Non-convex Optimization and Resource Allocation …major/colloques/nonconvex...Non-convex...

Non-convex Optimization and Resource Allocation

in Wireless Communication Networks

Ravi R. Mazumdar

School of Electrical and Computer EngineeringPurdue University

E-mail: [email protected]

Joint Work with Prof. Ness B. Shroff and Jang-Won Lee

ECE, Purdue University – p. 1/50

OutlineIntroduction and non-convexity

Joint power and rate allocation for the downlink in (CDMA)wireless systems

Opportunistic power scheduling for the downlink inmulti-server wireless systems

Conclusion and future work


MotivationTremendous growth in the number of users in communicationnetworks

Increasing demand on various services that can provide QoS

Scarce network resources

Need to efficiently design and engineer resource allocationschemes for heterogeneous services


MotivationMost services are elastic

can adjust the amount of resource consumption to somedegree

By appropriately exploiting the elasticity of servicescan maintain high efficiency and fairnesscan alleviate congestion within the network

Need appropriate model for the elasticity

Utilitydegree of user’s (service’s) satisfaction or performance byacquiring a certain amount of resourcedifferent elasticity with different utility functionsexample: expected throughput as a function of powerallocation in wireless system


Total system utilitymaximization

max

M∑

i=1

Ui(x̄)

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 problemcan be solved by using standard techniques

Otherwise,non-convex optimization problemdifficult to solve requiring a complex algorithm


Non-convexity inresource allocation

In general, three types of utility functionsconcave: traditional data services on the Internetsigmoidal-like (“S”): many wireless services and real-time

services on the Internetconvex: some wireless services

Resource allocation

Util

ity

ConcaveSigmoidalConvex


Non-convexity (cont’d)

0 200

1

γ

f(γ)

BPSKDPSKFSK

Packet transmission success probability


Non-convexity (cont’d)

Increasing demand for wireless and real-time servicesnon-concave utility functions becoming importantnon-convex optimization problem⇒ complex algorithm for a global optimum

Can we develop a simple algorithm for the approximation to theglobal optimum?


Inefficiency of naiveapproach

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

By allocating one unit to 10users and zero to one user:

10 units of total systemutility 0 1 2

1

U(x)V(x)

x*

Need resource allocation algorithms taking into account theproperties of non-concave functions


Dual approach

Primal problem

max

M∑

i=1

Ui(x̄)

s. t. gk(x̄) ≥ 0,

k = 1, 2, · · · , K

x̄ ∈ X

⇒

⇐

Dual problem

min Q(λ̄)

s. t. λ̄ ≥ 0̄,

Q(λ̄) = maxx̄∈X

{M∑

i=1

Ui(x̄) +K

∑

k=1

λkgk(x̄)}

convex optimization convex optimization

simpler constraints

smaller dimension

In many cases, the dual is easier to solve than the primal


Dual approach (cont’d)

Primal problem

max

M∑

i=1

Ui(x̄)

s. t. gk(x̄) ≥ 0,

k = 1, 2, · · · , K

x̄ ∈ X

⇒

6⇐

Dual problem

min Q(λ̄)

s. t. λ̄ ≥ 0̄,

Q(λ̄) = maxx̄∈X

{M∑

i=1

Ui(x̄) +K

∑

k=1

λkgk(x̄)}

non-convex optimiza-tion

convex optimization

simpler constraints

smaller dimension

May not guarantee the feasible and optimal primal solution


Part I

Joint power and rate allocation for thedownlink in (CDMA) wireless systems


Why joint power and rateallocation?

Power is fundamental radio resourcetrade off between performance of each user

Variable data ratetrade off between data rate and the probability of packettransmission success for a given power allocation

By jointly optimizing power and data rate allocation, thesystem performance can be further improved


Related workOh and Wasserman [MOBICOM99]: Uplink power and ratecontrol for a single class system without constraint on themaximum data rate

if applied to downlink, single server transmission isoptimal

Bedekar et al. [GLOBECOM99] and Berggren et al. [JSAC01]:Downlink power and rate control without constraint on themaximum data rate

single server transmission is optimal


Our workCDMA system that supports variable data rate by variablespreading gain

Downlink in a single cell

Snapshot of a time-slotConstant Path gain and interference level during thetime-slot

Base-station has the total transmission power limit PT

Each user i hasRmax

i : maximum data ratefi: function for packet transmission success probability


Signal to Interference andNoise Ratio (SINR)

SINR for user i

γi(Ri, P̄ ) =W

Ri

Pi

θ(∑M

m=1 Pm − Pi) + Ai

M : number of users in the cellW : chip rateθ: orthogonality factorPi: 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 iGi: path gain from the base-station to user i

SINR is a function of power and rate allocation


Packet transmissionsuccess probability: fi

fi is an increasing function of γi

For a given Ri, if∑M

m=1 Pm = PT , fi is

concave function,“S” function, orconvex function

of its own power allocation Pi

0 10

1

P

f(P

)

BPSKDPSKFSK


Problem formulation

(A) maxPi,Ri

M∑

i=1

Rifi(γi(Ri, P̄ ))

s. t.∑M

i=1 Pi ≤ PT

0 ≤ Pi ≤ PT , ∀i

0 ≤ Ri ≤ Rmaxi , i ∈ V

Ri = R∗i , i 6∈ V

V : a subset of users that have variable data rate

Rifi(γi(Ri, P̄ )): expected throughput of user i

Goal: Obtaining power and rate allocation that maximizes theexpected total system throughput with constraints on the totaltransmission power limit of the base-station and the maximum datarate of each user


Optimal rate allocation

To maximize the expected total system throughput, thebase-station must transmit at the maximum power limit

Redefine SINR for user i as

γi(Ri, Pi)4=

W

Ri

Pi

PT − Pi + Ai

=W

Ri

Pi∑M

j=1 Pj − Pi + Ai

= γi(Ri, P̄ )

For a given power allocation Pi, the optimal rate of user i,

R∗i (Pi) =

WPi

γ∗i(PT −Pi+Ai)

, if i ∈ V Pi ≤Rmax

i γ∗i (PT +Ai)W+Rmax

iγ∗

i

Rmaxi , if i ∈ V, Pi >

Rmaxi γ∗i (PT +Ai)W+Rmax

iγ∗

i

R∗i , if i 6∈ V,

where γ∗i = arg maxγ≥1{

1γfi(γ)}.


Equivalent powerallocation problem

(B) max

M∑

i=1

Ui(Pi)

s.t.∑M

i=1 Pi ≤ PT

0 ≤ Pi ≤ PT , ∀i,

Ui(Pi) =

Wγ∗

i

Pi

PT −Pi+Aifi(γ

∗i ), if i ∈ V, Pi ≤


iγ∗

i

Rmaxi fi(γi(R

maxi , Pi)), if i ∈ V, Pi >


iγ∗

i

R∗i f(γi(R

∗i , Pi)), if i 6∈ V

Ui(Pi) is a convex, concave, or “S” function of Pi.


Power allocationAmount of power maximizing net utility

Pi(λ) = argmax0≤Pi≤PT

{Ui(Pi) − λPi}

Maximum willingness to pay per unit power

λmaxi = min{λ ≥ 0 | max

0≤P≤PT

{Ui(P ) − λP} = 0}, ∀i

unique for each user i

if λ > λmaxi , then Pi(λ) = 0

if λ < λmaxi , then Pi(λ) > 0


Power allocation (cont’d)

Assume that λmax1 ≥ λmax

2 ≥ · · · ≥ λmaxM

User selectionSelect users from 1 to K that satisfies

K = max1≤j≤M

{

j∑

i=1

Pi(λmaxj ) ≤ PT }

Users are selected in a decreasing order of λmaxi

Power allocationFind λ∗ such that

∑Ki=1 Pi(λ

∗) = PT

Allocate power to each selected user i as Pi(λ∗)

Optimal power allocation for the selected users


Optimality

P̄ ∗: our power allocation

P̄ o: optimal power allocation

If∑M

i=1 Ui(γi(Poi )) → ∞ as M → ∞,

∑Mi=1 Ui(γi(P

∗i ))

∑Mi=1 Ui(γi(P o

i ))→ 1, as M → ∞

Our power allocation isasymptotically optimala good approximation of the optimal power allocation witha large number of small users


Multiple access strategy

If

Rmaxi Ai ≥

PT W

γ∗i

, ∀i,

single server transmission is optimal

when users have high maximum data rate or areexperiencing poor transmission environmentwhen there is no constraint on the maximum data rate

W : chip rate

γ∗

i: 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


Multiple access strategy(cont’d)

If∑M

i=1 Pi(λmaxM ) ≤ PT , selecting all users is optimal

If P1(λmax2 ) ≥ PT , selecting only user 1 is optimal

Otherwise, selecting a subset of users can be optimal

Condition for optimal multiple access strategy depends ontime-varying parameters such as

number of userstype of users (utility functions)channel condition of users

Static multiple access strategy could be inefficient

Need dynamic multiple access strategy (dynamic multi-servertransmission)


User selection strategy

If all users are homogeneous, selecting users according totransmission environment is optimal

higher priority to a user in a better transmissionenvironment

However, if users are heterogeneous, no simple optimal userselection strategy

Our user selection strategy provides a simple and unifiedselection strategy for heterogeneous users


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 higherpriority to be selected than user j if

Rmaxi > Rmax

j (maximum data rate),

fi(γ) > fj(γ), ∀γ (transmission scheme), or

Ai < Aj (transmission environment)

Our user selection strategy provides a simple and efficient selectionstrategy for heterogeneous users


Numerical resultsModel path gain considering distance loss and log-normallydistributed 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

BSBSBS

BS


Numerical results (cont’d)

Rmax1 1562.5 6250 25000

Selection ratio of class 1 0.501 0.388 0.198Selection ratio of class 2 0.568 0.392 0.020

Utility (Our)/Utility (Single) 3.415 3.854 1.016

f1 = f2

Rmax1 6= Rmax

2 (Rmax2 = 6250)

Selection ratio of class i: the ratio of the number of selectedusers to the number of users in class i



b1 2.5 3.5 4.5

Selection ratio of class 1 0.566 0.391 0.230Selection ratio of class 2 0.288 0.389 0.484


Rmax1 = Rmax

2

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 thanclass j



Ratio of class 1 0.4 0.6 0.8

Selection ratio of class 1 0.849 0.653 0.499Selection ratio of class 2 0.004 0 0


Rmax1 = Rmax

2

f1 = f2

Class 1: inner region

Class 2: outer region


Part II

Opportunistic power scheduling for thedownlink in multi-server wireless

systems


Why opportunisticscheduling?

Trade-off between efficiency and fairness due tomulti-class userstime-varying and location-dependent channel condition

Our previous problemhigh system efficiencyhowever, unfair to some (inefficient) users

Fairnessachieved by an appropriate scheduling scheme

Opportunistic scheduling considering each user’sdelay tolerancefairness or performance constrainttime-varying channel condition


Single-server vs.Multi-server

Single-server schedulingOnly one user can be scheduled in a time-slotIn every time-slot, must decide

which user must be selected

Multi-server schedulingMultiple users can be scheduled in a time-slotIn every time-slot, must decide

how many and which users must be selectedhow much power is allocated to each selected user

Most work studied single-server scheduling

However, single-server scheduling can be inefficientNeed dynamic multi-server scheduling


Related workSingle-server scheduling

Qualcomm’s HDR: proportional fairnessBorst and Whiting [INFOCOM01]: constraint on utilitybased fairnessLiu, Chong, and Shroff [JSAC01,COMNET03]: constraintson minimum performance, and utility and resource basedfairness

Multi-server schedulingKulkarni and Rosenberg [MSWiM03]: static multi-serverscheduling with independent interfacesLiu and Knightly [INFOCOM03]: dynamic multi-serverscheduling with constraint on utility based fairnessassuming orthogonality among users and linearrelationship between data rate and power allocation


Our workDynamic 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 usersstationary stochastic process with Prob{state s} = πs

⇒ time-varying channel condition of each user is modeled asa discrete state stationary stochastic process

Requirement for each userresource based fairnessutility based fairnessminimum performance


SINR and utility function

SINR for user i when system is in state s

γs,i(Ps,i) =NiPs,i

θ(PT − Ps,i) + As,i

Define

Us,i(Ps,i)4= Ui(γs,i(Ps,i))

The utility function varies randomly according to thechannel condition


Problem formulation withminimum performance

(C) maxPs,i

M∑

i=1

E{Ui}(=

M∑

i=1

S∑

s=1

πsUs,i(Ps,i))

s. t. E{Ui}(=

S∑

s=1

πsUs,i(Ps,i)) ≥ Ci, i = 1, 2, · · · , M

M∑

i=1

Ps,i ≤ PT , s = 1, 2, · · · , S

0 ≤ Ps,i ≤ PT , ∀s, i

Goal: Obtaining power scheduling that maximizes the expected totalsystem utility with constraints on the minimum expected utility foreach user and the total transmission power limit for the base-station


Problem with minimumperformance (cont’d)

Main difficulties

Feasibilityassume that the system has call admission controlensuring a feasible solution

Non-convexitydual approach

No knowledge for the underlying probability distribution a prioristochastic subgradient algorithm


Power scheduling

In each time-slot n, power is allocated to users by solving the dual of

(E) max

M∑

i=1

Ump

s(n),i(µ̄(n), Ps(n),i)

s. t.M∑

i=1

Ps(n),i ≤ PT

0 ≤ Ps(n),i ≤ PT , i = 1, 2, · · · , M

Ump

s(n),i(µ̄(n), Ps(n),i)

4= (1 + µ

(n)i )Us(n),i(Ps(n),i)

Similar to our previous problem


Power scheduling(Cont’d)

The utility function (µi) is adjusted to guarantee the minimumperformance constraint by using a stochastic subgradientalgorithm

µ(n+1)i = [µ

(n)i − α(n)v

(n)i ]+, ∀i

v(n)i = Us(n),i(P

∗s(n),i

(µ̄(n))) − Ci

stochastic subgradient of the dual

P ∗s(n),i

(µ̄(n)) is power allocation of user i in time-slot n

µ̄(n) converges to µ̄∗ that solves the dual problem


Feasibility

Always satisfies the constraint on total transmission power limit

If QM

→ 0 as M → ∞, then

∑

i∈H U∗i − Ci

M→ 0 as M → ∞

Q: expected number of users with the same channelconditionsH: set of users whose performance constraints are notsatisfiedU∗

i : expected utility of user i in our power scheduling inour power scheduling

Asymptotically feasible on average

Increase in the randomness of the system improves thedegree of users’ satisfaction


Optimality

If∑M

i=1 Uoi → ∞ and Q

M→ 0 as M → ∞, then

∑Mi=1 U∗

i∑M

i=1 Uoi

≥ 1 − ε and ε → 0 as M → ∞

Uoi : expected utility of user i in optimal power scheduling

If the above conditions are satisfied and U∗i ≥ Ci, ∀i, then

∑Mi=1 U∗

i∑M

i=1 Uoi

→ 1 as M → ∞

Asymptotically optimal


Numerical resultsThe 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 < d2 < d3 < d4

Performance comparison withNon-opportunistic schedulingGreedy scheduling



Comparison of average utilities (104 time-slots)

User 1 2 3 4 TotalNon-opportunistic 0.590 0.590 0.590 0.590 2.360

Greedy 0.973 0.964 0.796 0.168 2.901Our opportunistic 0.951 0.736 0.591 0.591 2.869



2 4 6 8 101

1.5

2

2.5

3

3.5

4

σ

Rat

io o

f ave

rage

tota

l sys

tem

util

ities

Ratio of average total system utility of our opportunistic powerscheduling to that of non-opportunistic power schedulingσ: standard deviation of each user’s channel condition


ConclusionUtility framework

suitable for resource allocation with multi-media and dataservicesa useful tool for resource allocation in the nextgenerations of communication networksnon-convex optimization problems in many cases

Dual approach providesefficient solution in many casessimple algorithm that can be easily implemented with a(distributed) network protocol


Conclusion (cont’d)

In wireless systems

Single server transmission is optimal only when all users havehigh data rate

In general, need dynamic multiple access (dynamicmulti-server system)

Trade-off between efficiency and fairnessOpportunistic scheduling achieves both of them

Randomness of the system could be beneficial to efficient andfair resource allocation, if appropriately exploited


Conclusion (cont’d)

Other problems

Pricing based base-station assignmentconsiders both transmission environment of the user andcongestion level of the base-station

Congestion control on the Internetalgorithms for concave utility functions cause instabilityand congestion in the presence of real-time services withnon-concave utility functionsself-regulating property stabilizes the system andalleviates congestion


Future workScheduling considering

user dynamicsnon-stationary environmentdelay or short-term fairness constraints

Resource allocation considering upper layer protocols (e.g.,TCP)

Resource allocation for uplink and multi-cellular system


Non-convex Optimization and Resource Allocation …major/colloques/nonconvex...Non-convex...

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Transcript of Non-convex Optimization and Resource Allocation …major/colloques/nonconvex...Non-convex...