexHarmonic MeanNP-hardConvexConvexBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201352Summary: Complexity of Resource Allocation Problems (2)Recall: All Utility Functions are SubjectivePragmatic approach: Select to enable efficient optimizationGood Choice: Any Problem with Polynomial complexityExample: Weighted max-min fairnessUse weights to adapt to other system needs

Bad Choice: Weighted Sum PerformanceGenerally NP-hard: Exponential complexity (in #users)Should be avoided Sometimes needed (virtual queuing techniques)

GeneralZero ForcingSingle AntennaSum PerformanceConvexMax-Min FairnessQuasi-ConvexQuasi-ConvexQuasi-ConvexEasy ProblemConvexConvexLinearProportional FairnessConvexConvexHarmonic MeanConvexConvexBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201353Summary: Complexity of Resource Allocation Problems (3)Complexity Analysis for Any Dynamic Cooperation ClustersSame optimization algorithms!Extra characteristics can sometime simplifyMulti-antenna transmission: More complex, higher performance

Ideal Joint TransmissionCoordinated BeamformingUnderlay Cognitive RadioBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201354Section: Subjective Resource AllocationQuestions?Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201355SectionStructure of Optimal BeamformingBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201356Parametrization of Optimal BeamformingLagrange multipliers of Easy problem

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201357Parametrization of Optimal Beamforming (2)

E. A. Jorswieck, E. G. Larsson, D. Danev, Complete characterization of the Pareto boundary for the MISO interference channel, IEEE Trans. on Signal Processing, 2008.E. Bjrnson, M. Bengtsson, B. Ottersten, Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers, IEEE Trans. on Signal Processing, 2012.EarlyworkState-of-the-artTradeoffMaximize signal vs. minimize interferenceHard to find optimal tradeoffBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201358Application 1: Generate Performance RegionPerformance Region is Generally UnknownCompact and normalPerhaps non-convex

Approach 1:Vary parameters in parametrizationApproach 2:Maximize sequence of utilities ()

A Posteriori Approach

Look at region at select operating pointBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201359Application 2: Heuristic BeamformingParametrization: Foundation for Low-Complexity BeamformingSelect parameters heuristically

One Approach Many NamesSet all parameters to same value:

Ideal joint transmission:

Proposed many times (since 1995):Transmit Wiener/MMSE filter, Regularized zero-forcing,Signal-to-leakage beamforming, Virtual SINR beamforming, Virtual-uplink MVDR beamforming, etc.

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201360Application 3: Behavior at low/high SNRsRecall: ParametrizationAssume total power constraint

Low SNR:Inverse Identity matrix: Beamforming in channel directionName: Maximum ratio transmission (MRT)

High SNR:Inverse Project orthogonal to co-usersName: Zero-forcing beamforming (ZFBF)

Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201361Application 3: Behavior at low/high SNRs (2)Example: 4-User Interference ChannelMaximize sum information rate, 4 antennas/transmitter

Four Strategies:Optimal beamforming(BRB algorithm)Transmit Wiener filterZFBF, MRT

ObservationsMRT good at low SNRZFBF good at high SNRWiener filter always goodBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201362Section: Structure of Optimal BeamformingQuestions?Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201363Summary: Part 1Bjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201364SummaryMulti-Cell Multi-Antenna Resource AllocationDivide power between users and spatial directionsSolve a multi-objective optimization problemPareto boundary: Set of efficient solutionsSubjective Utility FunctionSelection has fundamental impact on solvabilityMulti-antenna transmission: More possibilities higher complexityPragmatic approach: Select to enable efficient optimizationPolynomial complexity: Weighted max-min fairness etc.Not solvable in practice: Weighted sum performance etc.Structure of Optimal BeamformingSimple parametrization balance signal and interferenceFoundation for low-complexity beamformingEasy to generate Pareto boundaryBjrnson & Jorswieck: Coordinated Multi-Cell Systems8 September 201365Coffee BreakThank you for listening!Questions?

After the Break, Part 2 Application:Robustness to channel uncertaintyDistributed resource allocationTransceiver hardware impairmentsMulti-cast transmissionMulti-carrier systemsMulti-antenna usersDesign of cooperation clustersCognitive radio systemsPhysical layer security