Numerical comparison of decomposition algorithms for non-convex
sum-utility problems
Peiran Song
Jan. 11, 2013
This technical report is a supporting document to the paper
G. Scutari, F. Francisco, P. Song, D. P. Palomar and J. -S. Pang, “Decomposition by partial linearization: parallel
optimization of multi-agent systems”
In the following we will use the same notation as introduced in the above paper.
In this report we provide a detailed comparison of the following algorithms SJBR[1], SCALE (SCALE one step)[2],
WMMSE[3], and MDP[4] under different settings and scenarios. We start focusing on frequency-selective channels,
see Sec. 1; then we consider MIMO flat-fading channels, see Sec. 2.
1
Sum-Rate Maximization over SISO ICs
We compare the performance of SJBR[1], SCALE (SCALE one step)[2], WMMSE[3], and MDP[4] algorithms, in
terms of convergence speed and achievable sum-rate.
1.1
Numerical simulation setup
We simulated SISO frequency channels with N = 64 subcarriers; the channels are generated as FIR filters of order
L = 10, whose taps are i.i.d. Gaussian random variables with zero mean and variance 1/d3ij , where dij is the
distance between the transmitter j and the receiver i. For the sake of simplicity, we set dij = dji and dii = djj
for all i and j 6= i. We compare the aforementioned algorithms in both low interference (d , dij /dii = 3) and
high interference (d = 1) scenarios. We assume that all the users have the same power budget P , noise variance
σ 2 , and SN R = P/σ 2 = 3dB. All the algorithms are initialized by choosing the uniform power allocation, and
are terminated when (the absolute value) of the sum-rate error in two consecutive rounds becomes smaller than
a given accuracy criterion. Here we consider two different termination accuracies, namely, 10−3 and 10−6 . In all
the algorithms under consideration there is a bisection loop; the accuracy in the bisection loops is set to be one
magnitude higher than the termination accuracy (i.e. 10−4 and 10−7 , respectively). For the proposed algorithm,
the SJBR algorithm, we use the step size rule (Rule#1 [1, eq. (19)])
γ n = γ n−1 1 − γ n−1 , n = 1, 2, · · ·
(1)
where is set to 10−2 . The average is taken over 100 independent channel realizations.
1.2
Figures
In this subsection, we show the performance of the aforementioned algorithms for different network sizes, i.e.
Number of users equals to 5, 10, 15, 30, 50, 75, 100.
1
1.2.1
Termination accuracy 10−3
• d = 3 (Low interference scenario)
Convergence Speed (linear scale)
Convergence Speed (log scale)
4
90
80
70
Iterations
10
1400
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
2
10
Sum Rate (nats/channel use)
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
3
Iterations
Achievable Sum Rate
100
10
60
50
40
30
1
10
20
1200
1000
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
800
600
400
200
10
0
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
Number of Users
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
• d = 1 (High interfernce scenario)
Convergence Speed (log scale)
4
Achievable Sum Rate
Convergence Speed (linear scale)
400
200
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
180
160
140
3
Iteratons
Iterations
10
Sum Rate (nats/channel use)
10
120
100
80
2
10
60
40
350
300
250
200
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
150
100
20
1
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
1.2.2
50
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
Number of Users
Termination accuracy 10−6
• d = 3 (Low interference scenario)
Convergence Speed (log scale)
Convergence Speed (linear scale)
1400
350
1200
300
2
Iterations
Iterations
10
1
250
200
150
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
10
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
100
50
0
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Achievable Sum Rate
400
Sum Rate (nats/channel use)
3
10
800
600
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
400
200
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
1000
Number of Users
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
• d = 1 (High interference scenario)
Convergence Speed (log scale)
Convergence Speed (linear scale)
500
3
400
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
Iterations
400
Iterations
10
2
10
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
1
10
Achievable Sum Rate
600
Sum Rate (nats/channel use)
4
10
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
300
200
100
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
2
350
300
250
200
MDP
SJBR
WMMSE
SCALE
SCALE (one step)
150
100
50
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
1.2.3
Conclusions
The ’Sum-rate vs Number of Users’ figures show that all the algorithms reach the similar averaged sum-rate in
all scenarios and under different termination criteria. But the ’Iterations vs Number of Users’ figures clearly show
that the proposed algorithm, the SJBR, is much faster than all the others. The iteration gap between SJBR and
all the other algorithms, ranges from three times to one order of magnitude. Moreover, the higher the termination
accuracy the larger the gap. Note that SJBR, WMMSE, SCALE and SCALE one step are simultaneous-based
schemes, while MDP is a sequential algorithm.
1.2.4
A further test on iterations number
The figures above suggest that the iterations needed for SJBR algorithm to converge do not increase significantly
as the number of users increases. To further test this property, we simulated in the next plots where the Number
of Users are chosen from 5 up to 200.
Termination accuracy 10−3
• d = 3 (Low interference scenario)
Convergence Speed
80
Achievable Sum Rate
2500
70
Sum Rate (nats/channel use)
SJBR
WMMSE
SCALE (one step)
Iterations
60
50
40
30
20
10
0
20
40
60
80
100
120
140
160
180
SJBR
WMMSE
SCALE (one step)
2000
1500
1000
500
0
200
20
40
60
Number of Users
80
100
120
140
160
180
200
160
180
200
160
180
200
Number of Users
• d = 1 ( High interference scenario)
Convergence Speed
Achievable Sum Rate
220
450
SJBR
WMMSE
SCALE (one step)
180
400
Sum Rate (nats/channel use)
200
Iterations
160
140
120
100
80
60
350
300
250
200
150
100
40
20
SJBR
WMMSE
SCALE (one step)
20
40
60
80
100
120
140
160
180
50
200
20
40
60
Number of Users
80
100
120
140
Number of Users
Termination accuracy 10−6
• d = 3 (Low interference scenario)
Convergence Speed
Achievable Sum Rate
2500
450
400
Sum Rate (nats/channel use)
SJBR
WMMSE
SCALE (one step)
350
Iterations
300
250
200
150
100
SJBR
WMMSE
SCALE (one step)
2000
1500
1000
500
50
0
20
40
60
80
100
120
140
160
180
200
Number of Users
0
20
40
60
80
100
120
Number of Users
3
140
• d = 1 (High interference scenario)
Convergence Speed
Achievable Sum Rate
700
450
SJBR
WMMSE
SCALE (one step)
400
Sum Rate (nats/channel use)
600
Iterations
500
400
300
200
100
0
350
300
250
200
SJBR
WMMSE
SCALE (one step)
150
100
20
40
60
80
100
120
140
160
180
200
50
20
40
Number of Users
60
80
100
120
140
160
180
200
Number of Users
The above figures show that indeed the iterations number of SJBR algorithm does not increase significantly when
the network size increases.
2
Sum-Rate Maximization over MIMO ICs
We focus now on a MIMO scenario and compare the performance of the MIMO SJBR[1], WMMSE[5], and MDP[6].
The setup we simulated is described next.
2.1
Numerical simulation setup
We simulated uncorrelated fading channel model, where the coefficients are Gaussian distributed with zero mean
and variance 1/d3ij (dij = dji and dii = djj for all i and j 6= i). All the transmitters/receivers are equipped with 4
antennas. We compare the MIMO algorithms in three scenarios: the normalized distance d , dij /dii = 3, d = 2,
and d = 1. In the simulations, we assume that all the users have the same power budget P , noise covariance matrix
Rni = σ 2 I, and SN R = P/σ 2 = 3dB. All the algorithms are initialized by choosing a uniform power allocation
among the antennas, and are terminated when (the absolute value) of the sum-rate error in two consecutive rounds
becomes smaller than a given accuracy. Similarly to the SISO case, we consider two different termination accuracies
namely 10−3 , 10−6 . The accuracy in the bisection loops is one magnitude higher than the termination accuracy
(i.e. 10−4 and 10−7 , respectively). In SJBR algorithm we use the step size rule (1) with = 10−5 . The average is
taken over 100 independent channel realizations.
2.2
Figures
The figures in this subsection show the performance of the aforementioned MIMO algorithms in seven network sizes
(Number of users equals to 5, 10, 15, 30, 50, 75, 100), under different normalized distances and accuracies.
2.2.1
Termination accuracy 10−3
• d = 3 (Low interference scenario)
Convergence Speed (log scale)
Convergence Speed (linear scale)
140
40
MDP
SJBR
WMMSE
35
2
10
Iterations
Iterations
30
1
10
25
20
15
10
5
0
10
Achievable Sum Rate
45
MDP
SJBR
WMMSE
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
4
Sum Rate (nats/channels use)
3
10
120
MDP
SJBR
WMMSE
100
80
60
40
20
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
• d = 2 (Medium interference scenario)
Convergence Speed (log scale)
3
Convergence Speed (linear scale)
Achievable Sum Rate
50
65
45
MDP
SJBR
WMMSE
40
35
2
Iterations
Iterations
10
30
25
20
1
10
15
10
MDP
SJBR
WMMSE
5
0
10
60
Sum Rate (nats/channel use)
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
55
50
45
40
35
30
MDP
SJBR
WMMSE
25
20
15
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
Number of Users
• d = 1 (High interference scenario)
Convergence Speed (log scale)
4
Convergence Speed (linear scale)
Achievable Sum Rate
200
22
180
MDP
SJBR
WMMSE
MDP
SJBR
WMMSE
160
140
3
Iterations
Iterations
10
120
100
80
2
10
60
40
21
Sum Rate (nats/channel use)
10
20
1
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
19
18
17
16
15
MDP
SJBR
WMMSE
14
13
12
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
2.2.2
20
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
Number of Users
Termination accuracy 10−6
• d = 3 (Low interference scenario)
Convergence Speed (linear scale)
Convergence Speed (log scale)
3
10
140
180
MDP
SJBR
WMMSE
MDP
SJBR
WMMSE
160
Iterations
140
Iterations
Achievable Sum Rate
200
2
10
120
100
80
60
1
10
40
Sum Rate (nats/channel use)
4
10
120
100
80
60
40
MDP
SJBR
WMMSE
20
20
0
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
Number of Users
• d = 2 (Medium interference scenario)
Convergence Speed (log scale)
Convergence Speed (linear scale)
MDP
SJBR
WMMSE
65
180
MDP
SJBR
WMMSE
160
140
3
Iterations
Iterations
10
120
100
80
2
10
60
40
20
1
10
Achievable Sum Rate
200
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
0
60
Sum Rate (nats/channel use)
4
10
55
MDP
SJBR
WMMSE
50
45
40
35
30
25
20
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
5
15
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
• d = 1 (High interference scenario)
Convergence Speed (log scale)
4
Convergence Speed (linear scale)
Achievable Sum Rate
500
MDP
SJBR
WMMSE
22
MDP
SJBR
WMMSE
450
400
21
Sum Rate (nats/channel use)
10
Iterations
Iterations
350
3
10
300
250
200
150
100
50
2
10
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
19
18
17
16
15
MDP
SJBR
WMMSE
14
13
12
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
2.2.3
20
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Nuser of Users
Nuser of Users
Conclusions
The figures show that all the algorithms reach almost the same averaged sum-rate in all simulated scenarios. As in
the SISO case, in the MIMO setting, the proposed algorithm, SJBR, is faster than all the others; depending on the
specific scenario (and termination accuarcy) the iteration gap ranges from few iterations to one order of magnitude.
We recall here that SJBR and WMMSE are simultaneous-based schemes, whereas MDP is a sequential algorithm.
2.2.4
A further test on iterations number
We plot here similar curves as before, but increasing the network size till 200 users. All algorithms reach the same
sum rate (and thus the Sum Rate figures are omitted). The curves show that the convergence speed of the proposed
algorithm, the SJBR, does not increase significantly with respect to the number of users.
Termination accuracy 10−3
d=3 (Low interference scenario)
d=2 (Medium interference scenario)
100
90
100
90
SJBR
WMMSE
80
80
70
70
60
60
60
40
Iterations
70
50
50
40
50
40
30
30
30
20
20
20
10
10
0
20
40
60
80
100
120
140
160
180
0
200
SJBR
WMMSE
90
SJBR
WMMSE
80
Iterations
Iterations
d=1 (High interference scenario)
100
10
20
40
60
Number of Users
80
100
120
140
160
180
0
200
20
40
60
Number of Users
80
100
120
140
160
180
200
Number of Users
Termination accuracy 10−6
d=3 (Low interference scenario)
d=2 (Medium interference scenario)
200
180
200
SJBR
WMMSE
180
SJBR
WMMSE
180
160
140
140
140
120
120
120
100
80
Iterations
160
Iterations
160
Iterations
d=1 (High interference scenario)
200
100
80
100
80
60
60
60
40
40
40
20
20
20
0
20
40
60
80
100
120
Number of Users
140
160
180
200
0
20
40
60
80
100
120
Number of Users
6
140
160
180
200
0
SJBR
WMMSE
20
40
60
80
100
120
Number of Users
140
160
180
200
2.3
Choices of step-size rules and parameters for SJBR in MIMO ICs
The convergence of SJBR algorithm with a diminishing step size is guaranteed, if the step-size rule satisfies the
conditions in Theorem 3 [1]. Many step-size rules satisfy such conditions. A natural question is then to understand
how the performance of the SJBR are affected by the choice of the step-size rule, which is the goal of this section.
More specifically, here we compare the performance of SJBR algorithm using the following feasible diminishing
step-size rules
• Step-size rule 1: γ n = γ n−1 1 − γ n−1 , ∈ (0, 1) , n = 1, 2, · · ·
• Step-size rule 2: γ n =
γ n−1 + α
, α, β ∈ (0, 1) , α ≤ β, n = 1, 2, · · ·
1 + βn
• Step-size rule 3: γ n =
γ n−1 + α log (n)
√
, α, β ∈ (0, 1) , α ≤ β, n = 1, 2, · · ·
1+β n
Termination accuracy 10−3
2.3.1
d=1 (Low interference scenario)
d=1 (High interference scenario)
d=2 (Medium interference scenario)
15
100
35
Step−size rule 1
Step−size rule 2
Step−size rule 3
Step−size rule 1
Step−size rule 2
Step−size rule 3
30
Step−size rule 1
Step−size rule 2
Step−size rule 3
90
80
25
70
Iterations
Iterations
Iterations
10
20
15
5
60
50
40
30
10
20
5
10
0
10
20
30
40
50
60
70
80
90
0
100
Number of Users
10
20
30
40
50
60
70
80
90
0
100
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
Number of Users
When the termination accuracy is low, these three step-size rules lead to very similar performance (at least of
the given choice of the free parameters, namely: = 10−5 , α = 10−6 , β = 3 × 10−3 ).
Termination accuracy 10−6
2.3.2
d=3 (Low interference scenario)
d=2 (Medium interference scenario)
20
180
160
14
140
12
120
Iterations
16
10
8
700
Step−size rule 1
Step−size rule 2
Step−size rule 3
500
100
80
6
60
4
40
2
20
Step−size rule 1
Step−size rule 2
Step−size rule 3
600
Iterations
18
Iterations
d=1 (High interference scenario)
200
Step−size rule 1
Step−size rule 2
Step−size rule 3
400
300
200
100
0
10
20
30
40
50
60
Number of Users
70
80
90
100
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Number of Users
0
10
20
30
40
50
60
70
80
90
100
Number of Users
In “High interference” scenario (see the figure corresponding to d = 1), the situation is different: rule 3 seems to
need an ad-hoc tuning, which should depend on the network size. Rules 1 and 2 instead keep working quite good.
The performance of SJBR becomes more sensitive to the step size choices when the termination accuracy increases,
especially in the high interference scenario.
2.3.3
Parameter choices of Step size rule 1 γ n = γ n−1 1 − γ n−1 , ∈ (0, 1) , n = 1, 2, · · ·
Since Rule 1 worked quite well in all the scenarios that we simulated (and it has only one tuning parameter), it
seems to be a good winner candidate. It is then interesting to further study the behavior of the SJBR algorithm
based on this rule. In particular, we are interested to understand how sensitive the performance of SJBR is with
7
respect to the free parameter in (1). In the following we report the numerical results for the following choices of :
10−3 , 10−4 , 10−5 , 10−6 . The following tables show the iterations number and the achievable sum-rate of the SJBR
algorithm.
Termination accuracy 10−3
• d = 3 (Low interference scenario)
# of users = 5
# of users = 10
# of users = 15
# of users = 30
# of users = 50
# of users = 75
# of users = 100
= 10−6
iteration sum-rate
3.05
18.28
3.98
32.88
4.15
44.54
4.97
70.74
5.1
93.84
5.17
113.85
5.49
128.29
= 10−5
iteration sum-rate
3.05
18.28
3.98
32.88
4.14
44.54
4.97
70.74
5.1
93.84
5.17
113.85
5.49
128.29
= 10−4
iteration sum-rate
3.05
18.28
3.98
32.88
4.13
44.54
4.96
70.73
5.12
93.84
5.21
113.85
5.51
128.92
= 10−3
iteration sum-rate
3.06
18.28
4
32.88
4.17
44.54
5.01
70.74
5.13
93.84
5.24
113.85
5.52
128.92
= 10−5
iteration sum-rate
5.75
15.81
9.39
25.19
11.87
31.31
13.76
42.59
12.61
50.84
12.22
57.24
12
61.68
= 10−4
iteration sum-rate
5.78
15.81
9.4
25.19
11.83
31.31
13.76
42.59
12.57
50.84
12.26
57.24
12.03
61.68
= 10−3
iteration sum-rate
5.84
15.81
9.48
25.19
11.93
31.31
13.75
42.59
12.64
50.84
12.22
57.24
11.95
61.68
= 10−5
iteration sum-rate
32.88
12.75
48.65
16.79
47.82
17.88
45.75
19.49
46.93
20.56
49.22
21.39
49.72
21.83
= 10−4
iteration sum-rate
32.53
12.75
48.23
16.79
48.97
17.87
46.68
19.49
52.85
20.56
47.66
21.38
49.33
21.82
= 10−3
iteration sum-rate
32.98
12.75
49.3
16.79
50.07
17.88
47.29
19.49
53.71
20.56
48.78
21.39
50.86
21.83
= 10−5
iteration sum-rate
5.42
18.28
6.96
32.88
7.73
44.54
8.97
70.74
9.45
93.84
9.59
113.85
9.76
128.93
= 10−4
iteration sum-rate
5.42
18.28
6.96
32.88
7.73
44.54
8.96
70.74
9.4
93.84
9.55
113.85
9.74
128.93
= 10−3
iteration sum-rate
5.39
18.28
6.96
32.88
7.72
44.54
8.97
70.74
9.44
93.84
9.63
113.85
9.81
128.93
• d = 2 (Medium interference scenario)
# of users = 5
# of users = 10
# of users = 15
# of users = 30
# of users = 50
# of users = 75
# of users = 100
= 10−6
iteration sum-rate
5.75
15.81
9.37
25.19
11.9
31.31
13.75
42.59
12.65
50.84
12.23
57.24
11.99
61.68
• d = 1 (High interference scenario)
# of users = 5
# of users = 10
# of users = 15
# of users = 30
# of users = 50
# of users = 75
# of users = 100
= 10−6
iteration sum-rate
32.66
12.75
48.54
16.79
49.28
17.88
46.61
19.49
52.78
29.56
48.27
21.39
50.51
21.83
Termination accuracy 10−6
• d = 3 (Low interference scenario)
# of users = 5
# of users = 10
# of users = 15
# of users = 30
# of users = 50
# of users = 75
# of users = 100
= 10−6
iteration sum-rate
5.4
18.28
6.79
32.88
7.69
44.54
8.94
70.74
9.43
93.84
9.58
113.85
9.78
128.93
8
• d = 2 (Medium interference scenario)
# of users = 5
# of users = 10
# of users = 15
# of users = 30
# of users = 50
# of users = 75
# of users = 100
= 10−6
iteration sum-rate
13.52
15.81
25.77
25.19
32.13
31.31
41.63
42.59
34.98
50.84
30.46
57.24
28.69
61.68
= 10−5
iteration sum-rate
14.13
15.81
24.31
25.19
35.66
31.31
40.8
42.59
34.29
50.84
30.02
57.24
28.45
61.68
= 10−4
iteration sum-rate
13.47
15.81
25.9
25.19
32.09
31.31
41.65
42.59
35.06
50.84
30.38
57.24
28.77
61.68
= 10−3
iteration sum-rate
13.62
15.81
26.21
25.19
32.52
31.31
42.4
42.59
35.58
50.84
30.76
57.24
29.05
61.68
= 10−5
iteration sum-rate
114.77
12.77
170.25
16.86
152.59
17.94
119.63
19.57
115.21
20.65
114.44
21.47
114.3
21.91
= 10−4
iteration sum-rate
114.65
12.77
175.07
16.86
160.16
17.94
130.8
19.57
127.96
20.65
117.65
21.47
112.92
21.92
= 10−3
iteration sum-rate
121.85
12.77
197.93
16.86
177.79
17.94
140.63
19.57
136.82
20.65
124.28
21.47
118.51
21.92
• d = 1 (High interference scenario)
# of users = 5
# of users = 10
# of users = 15
# of users = 30
# of users = 50
# of users = 75
# of users = 100
= 10−6
iteration sum-rate
113.68
12.77
172.83
16.86
158.48
17.94
129.89
19.57
127.74
20.65
116.94
21.47
110.17
21.91
From the above experiment results, we observe that the performance of SJBR algorithm is not too sensitive to the
choice of the parameter , which makes the proposed algorithm very appealing in practical scenarios.
Description of the results for the WMMSE algorithm and SJBR algorithm in two scenarios, namely the interference
broadcasting channel (IBC) and the peer-to-peer interference channel (IC). Both scenarios as well as the WMMSE
are simulated using the matlab codes provided by Wei-Cheng Liao.
3
Scenario 1 (IBC)
3.1
Environment Setting
• Multicell cellular systems composed of 7 cell with 1 BS and K randomly generated MTs per cell, where BS =
Base Station and MT = Mobile Terminal.
• number of transmit antennas = number of receive antennas = 4
• SNR , Pt /σ 2 for all the links, where Pt =transmit power and σ 2 = 1 is the variance of the AWGN noise.
• Termination accuracy on the sum-rate= 1e − 2 (nats/s/Hz)
• tolerance of the bisection loop= 1e − 5
• parameters of SJBR:
– step-size rule: γ n = γ n−1 1 − γ n−1 , with γ 0 = 1 and = 1e − 3
– τ =0
• the initial point is generated randomly and set the same for both algorithms, i.e. V(0) for WMMSE and
H
Q(0) = V(0) V(0)
for SJBR
9
3.2
Results
We report next the following results for the two algorithms: i) average cpu-time required to reach convergence
(within the prescribed accuracy) vs. # of MTs in the system; ii) average # of iteration to reach convergence vs. #
of MTs in the system; and iii) average sum-rate vs. # of MTs in the system.
The average is taken over 1490 independent channel/users realizations. The above results are given for three
different values of the SNR, namely: SNR = 0dB, SNR = 3dB, and SNR = 10dB.
• SNR = 0dB
Average cpu−time vs. # of active MTs (SNR=0dB)
Average # of iterations vs. # of active MTs (SNR=0B)
3
130
WMMSE
SJBR
2.5
1.5
110
Sum rate (nats/s/Hz)
2
150
100
1
50
0.5
WMMSE
SJBR
120
200
# of iteration
cpu time per user (s)
Average sum−rate vs. # of active MTs (SNR=0dB)
250
WMMSE
SJBR
100
90
80
70
60
50
40
0
0
100
200
300
400
0
500
0
100
# of Mobile Terminals
200
300
400
30
500
0
100
# of Mobile Terminals
200
300
400
500
# of Mobile Terminals
• SNR = 3dB
Average cpu−time vs. # of active MTs (SNR=3dB)
Average # of iterations vs. # of active MTs (SNR=3B)
3.5
140
WMMSE
SJBR
WMMSE
SJBR
120
Sum rate (nats/s/Hz)
250
2.5
# of iteration
cpu time per user (s)
3
2
1.5
200
150
100
1
50
0.5
0
Average sum−rate vs. # of active MTs (SNR=3dB)
300
WMMSE
SJBR
0
100
200
300
400
0
500
100
80
60
40
0
100
# of Mobile Terminals
200
300
400
20
500
0
100
# of Mobile Terminals
200
300
400
500
# of Mobile Terminals
• SNR = 10dB
Average cpu−time vs. # of active MTs (SNR=10dB)
Average # of iterations vs. # of active MTs (SNR=10B)
4.5
Average sum−rate vs. # of active MTs (SNR=10dB)
400
WMMSE
SJBR
4
160
WMMSE
SJBR
350
WMMSE
SJBR
140
3
2.5
2
Sum rate (nats/s/Hz)
300
# of iteration
cpu time per user (s)
3.5
250
200
150
1.5
100
1
120
100
80
60
50
0.5
0
0
100
200
300
# of Mobile Terminals
400
500
0
0
100
200
300
# of Mobile Terminals
10
400
500
40
0
100
200
300
# of Mobile Terminals
400
500
4
Scenario 2 (IC)
4.1
Environment setting
• Multicell cellular systems composed of 7 cells; in each cell there are K active pairs BS-MT randomly placed.
• number of transmit antennas = number of receive antennas = 4
• SNR , Pt /σ 2 for all the links, where Pt =transmit power and σ 2 = 1 is the variance of the AWGN noise.
• Termination accuracy on the sum-rate= 1e − 2 (nats/s/Hz)
• tolerance of the bisection loop= 1e − 5
• parameters of SJBR:
– step-size rule: γ n = γ n−1 1 − γ n−1 , with γ 0 = 1 and = 1e − 3
– τ =0
• the initial point is generated randomly and set the same for both algorithms, i.e. V(0) for WMMSE and
H
Q(0) = V(0) V(0)
for SJBR
4.2
Results
We report next the following results for the two algorithms: i) average cpu-time required to reach convergence (within
the prescribed accuracy) vs. # of BS-MT pairs in the system; ii) average # of iteration to reach convergence vs.
# of BS-MT pairs in the system; and iii) average sum-rate vs. # of BS-MT pairs in the system.
The average is taken over 500 independent channel/users realizations. The above results are given for three
different values of the SNR, namely: SNR = 0dB, SNR = 3dB, and SNR = 10dB.
• SNR = 0dB
Average cpu−time vs. # of active BS−MT paris (SNR=0dB)
Average # of iterations vs. # of active BS−MT pairs (SNR=0B)
200
120
180
WMMSE
SJBR
WMMSE
SJBR
# of iteration
140
120
100
80
250
Sum rate (nats/s/Hz)
100
160
cpu time per user (s)
Average sum−rate vs. # of active BS−MT pairs (SNR=0dB)
300
WMMSE
SJBR
80
60
40
60
40
20
200
150
100
50
20
0
0
100
200
300
400
0
500
0
100
200
300
400
0
500
# of pairs
# of pairs
Average cpu−time vs. # of active BS−MT paris (SNR=3dB)
Average # of iterations vs. # of active BS−MT pairs (SNR=3B)
0
100
200
300
400
500
# of pairs
• SNR = 3dB
250
140
Average sum−rate vs. # of active BS−MT pairs (SNR=3dB)
350
WMMSE
SJBR
WMMSE
SJBR
WMMSE
SJBR
120
300
Sum rate (nats/s/Hz)
100
# of iteration
cpu time per user (s)
200
150
100
80
60
40
250
200
150
100
50
20
0
0
100
200
300
400
500
0
50
0
100
200
# of pairs
300
# of pairs
11
400
500
0
0
100
200
300
# of pairs
400
500
• SNR = 10dB
Average cpu−time vs. # of active BS−MT paris (SNR=10dB) Average # of iterations vs. # of active BS−MT pairs (SNR=10B) Average sum−rate vs. # of active BS−MT pairs (SNR=10dB)
300
180
WMMSE
SJBR
350
WMMSE
SJBR
160
WMMSE
SJBR
300
200
150
Sum rate (nats/s/Hz)
140
# of iteration
cpu time per user (s)
250
120
100
80
100
60
50
250
200
150
100
40
0
0
100
200
300
400
500
20
0
100
200
# of pairs
300
# of pairs
400
500
50
0
100
200
300
400
# of pairs
References
[1] G. Scutari, F. Francisco, P. Song, D. P. Palomar and J. -S. Pang, “Decomposition by partial linearization:
parallel optimization of multi-agent systems”, submitted to IEEE Trans on Signal Processing, 2013
[2] J. Papandriopoulos and J. S. Evans, “SCALE: a low-complexity distributed protocol for spectrum balancing in
multiuser dsl networks”, IEEE Trans on Information Theory, vol. 55, no.8, pp. 3711-3724, Aug, 2009
[3] M. Hong and Z. -Q. Luo, “Signal processing and optimal resource allocation for the interference channel”, Elsevier
e-Reference-Signal Processing, 2013. Available at http://arxiv.org/pdf/1206.5144v1.pdf.
[4] C. Shi, R. A. Berry amd M. L. Honig, “Monotonic convergence of distributed interference pricing in wireless
networks”, in Proc. IEEE international conference on Symposium on Information Theory (ISIT), vol. 3, pp.
1619-1623, July, 2009
[5] Q. Shi, M. Razaviyayn, Z. -Q. Luo and C. He, “An iteratively weighted MMSE approach to distributed sumutility maximization for a MIMO interfering broadcast channel”, IEEE Trans on Signal Processing, vol. 59, no.
9, pp. 4331-4340, Sept, 2011
[6] S.-J. Kim and G. B. Giannakis, “Optimal resource allocation for MIMO ad hoc cognitive radio networks”, IEEE
Trans on Information Theory, vol. 57, no. 5, May 2011
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