*p. c. weeraddana m. codreanu, m. latva-aho, ***a. ephremides * kth, royal institute of...

*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

19.04.23CWC | Centre For Wireless Communications 2

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

19.04.23CWC | Centre For Wireless Communications 3

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

19.04.23 CWC | Centre For Wireless Communications 4

Our contribution

04/19/23 CWC | Centre For Wireless Communications 5

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


each BS n carries out per BS optimizations

master problem objective

out-of-cell interference

Key Idea





upper bound for master objective;each coresponds to an approximated master problem in convex form; see (26)

Key Idea






A

subgradient method; BS-BS coordination

Key Idea






A


System model


: 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


: information symbol; ,

: power

signal vector transmitted by BS

: beamforming vector;

System model


signal received at

: channel; to

intra-cell interference


: cir. symm. complex Gaussian noise; variance

System model


received SINR of

: out-of-cell interference power; th BS to

intra-cell interference out-of-cell interference

System model


1 2

3

5

67

8


power, e.g.,

System model


received SINR of

: out-of-cell interference power; th BS to


System model


received SINR of


: set of out-of-cell interfering BSs that interferes

: out-of-cell interference power (complicating variables)

System model


1 2

3

5

67

8

e.g.,

System model


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


1 2

3

12

3 4

5

67

8

9

10 11

12



interference

e.g.,

System model


1 2

3

66

9 12



interference

e.g.,

Problem formulation


variables: and

Primal decomposition


variables:

subproblems (for all ) :

master problem:

variables:

Subproblem (BS optimization)


Subproblem


variables:

The problem above is NP-hard

Suboptimal methods, approximations

Subproblem: key idea


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



fixed

BS optimization:initialization

BS optimization:objective

approximation

BS optimization:GP

BS optimization:SOCP

YES

NO

Stopping criterionis satisfied ? STOP



fixed



approximation

BS optimization:GP


YES

NO




fixed

fixed



approximation

BS optimization:GP


YES

NO






approximation

BS optimization:GP


YES

NO




fixed

fixed



approximation

BS optimization:GP


YES

NO


Master problem


Master problem


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


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


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







A




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



approximation

BS optimization:GP


YES

NO



Integrate master problem & subproblemeach BS n carries out per BS optimizations




A


subgradient method:



Algorithm 1 (subproblem)

Algorithm 2 (overall problem)



approximation

BS optimization:GP


YES

NO


An example signaling frame structure


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


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


-> 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


= 50

distributed alg. [Emil-10]

22% subg.Alg.

subg.Alg.

Alg. 1 Alg. 1


Numerical Exampleseach BS n carries out per BS optimizations




A


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


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]


subgrad = 1


= 50


23%

Numerical Examples


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]


subgrad = 1


= 10


= 50


Numerical Examples


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]


subgrad = 1


= 50


Numerical Examples


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)


=2)

centralized alg. [Codreanu-07]distributed alg. [Emil-10]

Numerical Examples


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)


=1.5)


=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


*p. c. weeraddana m. codreanu, m. latva-aho, ***a. ephremides * kth, royal institute of...

Documents

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key idea

general wireless network

data streams of bs

cell interferencesystem

cell interference power

cell interference couplingp

*p. c. weeraddana **m. codreanu, **m. latva-aho, ***a. ephremides * kth, royal institute of...

*p. c. weeraddana m. codreanu, m. latva-aho, ***a. ephremides * kth, royal institute of...