Supélec, 2012-02-02
Optimization Concepts for
Resource Allocation in
Cellular Systems
Emil Björnson
PhD in Telecommunications
Signal Processing Lab
KTH Royal Institute of Technology
KTH in Stockholm
KTH was founded in 1827 and is
the largest of Sweden’s technical
universities.
Since 1917, activities have been
housed in central Stockholm, in
beautiful buildings which today
have the status of historical
monuments.
KTH is located on five campuses.
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A top European grant-earning university
• Europe’s most successful university in terms of
earning European Research Council Advanced Grant
funding for ”investigator-driven frontier research”
5 research projects awarded in 2008:
• Open silicon-based research platform for emerging devices
• Astrophysical Dynamos
• Atomic-Level Physics of Advanced Materials
• Agile MIMO Systems for Communications, Biomedicine, and
Defense
• Approximation of NP-hard optimization problems
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Emil Björnson
• 1983: Born in Malmö, Sweden
• 2007: Master of Science in
Engineering Mathematics,
Lund University, Sweden
• 2011-11-17: Defended doctoral
thesis in telecommunications,
KTH, Sweden
Multiantenna Cellular Communications
Channel Estimation, Feedback, and Resource Allocation
– Three Building Blocks of Physical Layer
– Mathematical Analysis and Optimization
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Background
• Cellular Communications
- Many transmitting multi-antenna base stations
- Many receiving single-antenna users
• Downlink Transmission
- Multiple transmit antennas – exploit spatial dimension
- Multiuser transmission
Pro: Higher performance, Con: Co-user interference
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Background (2): Multiple Cells
• Uncoordinated Cells:
+
+
−
−
Simple processing
Simple infrastructure
Uncontrolled interference
Or fractional frequency reuse
• Coordinated Cells:
+
−
−
−
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Controlled interference
Backhaul signaling
Computationally complex
Tight synchronization
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Background (3): Multiple Cells
• Dynamic User-Centric Coordination Clusters
- Inner Circle (Strong Channels):
Consider Transmitting to Users
- Outer Circle (Non-negligible Channels):
Avoid Interference to Users
- Can model any level of coordination
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Background (4)
• Resource Allocation
- Select users for transmission
- Design beamforming directions
- Allocate transmit power
• Optimize Resource Allocation
-
Maximize system performance
Satisfy system constraints (power, interference, fairness)
Any level of coordinated multipoint transmission
Robustness to uncertain channel information
• No mathematical details
- Focus on performance optimization concepts
- Assumption: Linear transmit/receive processing
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Outline
• What is Performance?
- Different user performance measures
- System Performance vs. user fairness
• Multi-user Performance Region
- How to interpret?
- How to choose operating point?
• Performance Optimization
- Geometrical interpretation of common formulations
- Right problem formulation = Easy to solve
• Low-complexity Strategies
- Exploit structure from optimal solution
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What is Performance?
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What is Performance?
• Service Quality
- Experienced by users (per-user level)
- Can also be measured at system-level
• Performance Based on
-
Average data rate
Latency
Coverage
Battery life
Etc.
• Simplified Performance Measures
- Necessary for optimization
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Single-user Performance Measures
• Mean Square Error (MSE)
- Difference: transmitted and received signal
- Easy to analyze
- Far from user perspective?
All improves
with SNR:
• Bit/Symbol Error Rate (BER/SER)
- Probability of error (for given data rate)
- Intuitive interpretation
- Complicated & ignores channel coding
Signal Power
Noise Power
Optimize
SNR instead!
• Data Rate
- Bits per ”channel use”
- Mutual information: perfect and long coding
- Still closest to reality?
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Multi-user Performance
• User Performance Measures
- Same measures – but one per user
• Performance Limitations
- Power Allocation
- Co-user interference:
SINR=
Signal Power
Interference + Noise Power
• Why Not Increase Power?
- Power = Money & Environmental Impact
- Reduce noise  Interference limited
• User Fairness
- New dimension of difficulty
- Heterogeneous user conditions
- Depends on performance measure
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Multi-user Performance Region
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Multi-user Performance Region
• Achievable Performance Region – 2 users
Performance
User 2
- Under power budget
Care about
user 2
Balance
between
users
Part of interest:
Upper boundary
Performance
Region
Care about
user 1
Performance User 1
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Multi-user Performance Region (3)
• Can it have any shape?
• No! Can prove that:
- Compact set
- Simply connected (No holes)
- Nice upper boundary
Normal set
Upper corner in region,
everything inside region
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Multi-user Performance Region (3)
• Possible Shapes of Region
- Convex, concave, or neither
- In general: Non-convex
- In any case: Region is unknown
Convex
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Concave
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Non-convex
Non-concave
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Multi-user Performance Region (3)
• Some Operating Points – Game Theory Names
Performance
User 2
Utilitarian point
(Max sum performance)
Egalitarian point
(Max fairness)
Single
user
point
Performance
Region
Which point
to choose?
Optimize:
Sum Performance?
Fairness?
Single user point
Performance User 1
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Performance Optimization
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System Performance versus Fairness
• Always Sacrifice Either
- Sum Performance
- User Fairness
- Or both: optimize something in between
• Two Standard Optimization Strategies
Starts from
Performance
Starts from
Fairness
- Maximize weighted sum performance:
maximize w1·R1 + w2·R2 + …
R1,R2,…
(w1 + w2+… = 1)
- Maximize performance with fairness-profile:
maximize Rsum
Rsum
subject to R1=a1·Rsum, R2=a2·Rsum, …
(a1 + a2+… = 1)
• Non-Convex Optimization Problems
- Generally hard to solve numerically
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The “Easy” Problem
• Given Point (R1,R2,…)
- Find transmit strategy that attains this point
- Minimize power usage
• Convex Problem
- Second-order cone or semi-definite program
- Global solution in polynomial time – use CVX, Yalmip
Single-cell
(total power)
• M. 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. Signal Processing, 2006.
Single-cell
(per ant. power)
• W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink
with per-antenna power constraints,” IEEE Trans. Signal Process., 2007.
Multi-cell
(general power,
robustness)
• E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic
Optimization Framework for Multicell MISO Systems,” IEEE Transactions on
Signal Processing, To appear.
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Exploiting the “Easy” Problem
• Easy: Achieve a Given Point
• Hard: Find a Good Point
Main part of
resource
allocation
• Shape of Performance Region
- Far from obvious – one dimension per user
Interference
Channel
Rate:
user 1
3 transmitters
w. 4 antennas
3 users
Rate: user 2
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Rate: user 3
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Geometric Optimization Interpretations
• Maximize Performance with Fairness Profile:
maximize Rsum
Rsum
subject to R1=a1·Rsum, R2=a2·Rsum, …
(a1 + a2+… = 1)
• Geometric Interpretation
- Search on line in direction (a1,a2,…) from origin
(a1,a2,…)·Rsum =(a1·Rsum,a2·Rsum,…)
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Geometric Optimization Interpretations (2)
• Simple line-search algorithm: Bisection
- Non-convex  Iterative convex (Quasi-convex)
1. Find start interval
2. Solve the “easy” problem
at midpoint
3. If feasible:
Remove lower half
Else: Remove upper half
4. Iterate
Subproblem: Convex optimization
Line-search: Linear convergence
One dimension (independ. #users)
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Geometric Optimization Interpretations (3)
• Maximize weighted sum performance:
maximize w1·R1 + w2·R2 + …
R1,R2,…
(w1 + w2+… = 1)
• Geometric interpretation
- Search on line w1·R1 + w2·R2 = max-value
Max-value is unknown!
-
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Distance from origin unknown
Line  hyperplane (dim: #user – 1)
Harder than fairness-profile problem!
Iterative search algorithm?
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Geometric Optimization Interpretations (4)
• Systematic Search Algorithm
- Concentrate on important parts of performance region
- Improve lower/upper bounds on optimum:
- Continue until
• Efficiently Solvable Subproblems
- Based on Fairness-profile problem
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Geometric Optimization Interpretations (5)
• Branch-Reduce-Bound (BRB) Algorithm
1.
2.
3.
4.
Cover region with a box
Divide the box into two sub-boxes
Remove parts with no solutions in
Search for solutions to improve bounds
(Based on Fairness-profile problem)
5. Continue with sub-box with largest value
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Geometric Optimization Interpretations (6)
Properties
- Global Convergence
- Accuracy ε>0 in finitely
many iterations
- Exponential complexity
only in #users
- Polynomial complexity
in other parameters
(#antennas/constraints)
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Geometric Optimization: Conclusions
• Fairness-Profile Approach:
Easy
- Quasi-Convex: Polynomial complexity
- Reason: Only one search dimension
• Weighted Sum Performance:
Difficult
- NP-hard: Exponential complexity (in #users)
- Reason: Optimizes both performance and fairness
• Every Weighted Sum = Some Fairness-Profile
- Easier to solve when posed as fairness-profile problem
- Parameter relationship non-obvious
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Geometric Optimization: References
• Line-Search Algorithm for Fairness-Profiles
• M. Mohseni, R. Zhang, and J. Cioffi, “Optimized transmission for fading
multiple-access and broadcast channels with multiple antennas,” IEEE J. Sel.
Areas Commun., vol. 24, no. 8, pp. 1627–1639, 2006.
• J. Lee and N. Jindal, “Symmetric capacity of MIMO downlink channels,” in
Proc. IEEE ISIT’06, 2006, pp. 1031–1035.
• E. Björnson, M. Bengtsson, and B. Ottersten, “Pareto Characterization of the
Multicell MIMO Performance Region With Simple Receivers,” IEEE Trans. on
Signal Processing, Submitted, 2011.
• BRB Algorithm
- Useful for more than weighted sum performance
- E.g. arithmetic, geometric, or harmonic mean performance
• H. Tuy, F. Al-Khayyal, and P. Thach, “Monotonic optimization: Branch and cut
methods,” Essays and Surveys in Global Optimization, Springer, 2005.
• E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic
Optimization Framework for Multicell MISO Systems,” IEEE Transactions on
Signal Processing, To appear.
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Low-complexity Strategies
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Low-complexity Strategies
• Hardware Limitations
- Polynomial complexity: Only slowly-varying channels
- Exponential complexity: Only suitable for benchmarking
• Heuristic Resource Allocation
- Find reasonable strategy with little effort
- Exploit available insight the optimal structure
• Parametrization of Upper Boundary
1. Select parameters in [0,1]
2. Get an strategy explicitly
- Can achieve any point on upper boundary
- Only necessary condition
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Low-complexity Strategies (2)
• Method 1: Interference-temperature Control
- Transmitters x (Receivers – 1) parameters
• X. Shang, B. Chen, and H. V. Poor, “Multi-user MISO interference channels
with single-user detection: Optimality of beamforming and the achievable rate
region,” IEEE Trans. Inf. Theory, 2011.
• R. Mochaourab, E. Jorswieck, “Optimal Beamforming in Interference Networks
with Perfect Local Channel Information,” IEEE Trans. Signal Processing, 2011.
• Method 2: Exploit Solution Structure of “Easy” Problem
-
Explicit strategy given by optimal Lagrange multipliers
Always same structure, but different parameters
Take Lagrange multipliers as our parameters!
Transmitters + Receivers – 1 parameters
• E. Björnson, M. Bengtsson, and B. Ottersten, “Pareto Characterization of the
Multicell MIMO Performance Region With Simple Receivers,” IEEE Trans. Signal
Processing, Submitted, 2011.
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Low-complexity Strategies (3)
• Number of Parameters
- Large difference for large problems
100
80
60
Method 1
40
Method 2
20
0
2
3
4
5
6
7
8
9
10
Number of Transmitters/Receivers
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Low-complexity Strategies (4)
• Parameterized Structure of Resource Allocation
- Beamforming directions maximize:
Signal Power at User 𝑘
µ𝑗 · Noise Power + 𝑙 λ𝑙 · Interference Power to User 𝑙
- Parameter λ𝑙 = Priority of User 𝑙
- Parameter µ𝑗 = Based on transmit power of Base station 𝑗
• Heuristic Selection:
- All Parameters = 1
- Called: Signal-to-leakage+noise ratio (SLNR) beamforming
- Known to work well!
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Example
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Example – Multicell Scenario
• Maximize Weighted Sum Rate
- Two base stations: 20 dBm output power
- Full, Partial, or No coordination
BRB algorithm
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Heuristic Parameters (=1)
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Summary
• Easy to Measure Single-user Performance
• Multi-user Performance Measures
- Sum performance vs. user fairness
• Performance Region
- All combinations of user performance
- Upper boundary: All efficient outcomes
- Explicit Parametrization: Low-complexity strategies
• Two Standard Optimization Strategies
- Maximize weighted sum performance
• Difficult to solve (optimally – heuristic approx. exists)
- Maximize performance with fairness profile
• Easy to solve (with line-search algorithm)
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Thank You for Listening!
Questions?
Papers and Presentations Available:
http://www.ee.kth.se/~emilbjo
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