Non-convex Optimization and Resource Allocation
in Wireless Communication Networks
Ravi R. Mazumdar
School of Electrical and Computer Engineering
Purdue University
E-mail: [email protected]
Joint Work with Prof. Ness B. Shroff and Jang-Won Lee
ECE, Purdue University – p. 1/50
Outline
Introduction and non-convexity
Joint power and rate allocation for the downlink in (CDMA)
wireless systems
Opportunistic power scheduling for the downlink in
multi-server wireless systems
Conclusion and future work
ECE, Purdue University – p. 2/50
Motivation
Tremendous growth in the number of users in communication
networks
Increasing demand on various services that can provide QoS
Scarce network resources
Need to efficiently design and engineer resource allocation
schemes for heterogeneous services
ECE, Purdue University – p. 3/50
Motivation
Most services are elastic
can adjust the amount of resource consumption to some
degree
By appropriately exploiting the elasticity of services
can maintain high efficiency and fairness
can alleviate congestion within the network
Need appropriate model for the elasticity
Utility
degree of user’s (service’s) satisfaction or performance by
acquiring a certain amount of resource
different elasticity with different utility functions
example: expected throughput as a function of power
allocation in wireless system
ECE, Purdue University – p. 4/50
Total system utility
maximization
max
M
X
Ui (x̄)
i=1
s. t.
gk (x̄) ≥ 0, k = 1, 2, · · · , K
x̄ ∈ X
If all Ui and gk are concave and X is a convex set,
convex optimization problem
can be solved by using standard techniques
Otherwise,
non-convex optimization problem
difficult to solve requiring a complex algorithm
ECE, Purdue University – p. 5/50
Non-convexity in
resource allocation
Utility
In general, three types of utility functions
concave: traditional data services on the Internet
sigmoidal-like (“S”): many wireless services and real-time
services on the Internet
convex: some wireless services
Concave
Sigmoidal
Convex
Resource allocation
ECE, Purdue University – p. 6/50
Non-convexity (cont’d)
f(γ)
1
0
BPSK
DPSK
FSK
0
γ
20
Packet transmission success probability
ECE, Purdue University – p. 7/50
Non-convexity (cont’d)
Increasing demand for wireless and real-time services
non-concave utility functions becoming important
non-convex optimization problem
⇒ complex algorithm for a global optimum
Can we develop a simple algorithm for the approximation to the
global optimum?
ECE, Purdue University – p. 8/50
Inefficiency of naive
approach
11 users and 10 units of a resource
Utility function for each user: U (x)
Approximate U (x) with concave function V (x)
With V (x), for each user,
x∗ = 10
11
however, U (x∗ ) = 0
zero total system utility
1
By allocating one unit to 10
U(x)
users and zero to one user:
V(x)
10 units of total system
x*1
0
2
utility
Need resource allocation algorithms taking into account the
properties of non-concave functions
ECE, Purdue University – p. 9/50
Dual approach
Primal problem
max
M
X
Ui (x̄)
Dual problem
⇒
i=1
s. t.
gk (x̄) ≥ 0,
k = 1, 2, · · · , K
⇐
convex optimization
Q(λ̄)
s. t.
λ̄ ≥ 0̄,
Q(λ̄) = max{
x̄∈X
x̄ ∈ X
min
M
X
i=1
Ui (x̄) +
K
X
λk gk (x̄)}
k=1
convex optimization
simpler constraints
smaller dimension
In many cases, the dual is easier to solve than the primal
ECE, Purdue University – p. 10/50
Dual approach (cont’d)
Primal problem
max
M
X
Ui (x̄)
Dual problem
⇒
i=1
s. t.
gk (x̄) ≥ 0,
k = 1, 2, · · · , K
6⇐
x̄ ∈ X
non-convex optimization
min
Q(λ̄)
s. t.
λ̄ ≥ 0̄,
Q(λ̄) = max{
x̄∈X
M
X
i=1
Ui (x̄) +
K
X
λk gk (x̄)}
k=1
convex optimization
simpler constraints
smaller dimension
May not guarantee the feasible and optimal primal solution
ECE, Purdue University – p. 11/50
Part I
Joint power and rate allocation for the
downlink in (CDMA) wireless systems
ECE, Purdue University – p. 12/50
Why joint power and rate
allocation?
Power is fundamental radio resource
trade off between performance of each user
Variable data rate
trade off between data rate and the probability of packet
transmission success for a given power allocation
By jointly optimizing power and data rate allocation, the
system performance can be further improved
ECE, Purdue University – p. 13/50
Related work
Oh and Wasserman [MOBICOM99]: Uplink power and rate
control for a single class system without constraint on the
maximum data rate
if applied to downlink, single server transmission is
optimal
Bedekar et al. [GLOBECOM99] and Berggren et al. [JSAC01]:
Downlink power and rate control without constraint on the
maximum data rate
single server transmission is optimal
ECE, Purdue University – p. 14/50
Our work
CDMA system that supports variable data rate by variable
spreading gain
Downlink in a single cell
Snapshot of a time-slot
Constant Path gain and interference level during the
time-slot
Base-station has the total transmission power limit PT
Each user i has
Rimax : maximum data rate
fi : function for packet transmission success probability
ECE, Purdue University – p. 15/50
Signal to Interference and
Noise Ratio (SINR)
SINR for user i
W
Pi
γi (Ri , P̄ ) =
P
Ri θ( M
m=1 Pm − Pi ) + Ai
M : number of users in the cell
W : chip rate
θ: orthogonality factor
Pi : power allocation for user i
Ri : data rate of user i
Ai = Ii /Gi : transmission environment of user i
Ii : background noise and intercell interference at user i
Gi : path gain from the base-station to user i
SINR is a function of power and rate allocation
ECE, Purdue University – p. 16/50
Packet transmission
success probability: fi
fi is an increasing function of γi
P
For a given Ri , if M
m=1 Pm = PT , fi is
concave function,
“S” function, or
convex function
of its own power allocation Pi
f(P)
1
BPSK
DPSK
FSK
0
P
10
ECE, Purdue University – p. 17/50
Problem formulation
(A)
max
Pi ,Ri
s. t.
M
X
Ri fi (γi (Ri , P̄ ))
i=1
PM
Pi ≤ P T
0 ≤ Pi ≤ PT ,
0 ≤ Ri ≤ Rimax ,
Ri = Ri∗ ,
i=1
∀i
i∈V
i 6∈ V
V : a subset of users that have variable data rate
Ri fi (γi (Ri , P̄ )): expected throughput of user i
Goal: Obtaining power and rate allocation that maximizes the
expected total system throughput with constraints on the total
transmission power limit of the base-station and the maximum data
rate of each user
ECE, Purdue University – p. 18/50
Optimal rate allocation
To maximize the expected total system throughput, the
base-station must transmit at the maximum power limit
Redefine SINR for user i as
W
W
Pi
Pi
γi (Ri , Pi )=
=
= γi (Ri , P̄ )
P
R i PT − P i + A i
Ri M
j=1 Pj − Pi + Ai
4
For a given power allocation Pi , the optimal rate of user i,
Ri∗ (Pi ) =







W Pi
∗
γi (PT −Pi +Ai ) ,
Rimax ,
Ri∗ ,
if i ∈
if i ∈
Rimax γi∗ (PT +Ai )
V Pi ≤ W +Rmax γ ∗
i
i
Rimax γi∗ (PT +Ai )
V, Pi > W +Rmax γ ∗
i
i
if i 6∈ V,
where γi∗ = arg maxγ≥1 { γ1 fi (γ)}.
ECE, Purdue University – p. 19/50
Equivalent power
allocation problem
(B)
max
M
X
Ui (Pi )
i=1
s.t.
Ui (Pi ) =







PM
Pi ≤ P T
0 ≤ Pi ≤ PT ,
i=1
Pi
W
∗
f
(γ
),
∗
i
i
γi PT −Pi +Ai
Rimax fi (γi (Rimax , Pi )),
Ri∗ f (γi (Ri∗ , Pi )),
∀i,
if i ∈ V, Pi ≤
if i ∈ V, Pi >
Rimax γi∗ (PT +Ai )
W +Rimax γi∗
Rimax γi∗ (PT +Ai )
W +Rimax γi∗
if i 6∈ V
Ui (Pi ) is a convex, concave, or “S” function of Pi .
ECE, Purdue University – p. 20/50
Power allocation
Amount of power maximizing net utility
Pi (λ) = argmax {Ui (Pi ) − λPi }
0≤Pi ≤PT
Maximum willingness to pay per unit power
λmax
= min{λ ≥ 0 |
i
max {Ui (P ) − λP } = 0}, ∀i
0≤P ≤PT
unique for each user i
if λ > λmax
, then Pi (λ) = 0
i
if λ < λmax
, then Pi (λ) > 0
i
ECE, Purdue University – p. 21/50
Power allocation (cont’d)
Assume that λmax
≥ λmax
≥ · · · ≥ λmax
1
2
M
User selection
Select users from 1 to K that satisfies
K = max {
1≤j≤M
j
X
Pi (λmax
) ≤ PT }
j
i=1
Users are selected in a decreasing order of λmax
i
Power allocation
PK
∗
Find λ such that i=1 Pi (λ∗ ) = PT
Allocate power to each selected user i as Pi (λ∗ )
Optimal power allocation for the selected users
ECE, Purdue University – p. 22/50
Optimality
P̄ ∗ : our power allocation
P̄ o : optimal power allocation
PM
If i=1 Ui (γi (Pio )) → ∞ as M → ∞,
PM
∗
U
(γ
(P
i
i
i ))
i=1
→ 1, as M → ∞
PM
o
i=1 Ui (γi (Pi ))
Our power allocation is
asymptotically optimal
a good approximation of the optimal power allocation with
a large number of small users
ECE, Purdue University – p. 23/50
Multiple access strategy
If
Rimax Ai
PT W
≥
, ∀i,
γi∗
single server transmission is optimal
when users have high maximum data rate or are
experiencing poor transmission environment
when there is no constraint on the maximum data rate
W : chip rate
γi∗ : constant that depends on fi
Ai = Ii /Gi : transmission environment of user i
Ii : intercell interference and background noise at user i
Gi : path gain from the base-station to user i
ECE, Purdue University – p. 24/50
Multiple access strategy
(cont’d)
If
PM
i=1
Pi (λmax
M ) ≤ PT , selecting all users is optimal
If P1 (λmax
) ≥ PT , selecting only user 1 is optimal
2
Otherwise, selecting a subset of users can be optimal
Condition for optimal multiple access strategy depends on
time-varying parameters such as
number of users
type of users (utility functions)
channel condition of users
Static multiple access strategy could be inefficient
Need dynamic multiple access strategy (dynamic multi-server
transmission)
ECE, Purdue University – p. 25/50
User selection strategy
If all users are homogeneous, selecting users according to
transmission environment is optimal
higher priority to a user in a better transmission
environment
However, if users are heterogeneous, no simple optimal user
selection strategy
Our user selection strategy provides a simple and unified
selection strategy for heterogeneous users
ECE, Purdue University – p. 26/50
User selection strategy
User i is called more efficient than user j if
Ui (γi (P )) ≥ Uj (γj (P )), ∀P
More efficient user has a higher priority to be selected
When other conditions are the same, user i has a higher
priority to be selected than user j if
Rimax > Rjmax (maximum data rate),
fi (γ) > fj (γ), ∀γ (transmission scheme), or
Ai < Aj (transmission environment)
Our user selection strategy provides a simple and efficient selection
strategy for heterogeneous users
ECE, Purdue University – p. 27/50
Numerical results
Model path gain considering distance loss and log-normally
distributed slow shadowing
Two classes of users, for a user in class i,
fi (γ) = ci {
1
1+
e−ai (γ−bi )
− di }
Compare with the single-server system
BS
BS
BS
BS
BS
BS
BS
BS
BS
ECE, Purdue University – p. 28/50
Numerical results (cont’d)
R1max
1562.5
6250
25000
Selection ratio of class 1
Selection ratio of class 2
Utility (Our)/Utility (Single)
0.501
0.568
3.415
0.388
0.392
3.854
0.198
0.020
1.016
f1 = f 2
R1max 6= R2max (R2max = 6250)
Selection ratio of class i: the ratio of the number of selected
users to the number of users in class i
ECE, Purdue University – p. 29/50
Numerical results (cont’d)
b1
2.5
3.5
4.5
Selection ratio of class 1
Selection ratio of class 2
Utility (Our)/Utility (Single)
0.566
0.288
4.196
0.391
0.389
3.852
0.230
0.484
3.525
R1max = R2max
f1 6= f2 (a1 = a2 , b2 = 3.5)
If bi < bj , then fi (γ) ≥ fj (γ), ∀γ
class i has a more efficient transmission scheme than
class j
ECE, Purdue University – p. 30/50
Numerical results (cont’d)
Ratio of class 1
0.4
0.6
0.8
Selection ratio of class 1
Selection ratio of class 2
Utility (Our)/Utility (Single)
0.849
0.004
3.409
0.653
0
3.912
0.499
0
3.980
R1max = R2max
f1 = f 2
Class 1: inner region
Class 2: outer region
ECE, Purdue University – p. 31/50
Part II
Opportunistic power scheduling for the
downlink in multi-server wireless
systems
ECE, Purdue University – p. 32/50
Why opportunistic
scheduling?
Trade-off between efficiency and fairness due to
multi-class users
time-varying and location-dependent channel condition
Our previous problem
high system efficiency
however, unfair to some (inefficient) users
Fairness
achieved by an appropriate scheduling scheme
Opportunistic scheduling considering each user’s
delay tolerance
fairness or performance constraint
time-varying channel condition
ECE, Purdue University – p. 33/50
Single-server vs.
Multi-server
Single-server scheduling
Only one user can be scheduled in a time-slot
In every time-slot, must decide
which user must be selected
Multi-server scheduling
Multiple users can be scheduled in a time-slot
In every time-slot, must decide
how many and which users must be selected
how much power is allocated to each selected user
Most work studied single-server scheduling
However, single-server scheduling can be inefficient
Need dynamic multi-server scheduling
ECE, Purdue University – p. 34/50
Related work
Single-server scheduling
Qualcomm’s HDR: proportional fairness
Borst and Whiting [INFOCOM01]: constraint on utility
based fairness
Liu, Chong, and Shroff [JSAC01,COMNET03]: constraints
on minimum performance, and utility and resource based
fairness
Multi-server scheduling
Kulkarni and Rosenberg [MSWiM03]: static multi-server
scheduling with independent interfaces
Liu and Knightly [INFOCOM03]: dynamic multi-server
scheduling with constraint on utility based fairness
assuming orthogonality among users and linear
relationship between data rate and power allocation
ECE, Purdue University – p. 35/50
Our work
Dynamic multi-server scheduling for downlink in a single cell
Allow users to interfere with each other
PT is total transmission power at the base-station
Utility function Ui for user i: convex, concave, or "S" function
In each time-slot, system is in one of the states {1, 2, · · · , S}
corresponds to channel conditions of all users
stationary stochastic process with Prob{state s} = πs
⇒ time-varying channel condition of each user is modeled as
a discrete state stationary stochastic process
Requirement for each user
resource based fairness
utility based fairness
minimum performance
ECE, Purdue University – p. 36/50
SINR and utility function
SINR for user i when system is in state s
Ni Ps,i
γs,i (Ps,i ) =
θ(PT − Ps,i ) + As,i
Define
4
Us,i (Ps,i )= Ui (γs,i (Ps,i ))
The utility function varies randomly according to the
channel condition
ECE, Purdue University – p. 37/50
Problem formulation with
minimum performance
(C)
max
Ps,i
M
X
E{Ui }(=
M X
S
X
πs Us,i (Ps,i ))
i=1 s=1
i=1
s. t. E{Ui }(=
S
X
πs Us,i (Ps,i )) ≥ Ci ,
i = 1, 2, · · · , M
s=1
M
X
Ps,i ≤ PT ,
s = 1, 2, · · · , S
i=1
0 ≤ Ps,i ≤ PT ,
∀s, i
Goal: Obtaining power scheduling that maximizes the expected total
system utility with constraints on the minimum expected utility for
each user and the total transmission power limit for the base-station
ECE, Purdue University – p. 38/50
Problem with minimum
performance (cont’d)
Main difficulties
Feasibility
assume that the system has call admission control
ensuring a feasible solution
Non-convexity
dual approach
No knowledge for the underlying probability distribution a priori
stochastic subgradient algorithm
ECE, Purdue University – p. 39/50
Power scheduling
In each time-slot n, power is allocated to users by solving the dual of
(E)
max
M
X
(n)
Usmp
, Ps(n) ,i )
(n) ,i (µ̄
i=1
s. t.
M
X
Ps(n) ,i ≤ PT
i=1
0 ≤ Ps(n) ,i ≤ PT ,
4
mp
(n)
Us(n) ,i (µ̄ , Ps(n) ,i )=
i = 1, 2, · · · , M
(n)
(1 + µi )Us(n) ,i (Ps(n) ,i )
Similar to our previous problem
ECE, Purdue University – p. 40/50
Power scheduling
(Cont’d)
The utility function (µi ) is adjusted to guarantee the minimum
performance constraint by using a stochastic subgradient
algorithm
(n+1)
µi
(n)
vi
(n)
= [µi
(n)
− α(n) vi ]+ , ∀i
= Us(n) ,i (Ps∗(n) ,i (µ̄(n) )) − Ci
stochastic subgradient of the dual
Ps∗(n) ,i (µ̄(n) ) is power allocation of user i in time-slot n
µ̄(n) converges to µ̄∗ that solves the dual problem
ECE, Purdue University – p. 41/50
Feasibility
Always satisfies the constraint on total transmission power limit
If
Q
M
→ 0 as M → ∞, then
P
i∈H
Ui∗ − Ci
→ 0 as M → ∞
M
Q: expected number of users with the same channel
conditions
H : set of users whose performance constraints are not
satisfied
Ui∗ : expected utility of user i in our power scheduling in
our power scheduling
Asymptotically feasible on average
Increase in the randomness of the system improves the
degree of users’ satisfaction
ECE, Purdue University – p. 42/50
Optimality
If
PM
o
i=1 Ui → ∞ and
PM
i=1
PM
i=1
Ui∗
Uio
Q
M
→ 0 as M → ∞, then
≥ 1 − and → 0 as M → ∞
Uio : expected utility of user i in optimal power scheduling
If the above conditions are satisfied and Ui∗ ≥ Ci , ∀i, then
PM
i=1
PM
i=1
Ui∗
Uio
→ 1 as M → ∞
Asymptotically optimal
ECE, Purdue University – p. 43/50
Numerical results
The same cellular model as our previous problem
Four users and each user i
same sigmoid utility function Ui (Ui (0) = 0 and Ui (∞) = 1)
same performance constraint Ci = 0.59
distance from the base-station to user i: di
d1 < d 2 < d 3 < d 4
Performance comparison with
Non-opportunistic scheduling
Greedy scheduling
ECE, Purdue University – p. 44/50
Numerical results (cont’d)
Comparison of average utilities (104 time-slots)
User
1
2
3
4
Total
Non-opportunistic 0.590 0.590 0.590 0.590 2.360
Greedy
0.973 0.964 0.796 0.168 2.901
Our opportunistic 0.951 0.736 0.591 0.591 2.869
ECE, Purdue University – p. 45/50
Ratio of average total system utilities
Numerical results (cont’d)
4
3.5
3
2.5
2
1.5
1
2
4
σ
6
8
10
Ratio of average total system utility of our opportunistic power
scheduling to that of non-opportunistic power scheduling
σ : standard deviation of each user’s channel condition
ECE, Purdue University – p. 46/50
Conclusion
Utility framework
suitable for resource allocation with multi-media and data
services
a useful tool for resource allocation in the next
generations of communication networks
non-convex optimization problems in many cases
Dual approach provides
efficient solution in many cases
simple algorithm that can be easily implemented with a
(distributed) network protocol
ECE, Purdue University – p. 47/50
Conclusion (cont’d)
In wireless systems
Single server transmission is optimal only when all users have
high data rate
In general, need dynamic multiple access (dynamic
multi-server system)
Trade-off between efficiency and fairness
Opportunistic scheduling achieves both of them
Randomness of the system could be beneficial to efficient and
fair resource allocation, if appropriately exploited
ECE, Purdue University – p. 48/50
Conclusion (cont’d)
Other problems
Pricing based base-station assignment
considers both transmission environment of the user and
congestion level of the base-station
Congestion control on the Internet
algorithms for concave utility functions cause instability
and congestion in the presence of real-time services with
non-concave utility functions
self-regulating property stabilizes the system and
alleviates congestion
ECE, Purdue University – p. 49/50
Future work
Scheduling considering
user dynamics
non-stationary environment
delay or short-term fairness constraints
Resource allocation considering upper layer protocols (e.g.,
TCP)
Resource allocation for uplink and multi-cellular system
ECE, Purdue University – p. 50/50