* p. c. weeraddana **m. codreanu, **m. latva-aho, ***a. ephremides
DESCRIPTION
Multicell MISO Downlink Weighted Sum-Rate Maximization: A Distributed Approach. * 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. - PowerPoint PPT PresentationTRANSCRIPT
*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-RateMaximization: A Distributed Approach
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
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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
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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
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Our contribution
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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
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
Key Idea
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each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
System model
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: 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
12
3 45
67
8
9
10 11
12
System model
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: information symbol; ,
: power
signal vector transmitted by BS
: beamforming vector;
System model
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signal received at
: channel; to
intra-cell interference
out-of-cell interference
: cir. symm. complex Gaussian noise; variance
System model
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received SINR of
: out-of-cell interference power; th BS to
intra-cell interference out-of-cell interference
System model
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1 2
3
5
67
8
out-of-cell interference
power, e.g.,
System model
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1 2
3
5
67
8
out-of-cell interference
power, e.g.,
System model
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1 2
3
5
67
8
out-of-cell interference
power, e.g.,
System model
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1 2
3
5
67
8
out-of-cell interference
power, e.g.,
System model
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1 2
3
5
67
8
out-of-cell interference
power, e.g.,
System model
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received SINR of
: out-of-cell interference power; th BS to
intra-cell interference out-of-cell interference
System model
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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)
System model
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1 2
3
12
3 4
5
67
8
9
10 11
12
: set of d.s. that are
subject to out-of-cell
interference
e.g.,
System model
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1 2
3
12
3 4
5
67
8
9
10 11
12
: set of d.s. that are
subject to out-of-cell
interference
e.g.,
System model
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1 2
3
66
9 12
: set of d.s. that are
subject to out-of-cell
interference
e.g.,
Primal decomposition
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variables:
subproblems (for all ) :
master problem:
variables:
Subproblem
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variables:
The problem above is NP-hard
Suboptimal methods, approximations
Subproblem: key idea
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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
Subproblem: key idea
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fixed
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Subproblem: key idea
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fixed
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Subproblem: key idea
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fixed
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Subproblem: key idea
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fixed
fixed
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Subproblem: key idea
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BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Subproblem: key idea
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fixed
fixed
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Subproblem: key idea
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fixed
fixed
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
Master problem
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variables:
Recall: subproblems are NP-hard -> we cannot even compute the master objective value
Suboptimal methods, approximations
Problem is nonconvex -> subgradient method alone fails
Master problem: key idea
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The method is inspired from sequential convex approximation (upper bound) techniques
subgradient method is adopted to solve the resulting convex problems
Integrate master problem & subproblem
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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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Integrate master problem & subproblem
each BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
Integrate master problem & subproblem
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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
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
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Integrate master problem & subproblemeach BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
subgradient method:
Integrate master problem & subproblem
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Algorithm 1 (subproblem)
Algorithm 2 (overall problem)
BS optimization:initialization
BS optimization:objective
approximation
BS optimization:GP
BS optimization:SOCP
YES
NO
Stopping criterionis satisfied ? STOP
An example signaling frame structure
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Algorithm 2 (overall problem)
BS coordination windows:time periods for which per BS stopping criterion is satisfied
signalling phase
data transmission phase
1 2
BS optimization windows:time periods for which per BS stopping criterion is not satisfied
3
Initial signalling window: used for Algorithm 2 initialization
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
Numerical Examples
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channel gains:
: distance from to : small scale fading coefficients
-10 -5 0 5 10 15 20 25
-10
-5
0
5
10
X-Coordinate [distance unit]
Y-C
oo
rdin
ate
[dis
tan
ce u
nit
]
1
2
3
4
5
6
7
8
1 2
-10 -5 0 5 10 15 20 25
-10
-5
0
5
10
15
20
X-Coordinate [distance unit]
Y-C
oo
rdin
ate
[dis
tan
ce u
nit
]
1
2
3
4
56, 7
8
9
101112
12
3
SNR operating point:
Numerical Examples
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-> 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
1 2
20 40 60 80 100 1208
9
10
11
12
13
14
15
(Algorithm 1 iterations + subgradient iterations)
WS
R v
alue
[bi
ts/s
ec/H
z]
WMMSE [Luo-11]centralized alg. [Codreanu-07]proposed Alg. 2, J
subgrad = 1
proposed Alg. 2, Jsubgrad
= 10
proposed Alg. 2, Jsubgrad
= 50
distributed alg. [Emil-10]
22% subg.Alg.
subg.Alg.
Alg. 1 Alg. 1
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Numerical Exampleseach BS n carries out per BS optimizations
master problem objective
out-of-cell interference
upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)
A
subgradient method; BS-BS coordination
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
Numerical Examples
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same behavior
smaller performs better compared to large
12
3
20 40 60 80 100 12018
20
22
24
26
28
30
32
(Algorithm 1 iterations + subgradient iterations )
WS
R v
alue
[bi
ts/s
ec/H
z]
WMMSE [Luo-11]centralized alg. [Codreanu-07]proposed Alg. 2, J
subgrad = 1
proposed Alg. 2, Jsubgrad
= 50
distributed alg. [Emil-10]
23%
Numerical Examples
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algorithm performs better than the centralized algorithm
not surprising since both algorithms are suboptimal algorithms
1 2
20 40 60 80 100 1208
9
10
11
12
13
14
(Algorithm 1 iterations + subgradient iterations )
WS
R v
alue
[bi
ts/s
ec/H
z]
WMMSE [Luo-11]centralized alg. [Codreanu-07]proposed Alg. 2, J
subgrad = 1
proposed Alg. 2, Jsubgrad
= 10
proposed Alg. 2, Jsubgrad
= 50
distributed alg. [Emil-10]
Numerical Examples
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12
3
algorithm performs better than the centralized algorithm
not surprising since both algorithms are suboptimal algorithms
20 40 60 80 100 12016
18
20
22
24
26
28
30
(Algorithm 1 iterations + subgradient iterations)
WS
R v
alue
[bi
ts/s
ec/H
z]
WMMSE [Luo-11]centralized alg. [Codreanu-07]proposed Alg. 2, J
subgrad = 1
proposed Alg. 2, Jsubgrad
= 50
distributed alg. [Emil-10]
Numerical Examples
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1 2
; 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
0 2 4 6 8 108
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
subgradient iterations (for Alg. 2) / iterations (for WMMSE alg.)
Ave
rage
WS
R v
alue
[bi
ts/s
ec/H
z]
WMMSE [Luo-11]proposed Alg. 2WMMSE [Luo]: 1.0% error in cov. estimation (i.e., e
cov=1)
WMMSE [Luo]: 1.5% error in cov. estimation (i.e., ecov
=1.5)
WMMSE [Luo]: 2.0% error in cov. estimation (i.e., ecov
=2)
centralized alg. [Codreanu-07]distributed alg. [Emil-10]
Numerical Examples
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12
3
; 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
0 2 4 6 8 10
18
20
22
24
26
28
30
subgradient iterations (for Alg. 2) / iterations (for WMMSE alg.)
Ave
rage W
SR
valu
e [
bits
/sec/
Hz]
WMMSE [Luo-11]proposed Alg. 2WMMSE [Luo]: 1.0% error in cov. estimation (i.e., e
cov=1)
WMMSE [Luo]: 1.5% error in cov. estimation (i.e., ecov
=1.5)
WMMSE [Luo]: 2.0% error in cov. estimation (i.e., ecov
=2)
centralized alg. [Codreanu-07]distributed alg. [Emil-10]
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
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