Nonconvex Wireless Power Control
Chee Wei Tan
CS 8292 : Advanced Topics in Convex Optimization and its Applications
Fall 2010
Outline
• Max-min weighted SIR optimization
• Nonlinear Perron-Frobenius Theorem and Algorithm
• Sum Rate Maximization
1
Interplay of Mathematical Tools
2
Max-Min Weighted SIR
• Downlink case: consider maxp≥0
X
SIRl(p)
min
subject to
pl ≤ P̄ .
l
βl
l
Theorem 1. The optimal solution is such that the value SIRl/βl for all users
are equal. The optimal weighted max-min SIR is given by
1
γ =
,
ρ(diag(β)B)
∗
where
>
B = F + (1/P̄ )v1
Further, the optimal p, denoted by p∗, is tx(diag(β)B) for some constant
t > 0.
C. W. Tan, M. Chiang & R. Srikant, , Maximizing Sum Rate and Minimizing MSE on Multiuser
Downlink: Optimality, Fast Algorithms, and Equivalence via Max-min SIR, IEEE ISIT, 2009
3
Nonlinear Perron-Frobenius Theory
• Find (λ̌, š) in
λs = As + b,
λ ∈ R,
s ≥ 0,
ksk = 1,
where A and b is a square irreducible nonnegative matrix and
nonnegative vector, respectively and k · k a monotone vector norm.
>
• (λ̌, š) is Perron-Frobenius eigenvalue-vector pair of A + bc∗ ,
where
>
c∗ = arg max ρ(A + bc ),
kck∗=1
where k · k∗ is the dual norm of k · k, and š = (Aš + b)/kAš + bk.
V. D. Blondel, L. Ninove and P. Van Dooren, An affine eigenvalue problem on the nonnegative
orthant, Linear Algebra & its Applications, 2005
4
A Fast Max-min SIR Algorithm
• Based on Nonlinear Perron-Frobenius Theory
• Algorithm 1.
1. Update power p(k + 1):
pl(k + 1) =
βl
pl(k) ∀ l.
SIRl(p(k))
2. Normalize p(k + 1):
pl(k + 1) ← pl(k + 1)/
X
pj (k + 1) · P̄ ∀ l.
j
• Theorem 2. Starting from any initial point p(0), p(k) in Algorithm 1
converges geometrically fast to
x(diag(β)B) (unique up to a scaling constant).
5
Problem: Maximize Sum Shannon Rates
?
• Find p = arg max
0≤p≤p
X
>
wl log(1 + SIRl(p)) where 1 w = 1
l
• Characterize the achievable rate region: rl = log(1 + SIRl(p)) ∀ l
• Two-User case:
max w1 log 1 +
subject to:
G11p1
G12p2 + n1
G22p2
+ w2 log 1 +
G21p1 + n2
0 ≤ p1 ≤ p̄1, 0 ≤ p2 ≤ p̄2
6
Solution Map
Tan, Friedland and Low, Spectrum Management in Multiuser Cognitive Wireless Networks:
Optimality and Algorithm,
IEEE Journal on Selected Areas in Communications (JSAC), 2011
7
Sum Shannon Rate Global Optimization
• Nonlinear map between power p and SIR γ = exp(γ̃):
−1
p? = (I − diag(exp(γ̃ ?))F)
diag(exp(γ̃ ?))v
(5)
• Constraints over p ≤ p̄ into γ̃
• Convert into convex maximization (dB domain):
P
maximize
l wl log(1 + exp(γ̃l ))
>
subject to log ρ(diag(exp(γ̃))(F + (1/p̄l)vel )) ≤ 0 ∀ l,
variables: γ̃l, ∀ l.
8
Nonnegative Matrix Theory: Minimax Theorem
• Theorem 3. Friedland-Karlin inequality [FriedlandKarlin’75]:
irreducible nonnegative matrix A,
Y
xl yl
((Az)l/zl)
For any
≥ ρ(A)
l
for all strictly positive z, where x and y are the Perron and left eigenvectors of
A respectively. Equality holds in (7) if and only if z = ax for some positive a.
• Donsker-Varadhan’s variational principle (1975):
maxλ≥0,1> λ=1 min
p≥0
X
l
X (Ap)l
(Ap)l
= min maxλ≥0,1> λ=1
λl
λl
p
≥
0
pl
pl
l
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Sum Shannon Rate Global Optimization
• Convex Maximization (dB domain):
P
maximize
l wl log(1 + exp(γ̃l ))
>
subject to log ρ(diag(exp(γ̃))(F + (1/p̄l)vel )) ≤ 0 ∀ l,
variables: γ̃l, ∀ l.
• Relaxation of the constraint set by the Friedland-Karlin Inequalities:
Y
x (A)yl (A)
γl l
ρ(A) ≤ ρ(diag(γ)A)
l
X
xl(A)yl(A)γ̃l + log ρ(A) ≤ log ρ(diag(exp(γ̃))A)
(dB domain).
l
• Outer approximation algorithm (Kelley’s cutting planes)
10
Global Optimizing Sum Rate: Examples
• lim min
k→∞
−1
I − diag(exp(γ̃ k ))F
diag(exp(γ̃ k ))v, p̄l = p?
• Fast convergence in numerical examples
Rate region
7
←2
←3
r2 (nats/symbol), w2
6
5
4
←5
3
2
←7
←9
←10
←11
←8
←6
←4
w =x◦y
1
0
0
2
←1
4
r1 (nats/symbol), w1
6
11
Global Optimizing Sum Rate: Examples
• Efficient and fast for small to medium-sized networks
Problem
size
Maximal number of
generated vertices
Number of
iterations
CPU time
(minutes)
2
15
12
0.062
4
139
760
4.1
6
14022
1238
83
8
283681
1968
468
Table 1: A comparison of the typical convergence and complexity statistics with
the problem size. The CPU time is computed based on an implementation on a
64-bit Sun/Solaris 10 (SunOS 5.10) computer.
12
Summary
• Intriguing link between nonlinear Perron-Frobenius theory and geometric
programming
• Distributed algorithm vs. centralized algorithm: configuration (no. of tuning
parameters), feasibility, convergence, scalability etc.
Reading assignment:
• M. Chiang, P. Hande, T. Lan and C. W. Tan, Power Control by Geometric
Programming, IEEE Trans. on Wireless Communications, 6(7), pp. 2640 2651, 2007.
• C. W. Tan, M. Chiang and R. Srikant, Maximizing Sum Rate and Minimizing
MSE on Multiuser Downlink: Optimality, Fast Algorithms, and Equivalence via
Max-min SIR, IEEE ISIT, 2009.
See also Convex Optimization Textbook’s Extra Exercises A12.3
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