2640 ieee transactions on wireless ...chiangm/gppower.pdf2640 ieee transactions on wireless...

2640 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 7, JULY 2007

Power Control By Geometric ProgrammingMung Chiang, Chee Wei Tan, Daniel P. Palomar, Daniel O’Neill, and David Julian

Abstract— In wireless cellular or ad hoc networks whereQuality of Service (QoS) is interference-limited, a variety ofpower control problems can be formulated as nonlinear op-timization with a system-wide objective, e.g., maximizing thetotal system throughput or the worst user throughput, subject toQoS constraints from individual users, e.g., on data rate, delay,and outage probability. We show that in the high Signal-to-Interference Ratios (SIR) regime, these nonlinear and apparentlydifficult, nonconvex optimization problems can be transformedinto convex optimization problems in the form of geometricprogramming; hence they can be very efficiently solved forglobal optimality even with a large number of users. In themedium to low SIR regime, some of these constrained nonlinearoptimization of power control cannot be turned into tractableconvex formulations, but a heuristic can be used to computein most cases the optimal solution by solving a series ofgeometric programs through the approach of successive convexapproximation. While efficient and robust algorithms have beenextensively studied for centralized solutions of geometric pro-grams, distributed algorithms have not been explored before. Wepresent a systematic method of distributed algorithms for powercontrol that is geometric-programming-based. These techniquesfor power control, together with their implications to admissioncontrol and pricing in wireless networks, are illustrated throughseveral numerical examples.

Index Terms— Convex optimization, CDMA power control,Distributed algorithms.

I. INTRODUCTION

DUE to the broadcast nature of radio transmission, datarates and other Quality of Service (QoS) in a wireless

network are affected by interference. This is particularlyimportant in Code Division Multiple Access (CDMA) systemswhere users transmit at the same time over the same frequencybands and their spreading codes are not perfectly orthogonal.Transmit power control is often used to tackle this problem ofsignal interference. In this paper, we study how to optimizeover the transmit powers to create the optimal set of Signal-to-Interference Ratios (SIR) on wireless links. Optimalityhere can be with respect to a variety of objectives, such asmaximizing a system-wide efficiency metric (e.g., the totalsystem throughput), or maximizing a Quality of Service (QoS)

Manuscript received December 23, 2005; revised August 11, 2006; ac-cepted October 22, 2006. The associate editor coordinating the review of thispaper and approving it for publication was Q. Zhang.

M. Chiang and C. W. Tan are with the Department of Electrical Engi-neering, Princeton University, Princeton, NJ 08544 USA (e-mail: {chiangm,cheetan}@princeton.edu).

D. P. Palomar was with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544. He is now with the Department of Electronicand Computer Engineering, Hong Kong University of Science and Technol-ogy, Clear Water Bay, Kowloon, Hong Kong (e-mail: [email protected]).

D. O’Neill is with Sun Microsystems, Santa Clara, CA 95054 USA (e-mail:[email protected]).

D. Julian is with Qualcomm, San Diego, CA 92121 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2007.05960.

metric for a user in the highest QoS class, or maximizing aQoS metric for the user with the minimum QoS metric value(i.e., a maxmin optimization).

While the objective represents a system-wide goal to beoptimized, individual users’ QoS requirements also need to besatisfied. Any power allocation must therefore be constrainedby a feasible set formed by these minimum requirementsfrom the users. Such a constrained optimization captures thetradeoff between user-centric constraints and some network-centric objective. Because a higher power level from onetransmitter increases the interference levels at other receivers,there may not be any feasible power allocation to satisfythe requirements from all the users. Sometimes an existingset of requirements can be satisfied, but when a new user isadmitted into the system, there exists no more feasible powercontrol solutions, or the maximized objective is reduced dueto the tightening of the constraint set, leading to the need foradmission control and admission pricing, respectively.

Because many QoS metrics are nonlinear functions ofSIR, which is in turn a nonlinear (and neither convex norconcave) function of transmit powers, in general power controloptimization or feasibility problems are difficult nonlinear op-timization problems that may appear to be NP-hard problems.This paper shows that, when SIR is much larger than 0dB, aclass of nonlinear optimization called Geometric Programming(GP) can be used to efficiently compute the globally optimalpower control in many of these problems, and efficientlydetermine the feasibility of user requirements by returningeither a feasible (and indeed optimal) set of powers or a certifi-cate of infeasibility. This also leads to an effective admissioncontrol and admission pricing method. The key observationis that despite the apparent nonconvexity, through logarithmicchange of variable the GP technique turns these constrainedoptimization of power control into convex optimization, whichis intrinsically tractable despite its nonlinearity in objectiveand constraints. When SIR is comparable to or below 0dB,the power control problems are truly nonconvex with noefficient and global solution methods. In this case, we presenta heuristic that is provably convergent and empirically oftencomputes the globally optimal power allocation by solving asequence of GPs through the approach of successive convexapproximations.

The GP approach reveals the hidden convexity structure,which implies efficient solution methods, in power controlproblems with nonlinear objective functions and specific SIRconstraints. It also clearly differentiates the tractable formula-tions in high-SIR regime from the intractable ones in low-SIRregime. In contrast to the classical Foschini-Miljanic powercontrol [8] and many of the related work, the power controlproblems in this paper have nonlinear objective functions in

1536-1276/07$25.00 c© 2007 IEEE

CHIANG et al.: POWER CONTROL BY GEOMETRIC PROGRAMMING 2641

terms of system performance and are nonconvex optimization.In contrast to more recent work on optimal SIR assignment(e.g., [16]), the formulations here have hard QoS constraints,such as minimum SIR targets. Therefore, the range of powercontrol problems that can be efficiently solved is widened.

Power control by GP is applicable to formulations inboth cellular networks with single-hop transmission betweenmobile users and base stations, and ad hoc networks withmulithop transmission among the nodes, as illustrated throughseveral numerical examples in this paper. Traditionally, GP issolved by centralized computation through the highly efficientinterior point methods. In this paper we present a new resulton how GP can be solved distributively with message passing,which has independent value to general maximization ofcoupled objective, and apply it to power control problems.

The rest of this paper is organized as follows. In sectionII, we provide a concise introduction to GP. Section IIIgeneralizes the results in [10] with several subsections, eachdiscussing GP based power control with different representa-tive formulations in cellular and multihop networks that canbe transformed into convex problems. Then, generalizing thenew results in [21], we present two extensions overcoming thetwo main limitations in [10]: solution method for nonconvexpower control in low-SIR regime in section IV and distributedalgorithm in section V.

II. GEOMETRIC PROGRAMMING

GP is a class of nonlinear, nonconvex optimization problemswith many useful theoretical and computational properties.Since a GP can be turned into a convex optimization problem1,a local optimum is also a global optimum, Lagrange dualitygap is zero under mild conditions, and a global optimumcan always be computed very efficiently. Numerical efficiencyholds both in theory and in practice: interior point methodsapplied to GP have provably polynomial time complexity[14], and are very fast in practice with high-quality softwaredownloadable from the Internet (e.g., the MOSEK package).Convexity and duality properties of GP are well understood,and large-scale, robust numerical solvers for GP are available.Furthermore, special structures in GP and its Lagrange dualproblem lead to distributed algorithms, physical interpreta-tions, and computational acceleration even beyond the genericresults for convex optimization. A detailed tutorial of GP andcomprehensive survey of its recent applications to communi-cation systems can be found in [7]. This section contains abrief introduction of GP terminology for applications to beshown in the next three sections on power control problems.

There are two equivalent forms of GP: standard form andconvex form. The first is a constrained optimization of a typeof function called posynomial, and the second form is obtainedfrom the first through a logarithmic change of variable.

We first define a monomial as a function f : Rn++ → R:

f(x) = d xa(1)

1 xa(2)

2 . . . xa(n)

n

1Minimizing a convex objective function subject to upper bound inequalityconstraints on convex constraint functions and linear equality constraints is aconvex optimization problem.

where the multiplicative constant d ≥ 0 and the exponentialconstants a(j) ∈ R, j = 1, 2, . . . , n. A sum of monomials,indexed by k below, is called a posynomial:

f(x) =K∑

k=1

dkxa(1)k

1 xa(2)k

2 . . . xa(n)k

n .

where dk ≥ 0, k = 1, 2, . . . , K, and a(j)k ∈ R, j =

1, 2, . . . , n, k = 1, 2, . . . , K . For example, 2x−π1 x0.5

2 +3x1x1003

is a posynomial in x, x1−x2 is not a posynomial, and x1/x2

is a monomial, thus also a posynomial.Minimizing a posynomial subject to posynomial upper

bound inequality constraints and monomial equality con-straints is called GP in standard form:

minimize f0(x)subject to fi(x) ≤ 1, i = 1, 2, . . . , m,

hl(x) = 1, l = 1, 2, . . . , M(1)

where fi, i = 0, 1, . . . , m, are posynomials: fi(x) =∑Ki

k=1 dikxa(1)ik

1 xa(2)ik

2 . . . xa(n)ik

n , and hl, l = 1, 2, . . . , M are

monomials: hl(x) = dlxa(1)l

1 xa(2)l

2 . . . xa(n)l

n .GP in standard form is not a convex optimization problem,

because posynomials are not convex functions. However,with a logarithmic change of the variables and multiplicativeconstants: yi = log xi, bik = log dik, bl = log dl, and alogarithmic change of the functions’ values, we can turn itinto the following equivalent problem in y:

minimize p0(y) = log∑K0

k=1 exp(aT0ky + b0k)

subject to pi(y) = log∑Ki

k=1 exp(aTiky + bik) ≤ 0, ∀i,

ql(y) = aTl y + bl = 0, l = 1, 2, . . . , M.

(2)This is referred to as GP in convex form, which is a convexoptimization problem since it can be verified that the log-sum-exp function is convex [5].

In summary, GP is a nonlinear, nonconvex optimizationproblem that can be transformed into a nonlinear, convex prob-lem. GP in standard form can be used to formulate networkresource allocation problems with nonlinear objectives undernonlinear QoS constraints. The basic idea is that resources areoften allocated proportional to some parameters, and whenresource allocations are optimized over these parameters,we are maximizing an inverted posynomial subject to lowerbounds on other inverted posynomials, which are equivalentto GP in standard form.

III. POWER CONTROL BY

GEOMETRIC PROGRAMMING: CONVEX CASE

Various schemes for power control, centralized or distrib-uted, have been extensively studied since 1990s based ondifferent transmission models and application needs, e.g., in[2], [8], [13], [19], [20], [23]. This section summarizes the newapproach of formulating power control problems through GP.The key advantage is that globally optimal power allocationscan be efficiently computed for a variety of nonlinear system-wide objectives and user QoS constraints, even when thesenonlinear problems appear to be nonconvex optimization.


A. Basic model

Consider a wireless (cellular or multihop) network with nlogical transmitter/receiver pairs. Transmit powers are denotedas P1, . . . , Pn. In the cellular uplink case, all logical receiversmay reside in the same physical receiver, i.e., the base station.In the multihop case, since the transmission environment canbe different on the links comprising an end-to-end path, powercontrol schemes must consider each link along a flow’s path.

Under Rayleigh fading, the power received from transmitterj at receiver i is given by GijFijPj where Gij ≥ 0 representsthe path gain (it may also encompass antenna gain and codinggain) that is often modeled as proportional to d−γ

ij , wheredij denotes distance, γ is the power fall-off factor, and Fij

models Rayleigh fading and are independent and exponentiallydistributed with unit mean. The distribution of the receivedpower from transmitter j at receiver i is then exponential withmean value E [GijFijPj ] = GijPj . The SIR for the receiveron logical link i is

SIRi =PiGiiFii∑N

j �=i PjGijFij + ni

(3)

where ni is the noise power for receiver i.The constellation size M used by a link can be

closely approximated for M-ary Quadrature Amplitude Mod-ulation (MQAM) modulations as follows: M = 1 +(−φ1SIR)/(ln(φ2BER)), where BER is the bit error rateand φ1, φ2 are constants that depend on the modulationtype [9]. Defining T as the symbol period and K =(−φ1)/(ln(φ2BER)) leads to an expression of the data rateRi on the ith link as a function of the SIR: Ri = 1

T log2(1 +KSIRi), which can be approximated as

Ri =1T

log2(KSIRi) (4)

when KSIR is much larger than 1. This approximation isreasonable either when the signal level is much higher than theinterference level or, in CDMA systems, when the spreadinggain is large. For notational simplicity in the rest of this paper,we redefine Gii as K times the original Gii, thus absorbingconstant K into the definition of SIR.

The aggregate data rate for the system can then be writtenas

Rsystem =∑

i

Ri =1T

log2

[∏i

SIRi

].

So in the high SIR regime, aggregate data rate maximizationis equivalent to maximizing a product of SIR. The systemthroughput is the aggregate data rate supportable by the systemgiven a set of users with specified QoS requirements.

Outage probability is another important QoS parameterfor reliable communication in wireless networks. A channeloutage is declared and packets lost when the received SIRfalls below a given threshold SIRth, often computed from theBER requirement. Most systems are interference dominatedand the thermal noise is relatively small, thus the ith linkoutage probability is

Po,i = Prob{SIRi ≤ SIRth}= Prob{GiiFiiPi ≤ SIRth

∑j �=i

GijFijPj}.

The outage probability can be expressed as Po,i = 1 −∏j �=i 1/(1 + SIRthGijPj

GiiPi) [11], which means that the upper

bound Po,i ≤ Po,i,max can be written as an upper bound ona posynomial in P:∏

j �=i

(1 +

SIRthGijPj

GiiPi

)≤ 1

1 − Po,i,max. (5)

B. Cellular wireless networks

We first present how GP-based power control applies tocellular wireless networks with one-hop transmission from Nusers to a base station, extending the scope of power control bythe classical solution in CDMA systems that equalizes SIRs,and those by the iterative algorithms (e.g., in [2], [8], [13])that minimize total power (a linear objective function) subjectto SIR constraints.

We start the discussion on the suite of power control prob-lem formulations with a simple objective function and simpleconstraints. The following constrained problem of maximizingthe SIR of a particular user i∗ is a GP:

maximize Ri∗(P)subject to Ri(P) ≥ Ri,min, ∀i,

Pi1Gi1 = Pi2Gi2,0 ≤ Pi ≤ Pi,max, ∀i.

The first constraint, equivalent to SIRi ≥ SIRi,min, setsa floor on the SIR of other users and protects these usersfrom user i∗ increasing its transmit power excessively. Thesecond constraint reflects the classical power control criterionin solving the near-far problem in CDMA systems: the ex-pected received power from one transmitter i1 must equal thatfrom another i2. The third constraint is regulatory or systemlimitations on transmit powers. All constraints can be verifiedto be inequality upper bounds on posynomials in transmitpower vector P.

Alternatively, we can use GP to maximize the minimumrate among all users. The maxmin fairness objective:

maximize mini

{Ri(P)}can also be accommodated in GP-based power control becauseit can be turned into equivalently maximizing an auxiliaryvariable t such that SIRi(P) ≥ exp(t), ∀i, which hasposynomial objective and constraints in (P, t).

Example 1. A simple system comprised of five users isused for a numerical example. The five users are spaced atdistances d of 1, 5, 10, 15, and 20 units from the base station.The power fall-off factor γ = 4. Each user has a maximumpower constraint of Pmax = 0.5mW . The noise power is0.5μW for all users. The SIR of all users, other than theuser we are optimizing for, must be greater than a commonthreshold SIR level β. In different experiments, β is varied toobserve the effect on the optimized user’s SIR. This is doneindependently for the near user at d = 1, a medium distanceuser at d = 15, and the far user at d = 20. The results areplotted in Figure 1.

Several interesting effects are illustrated. First, when therequired threshold SIR in the constraints is sufficiently high,there are no feasible power control solutions. At moderate


−5 0 5 10−20

−15

−10

−5

0

5

10

15

20

Threshold SIR (dB)

Opt

imiz

ed S

IR (

dB)

Optimized SIR vs. Threshold SIR

nearmediumfar

Fig. 1. Constrained optimization of power control in a cellular network(Example 1).

threshold SIR, as β is decreased, the optimized SIR initiallyincreases rapidly. This is because it is allowed to increaseits own power by the sum of the power reductions in thefour other users, and the noise is relatively insignificant. Atlow threshold SIR, the noise becomes more significant andthe power trade-off from the other users less significant, sothe curve starts to bend over. Eventually, the optimized userreaches its upper bound on power and cannot utilize the excesspower allowed by the lower threshold SIR for other users. Thisis exhibited by the transition from a sharp bend in the curveto a much shallower sloped curve.

We now proceed to show that GP can also be applied tothe problem formulations with an overall system objective oftotal system throughput, under both user data rate constraintsand outage probability constraints.

The following constrained problem of maximizing systemthroughput is a GP:

maximize Rsystem(P)subject to Ri(P) ≥ Ri,min, ∀i,

Po,i(P) ≤ Po,i,max, ∀i,0 ≤ Pi ≤ Pi,max, ∀i

(6)

where the optimization variables are the transmit powers P.The objective is equivalent to minimizing the posynomial∏

i ISRi, where ISR is 1/SIR. Each ISR is a posynomial in Pand the product of posynomials is again a posynomial. Thefirst constraint is from the data rate demand Ri,min by eachuser. The second constraint represents the outage probabilityupper bounds Po,i,max. These inequality constraints put upperbounds on posynomials of P, as can be readily verifiedthrough (4) and (5). Thus (6) is indeed a GP, and efficientlysolvable for global optimality.

There are several obvious variations of problem (6) thatcan be solved by GP, e.g., we can lower bound Rsystem asa constraint and maximize Ri∗ for a particular user i∗, orhave a total power

∑i Pi constraint or objective function. The

objective function to be maximized can also be generalized toa weighted sum of data rates:

∑i wiRi where w � 0 is a

20m

20mA

C

B

D

1 2

3 4

Fig. 2. A small wireless multihop network (Example 2).

given weight vector. This is still a GP because maximizing∑i wi log SIRi is equivalent to maximizing log

∏i SIRwi

i ,which is in turn equivalent to minimizing

∏i ISRwi

i . Now useauxiliary variables {ti}, and minimize

∏i twi

i over the originalconstraints in (6) plus the additional constraints ISRi. ≤ ti forall i. This is readily verified to be a GP, and is equivalent tothe original problem.

Generalizing the above discussions and observing that high-SIR assumption is needed for GP formulation only when thereare sums of log(1+SIR) in the optimization problem, we havethe following summary.

Proposition 1: In the high-SIR regime, any combinationof objectives (A)-(E) and constraints (a)-(e) in Table I isa power control problem that can be solved by GP, i.e.,can be transformed into a convex optimization with efficientalgorithms to compute the globally optimal power vector.When objectives (C)-(D) and constraints (c)-(d) do not appear,the power control optimization problem can be solved by GPin any SIR regime.

In addition to efficient computation of the globally optimalpower allocation with nonlinear objectives and constraints,GP can also be used for admission control based on feasi-bility study described in [7], and for determining which QoSconstraint is a performance bottleneck, i.e., met tightly at theoptimal power allocation.2

C. Extensions: Multihop wireless networks

In wireless multihop networks, system throughput may bemeasured either by end-to-end transport layer utilities or bylink layer aggregate throughput. GP application to the firstapproach has appeared in [6], and we focus on the second(and easier) approach in this section. Formulations in TableI can be readily extended to multihop case by indexing eachlogical link as i. An example is presented below for the totalthroughput maximization formulation (6).

Example 2. Consider a simple four node multihop networkshown in Figure 2. There are two connections A → B → D

2This is because most GP solution algorithms solve both the primal GP andits Lagrange dual problem: by complementary slackness condition, a resourceconstraint is tight at optimal power allocation when the corresponding optimaldual variable is non-zero.


TABLE I

SUITE OF POWER CONTROL OPTIMIZATION SOLVABLE BY GP

Objective Function Constraints

(A) Maximize Ri∗ (specific user) (a) Ri ≥ Ri,min (rate constraint)

(B) Maximize mini Ri (worst-case user) (b) Pi1Gi1 = Pi2Gi2 (near-far constraint)

(C) Maximize∑

iRi (total throughput) (c)

∑iRi ≥ Rsystem,min (total throughput constraint)

(D) Maximize∑

iwiRi (weighted rate sum) (d) Po,i ≤ Po,i,max (outage prob. constraint)

(E) Minimize∑

iPi (total power) (e) 0 ≤ Pi ≤ Pi,max (power constraint)

and A → C → D. Nodes A and D, as well as B andC, are separated by a distance of 20m. Path gain between atransmitter and a receiver has a common fall-off factor γ = 4.Each link has a maximum transmit power of 1mW. All nodesuse MQAM modulation. The minimum data rate for eachconnection is 100bps, and the target BER is 10−3. AssumingRayleigh fading, we require outage probability be smaller than0.1 on all links for an SIR threshold of 10dB. Spreading gain is200. Using GP formulation (6), we find the maximized systemthroughput R∗ = 216.8kbps, R∗

i = 54.2kbps for each link,P ∗

1 = P ∗3 = 0.709mW and P ∗

2 = P ∗4 = 1mW. The resulting

optimized SIR is 21.7dB on each link. For this topology, wealso consider an illustrative example of admission control andpricing. Three new users U1, U2, and U3 are going to arriveto the network in order. U1 and U2 require 30kbps sent alongthe upper path A → B → D, while U3 requires 10kbps sentfrom A → B. All three users require the outage probabilityto be less than 0.1. When U1 arrives at the system, its priceis the baseline price. Next, U2 arrives, and its QoS demandsdecrease the maximum system throughput from 216.82kbps to116.63kbps, so its price is the baseline price plus an amountproportional to the reduction in system throughput. Finally,U3 arrives, and its QoS demands produce no feasible powerallocation solution, so she is not admitted to the system.

D. Extensions: Queuing models

We now turn to delay and buffer overflow properties tobe included in constraints or objective function of powercontrol optimization. The average delay a packet experiencestraversing a network is an important design considerationin some applications. Queuing delay is often the primarysource of delay, particularly for bursty data traffic in multihopnetworks. A node i first buffers the received packets in aqueue and then transmits these packets at a rate R set bythe SIR on the egress link, which is in turn determined by thetransmit powers P. A FIFO queuing discipline is used here forsimplicity. Routing is assumed to be fixed, and is feed-forwardwith all packets visiting a node at most once.

Packet traffic entering the multihop network at the trans-mitter of link i is assumed to be Poisson with parameter λi

and to have an exponentially distributed length with parameterΓ. Using the model of an M/M/1 queue as in [3], theprobability of transmitter i having a backlog of Ni = kpackets to transmit is well-known to be Prob{Ni = k} =(1 − ρ)ρk where ρ = λi/ΓRi(P), and the expected delay is1/(ΓRi(P)−λi). Under the feed-forward routing and Poisson

input assumptions, Burke’s theorem in [3] can be applied.Thus the total packet arrival rate at node i is Λi =

∑j∈I(i) λj

where I(i) is the set of connections traversing node i. Theexpected delay D̄i can be written as

D̄i =1

ΓRi(P) − Λi. (7)

A bound D̄i,max on D̄i can thus be written as1/(Γ log2(SIRi)/T − Λi) ≤ D̄i,max, or equivalently,ISRi(P) ≤ 2−T (D̄−1

max+Λi)/Γ, an upper bound on a posynomialISR of P that is allowed in GP formulations.

The probability PBO of dropping a packet due to bufferoverflow at a node is also important in several applications.It is again a function of P and can be written as PBO,i =Prob{Ni > B} = ρB+1 where B is the buffer sizeand ρ = Λi/(ΓRi(P)). Setting an upper bound PBO,i,max

on the buffer overflow probability also gives a posynomiallower bound constraint in P: ISRi(P) ≤ 2−Ψ where Ψ =(TΛi)/(Γ(PBO,i,max)

1B+1 ), which is allowed in GP formula-

tions. In summary, we have the following.Proposition 2: The following nonlinear problem of opti-

mizing powers to maximize system throughput in the high-SIRregime, subject to constraints on outage probability, expecteddelay, and the probability of buffer overflow, is a GP:

maximize Rsystem(P)subject to D̄i(P) ≤ D̄i,max, ∀i,

PBO,i(P) ≤ PBO,i,max, ∀i,Any combination of constraints (a)-(e)in Table (I)

(8)

where the optimization variables are the transmit powers P.Example 3. Consider a numerical example of the opti-

mal tradeoff between maximizing the system throughput andbounding the expected delay for the network shown in Figure3. There are six nodes, eight links, and five multihop connec-tions. All sources are Poisson with intensity λi = 200 pack-ets per second, and exponentially distributed packet lengthswith an expectation of of 100 bits. The nodes use CDMAtransmission scheme with a symbol rate of 10k symbols persecond and the spreading gain is 200. Transmit powers arelimited to 1mW and the target BER is 10−3. The path lossmatrix is calculated based on a power falloff of d−4 with thedistance d, and a separation of 10m between any adjacentnodes along the perimeter of the network. Figure 4 showsthe maximized system throughput for different upper boundnumerical values in the expected delay constraints, obtained


A

B C

D

E F

1

2

3

4

5

6

7 8

Fig. 3. Topology and flows in a multihop wireless network (Example 3).

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1220

230

240

250

260

270

280

290Optimized throughput for different delay bounds

Delay bound (s)

Max

imiz

ed s

yste

m th

roug

hput

(kb

ps)

Fig. 4. Optimal tradeoff between maximized system throughput and averagedelay constraint (Example 3).

by solving a sequence of GPs, one for each point on the curve.There is no feasible power allocation to achieve delay smallerthan 0.036s. As the delay bound is relaxed, the maximizedsystem throughput increases sharply first, then more slowlyuntil the delay constraints are no longer active. GP efficientlyreturns the globally Pareto-optimal tradeoff curve betweensystem throughput and queuing delay.

There are two main limitations in the GP-based powercontrol methods discussed so far: high-SIR assumption andcentralized computation. Both can be overcome as discussedin the next two sections.

IV. POWER CONTROL BY

GEOMETRIC PROGRAMMING: NON-CONVEX CASE

If we maximize the total throughput Rsystem in the mediumto low SIR case, i.e., when SIR is not much larger than 0dB,the approximation of log(1 + SIR) as log SIR does not hold.Unlike SIR, which is an inverted posynomial, 1+SIR is not aninverted posynomial. Instead, 1/(1 + SIR) is a ratio betweentwo posynomials:

f(P)g(P)

=

∑j �=i GijPj + ni∑j GijPj + ni

. (9)

Minimizing or upper bounding a ratio between two posyn-omials belongs to a truly nonconvex class of problems knownas Complementary GP [1], [7] that is an intractable NP-hardproblem. An equivalent generalization of GP is SignomialProgramming (SP) [1], [7]: minimizing a signomial subjectto upper bound inequality constraints on signomials, where asignomial s(x) is a sum of monomials, possibly with negativemultiplicative coefficients: s(x) =

∑Ni=1 cigi(x) where c ∈

RN and gi(x) are monomials.3

A. Successive convex approximation method

Consider the following nonconvex problem:

minimize f0(x)subject to fi(x) ≤ 1, i = 1, 2, . . . , m,

(10)

where f0 is convex without loss of generality4, but thefi(x)’s, ∀i are nonconvex. Since directly solving this problemis difficult, we will solve it by a series of approximationsf̃i(x) ≈ fi(x), ∀x, each of which can be optimally solved inan easy way. It turns out that if the approximations satisfythe following three properties, then the solutions of this seriesof approximations converge to a point satisfying the Karush-Kuhn-Tucker (KKT) conditions of the original problem [12]:

(1) fi(x) ≤ f̃i(x) for all x,(2) fi(x0) = f̃i(x0) where x0 is the optimal solution of the

approximated problem in the previous iteration,(3) ∇fi(x0) = ∇f̃i(x0).Condition (1) guarantees that the approximation f̃i(x) is

tightening the constraints in (10), and any solution of theapproximated problem will be a feasible point of the originalproblem in (10). Condition (2) guarantees that the solution ofeach approximated problem will decrease the cost function:f0(x(k)) ≤ f0(x(k−1)), where x(k) is the solution to the k-thapproximated problem. Condition (3) guarantees that the KKTconditions of the original problem in (10) will be satisfied afterthe series of approximations converges.

The following algorithm describes the generic successiveapproximation approach.

Algorithm Successive approximation to a nonconvex prob-lem

Input Method to approximate fi(x) with f̃i(x) , ∀i, aroundsome point of interest x0.

Output A vector that satisfies the KKT conditions of theoriginal problem.

0) Choose an initial feasible point x(0) and set k = 1.1) Form the k-th approximated problem of (10) based on

approximating fi(x) with f̃i(x) around the previous pointx(k−1).

2) Solve the k-th approximated problem to obtain x(k).3) Increment k and go to step 2 until convergence to a

stationary point.

3An SP can always be converted into a Complementary GP, because aninequality in SP, which can be written as fi1(x)−fi2(x) ≤ 1, where fi1, fi2

are posynomials, is equivalent to an inequality fi1(x)/(1 + fi2(x)) ≤ 1 inComplementary GP. Similarly, an upper bound on a ratio of posynomials inComplementary GP can be rewritten as an SP inequality constraint.

4If f0 is nonconvex, we can move the objective function tothe constraints by introducing auxiliary scalar variable t and writingminimize t subject to the additional constraint f0(x) − t ≤ 0.


B. Examples of successive convex approximation

1) Logarithmic approximation for GP: In [17], [18], anonconvex problem involving the function log(1 + SIR) isapproximated by a + b log(SIR) for some a and b that satisfythe above three conditions.

2) Single condensation method for GP: ComplementaryGPs involve upper bounds on the ratio of posynomials as in(9); they can be turned into GPs by approximating the denomi-nator of the ratio of posynomials, g(x), with a monomial g̃(x),but leaving the numerator f(x) as a posynomial.

Lemma 1: Let g(x) =∑

i ui(x) be a posynomial. Then

g(x) ≥ g̃(x) =∏

i

(ui(x)αi

)αi

. (11)

If, in addition, αi = ui(x0)/g(x0), ∀i, for any fixed positivex0, then g̃(x0) = g(x0), and g̃(x0) is the best local monomialapproximation to g(x0) near x0 in the sense of first orderTaylor approximation.

Proof: The arithmetic-geometric mean inequality statesthat

∑i αivi ≥ ∏

i vαi

i , where v 0 and α � 0, 1T α =1. Letting ui = αivi, we can write this basic inequality as∑

i ui ≥∏

i (ui/αi)αi . The inequality becomes an equality if

we let αi = ui/∑

i ui, ∀i, which satisfies the condition thatα � 0 and 1T α = 1. The best local monomial approximationg̃(x0) of g(x0) near x0 can be easily verified [4].

Proposition 3: The approximation of a ratio of posynomialsf(x)/g(x) with f(x)/g̃(x), where g̃(x) is the monomialapproximation of g(x) using the arithmetic-geometric meanapproximation of Lemma 1, satisfies the three conditions forthe convergence of the successive approximation method.

Proof: Conditions (1) and (2) are clearly satisfied sinceg(x) ≥ g̃(x) and g̃(x0) = g(x0) (Lemma 1). Condition (3) iseasily verified by taking derivatives of g(x) and g̃(x).

Suppose we want to minimize f0(x) subject to an equalityconstraint on a ratio of posynomials f(x)/g(x) = 1. Then,we can rewrite this as minimizing f0(x) + φt subject tof(x)/g(x) ≤ 1 and f(x)/g(x) ≥ 1 − t where φ is a suf-ficiently large number to guarantee that the optimum solutionwill have t ≈ 0.5 The second constraint can be rewritten asg(x)/(f(x) + tg(x)) ≤ 1 where we can apply the singlecondensation method to the denominators of both constraints.

3) Double condensation method for GP[1]: Another choiceof approximation is to make a double monomial approxi-mation for both the denominator and numerator in (9). Wecan still use the arithmetic-geometric mean approximation ofLemma 1 as a monomial approximation for the denominator.But, Lemma 1 cannot be used as a monomial approximationfor the numerator. To satisfy the three conditions for theconvergence of the successive approximation method, a mono-mial approximation for the numerator f(x) should satisfyf(x) ≤ f̃(x).

C. Applications to power control

Figure 5 shows a block diagram of the approach of GP-based power control for general SIR regime. In the high SIRregime, we need to solve only one GP. In the medium to low

5In our numerical analysis, we use φ = 1 + k at the k-th iteration.

(High SIR) OriginalProblem

Solve1 GP

(Mediumto

Low SIR)

OriginalProblem

SPComplementaryGP (Condensed)

Solve1 GP

�

� � �

�

Fig. 5. GP-based power control for general SIR regime.

SIR regimes, we solve truly nonconvex power control prob-lems that cannot be turned into convex formulation through aseries of GPs.

GP-based power control problems in the medium to low SIRregimes become SP (or, equivalently, Complementary GP),which can be solved by the single or double condensationmethod. We focus on the single condensation method here.Consider a representative problem formulation of maximizingtotal system throughput in a cellular wireless network subjectto user rate and outage probability constraints in problem (6),which can be explicitly written out as

minimize∏N

i=11

1+SIRi

subject to (2TRi,min − 1) 1SIRi

≤ 1, ∀i,

(SIRth)N−1(1 − Po,i,max)∏N

j �=iGijPj

GiiPi≤ 1, ∀i,

Pi(Pi,max)−1 ≤ 1, ∀i.(12)

All the constraints are posynomials. However, the objectiveis not a posynomial, but a ratio between two posynomialsas in (9). This power control problem can be solved by thecondensation method by solving a series of GPs. Specifically,we have the following single-condensation algorithm:

Algorithm Single condensation GP power controlInput An initial feasible power vector P.Output A power allocation that satisfies the KKT condi-

tions.1) Evaluate the denominator posynomial of the objective

function in (12) with the given P.2) Compute for each term i in this posynomial,

αi =value of ith term in posynomial

value of posynomial.

3) Condense the denominator posynomial of the (12) ob-jective function into a monomial using (11) with weights αi.

4) Solve the resulting GP using an interior point method.5) Go to step 1 using P of step 4.6) Terminate the kth loop if ‖ P(k) − P(k−1) ‖≤ ε where

ε is the error tolerance for exit condition.As condensing the objective in the above problem gives

us an underestimate of the objective value, each GP in thecondensation iteration loop tries to improve the accuracy of theapproximation to a particular minimum in the original feasibleregion. All three conditions for convergence in subsection IV.Aare satisfied, and the algorithm is provably convergent. Em-pirically through extensive numerical experiments, we observethat it often compute the globally optimal power allocation.

Example 4. We consider a wireless cellular network with 3users. Let T = 10−6s, Gii = 1.5, and generate Gij , i �= j, as


0 50 100 150 200 250 300 350 400 450 5005050

5100

5150

5200

5250

5300

Experiment index

Tota

l sys

tem

thro

ughp

ut a

chie

ved

Optimized total system throughput

Fig. 6. Maximized total system throughput achieved by (single) condensationmethod for 500 different initial feasible vectors (Example 4). Each pointrepresents a different experiment with a different initial power vector.

independent random variables uniformly distributed between0 and 0.3. Threshold SIR is SIRth = −10dB, and minimaldata rate requirements are 100 kbps, 600 kbps and 1000 kbpsfor logical links 1, 2 and 3 respectively. Maximal outageprobabilities are 0.01 for all links, and maximal transmitpowers are 3mW, 4mW and 5mW for link 1, 2 and 3,respectively. For each instance of SP power control (12),we pick a random initial feasible power vector P uniformlybetween 0 and Pmax. Figure 6 compares the maximizedtotal network throughput achieved over five hundred sets ofexperiments with different initial vectors. With the (single)condensation method, SP converges to different optima overthe entire set of experiments, achieving (or coming very closeto) the global optimum at 5290 bps 96% of the time anda local optimum at 5060 bps 4% of the time, thus is verylikely to converge to or very close to the global optimum. Theaverage number of GP iterations required by the condensationmethod over the same set of experiments is 15 if an extremelytight exit condition is picked for SP condensation iteration:ε = 1 × 10−10. This average can be substantially reduced byusing a larger ε, e.g., increasing ε to 1 × 10−2 requires onaverage only 4 GPs.

We have thus far discussed a power control problem (12)where the objective function needs to be condensed. Themethod is also applicable if some constraint functions aresignomials and need to be condensed. For example, considerthe case of differentiated services where a user expects toobtain a predicted QoS relatively better than the other users.We may have a proportional delay differentiation model wherea user who pays more tariff obtains a delay proportionallylower as compared to users who pay less. Then for a particularratio between any user i and j, σij , we have

Di

Dj

= σij , (13)

which, by (7), is equivalent to

1 + SIRj

(1 + SIRi)σij= 2(λj−σijλi)T/Γ. (14)

TABLE II

EXAMPLES 4 AND 5. ROW 2 SHOWS EXAMPLE 4 USING THE SINGLE

CONDENSATION ON SYSTEM THROUGHPUT MAXIMIZATION ONLY, AND

ROW 1 SHOWS THE OPTIMAL SOLUTIONS FOUND BY EXHAUSTIVE

SEARCH. ROW 4 SHOWS EXAMPLE 5 WITH AN ADDITIONAL DIFFSERV

CONSTRAINT D1/D3 = 1, WHICH ALSO RECOVERS THE NEW OPTIMAL

SOLUTION OF 6624 KBPS AS VERIFIED BY EXHAUSTIVE SEARCH, AND

ROW 3 SHOWS THE OPTIMAL SOLUTIONS FOUND BY EXHAUSTIVE

SEARCH.

Method System throughput P ∗1 P ∗

2 P ∗3

Exhaustive 6626 kbps 0.65 0.77 0.79search

Single 6626 kbps by 0.65 0.77 0.78condensation solving 17 GPs

Exhaustive 6624 kbps 0.68 0.75 0.68search

With DiffServ 6624 kbps by 0.68 0.75 0.68constraint solving 17 GPs

0 2 4 6 8 10 12 14 16 180.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Number of iterations

Pow

er (m

W)

Transmit power for each user

User 1User 2User 3

Fig. 7. Convergence of power variables (Example 5).

The denominator on the left hand side is a ratio betweenposynomials raised to a positive power. Therefore, the singlecondensation method can be readily used to solve the pro-portional delay differentiation problem because the functionon the left hand side can be condensed using successiveapproximation on an equality constraint of a posynomial inSection IV-B.

Example 5. We consider the wireless cellular network inExample 4 with an additional constraint D1/D3 = 1. Thearrival rates of each user at base station is measured andinput as network parameters into (14). Figures 7 and 8 showthe convergence towards satisfying all the QoS constraintsincluding the DiffServ constraint. As shown on the figures,the convergence is fast, with the power allocations very closeto the optimal power allocation by the 8th GP iteration. Also,Table II summarizes the optimizers for Examples 4 and 5 usingε = 1 × 10−10.


0 2 4 6 8 10 12 14 16 186460

6480

6500

6520

6540

6560

6580

6600

6620

6640

Tota

l sys

tem

thro

ughp

ut a

chie

ved

Number of iterations

Optimized total system throughput

Fig. 8. Convergence of total system throughput (Example 5).

V. DISTRIBUTED IMPLEMENTATION

A limitation for GP based power control in ad hoc networkswithout base stations is the need for centralized computation(e.g., by interior point methods). The GP formulations ofpower control problems can also be solved by a new method ofdistributed algorithm for GP. The basic idea is that each usersolves its own local optimization problem and the couplingamong users is taken care of by message passing among theusers. Interestingly, the special structure of coupling for theproblem at hand (essentially, all coupling among the logicallinks can be lumped together using interference terms) allowsone to further reduce the amount of message passing amongthe users. Specifically, we use a dual decomposition method todecompose a GP into smaller subproblems whose solutions arejointly and iteratively coordinated by the use of dual variables.The key step is to introduce auxiliary variables and to addextra equality constraints, thus transferring the coupling in theobjective to coupling in the constraints, which can be solvedby introducing ‘consistency pricing’. We illustrate this ideathrough an unconstrained GP followed by an application ofthe technique to power control.

A. Distributed algorithm for GP

Suppose we have the following unconstrained standard formGP in x 0:

minimize∑

i fi(xi, {xj}j∈I(i)) (15)

where I(i) is the set of users coupled with the ith user, and xi

denotes the local variable of the ith user, {xj}j∈I(i) denote thecoupled variables from other users, and fi is either a monomialor posynomial. Making a change of variable yi = log xi, ∀i,in the original problem, we obtain

minimize∑

i fi(eyi , {eyj}j∈I(i)).

We now rewrite the problem by introducing auxiliary vari-ables yij for the coupled arguments and additional equality

constraints to enforce consistency:

minimize∑

i fi(eyi , {eyij}j∈I(i))subject to yij = yj , ∀j ∈ I(i), ∀i.

(16)

Each ith user controls the local variables (yi, {yij}j∈I(i)).Next, the Lagrangian of (16) is formed as

L({yi}, {yij}; {γij})=

∑i

fi(eyi , {eyij}j∈I(i)) +∑

i

∑j∈I(i)

γij(yj − yij)

=∑

i

Li(yi, {yij}; {γij})

where

Li(yi, {yij}; {γij}) (17)

= fi(eyi , {eyij}j∈I(i)) +

( ∑j:i∈I(j)

γji

)yi −

∑j∈I(i)

γijyij .

The minimization of the Lagrangian with respect to theprimal variables ({yi}, {yij}) can be done simultaneously anddistributively by each user in parallel. In the more general casewhere the original problem (15) is constrained, the additionalconstraints can be included in the minimization at each Li.

In addition, the following master Lagrange dual problem hasto be solved to obtain the optimal dual variables or consistencyprices {γij}:

max{γij}

g({γij}) (18)

where

g({γij}) =∑

i

minyi,{yij}

Li(yi, {yij}; {γij}).

Note that the transformed primal problem (16) is convex,and the duality gap is zero under mild conditions; hencethe Lagrange dual problem (18) indeed solves the originalstandard GP problem (15). A simple way to solve the maxi-mization in (18) is with the following subgradient update forthe consistency prices:

γij(t + 1) = γij(t) + δ(t)(yj(t) − yij(t)). (19)

Appropriate choice of the stepsize δ(t) > 0, e.g., δ(t) = δ0/tfor some constant δ0 > 0, leads to convergence of the dualalgorithm [7].

Summarizing, the ith user has to: i) minimize the La-grangian Li in (18) involving only local variables uponreceiving the updated dual variables {γji, j : i ∈ I(j)} (notethat {γij , j ∈ I(i)} are local dual variables), and ii) updatethe local consistency prices {γij , j ∈ I(i)} with (19), andbroadcast the updated prices to the coupled users.

B. Applications to power control

As an illustrative example, we maximize the total systemthroughput in the high SIR regime with constraints local toeach user. If we directly applied the distributed approachdescribed in the last subsection, the resulting algorithm wouldnot be very practical since it would require knowledge byeach user of the interfering channels and interfering transmitpowers, which would translate into a large amount of message


passing. To obtain a practical distributed solution, we canleverage the structures of power control problems at hand,and instead keep a local copy of each of the effective receivedpowers PR

ij = GijPj . Again using problem (6) as an exampleformulation and assuming high SIR, we can write the problemas following (after the logarithmic change of variable x̃ =log x):

minimize∑

i log(G−1

ii exp(−P̃i)(∑

j �=i exp(P̃Rij ) + σ2

))subject to P̃R

ij = G̃ij + P̃j ,

Constraints local to each user, i.e., (a),(d) and (e)in Table (I).

(20)The partial Lagrangian is

L =∑

i

log

⎛⎝G−1

ii exp(−P̃i)

⎛⎝∑

j �=i

exp(P̃Rij ) + σ2

⎞⎠⎞⎠+

∑i

∑j �=i

γij

(P̃R

ij −(G̃ij + P̃j

)), (21)

and the separable property of the Lagrangian in (21) canbe exploited to determine the optimal power allocation P∗

distributively.6 The distributed power control algorithm issummarized as follows.

Algorithm Distributed power allocation update to maximizeRsystem.

Input Each ith user sets γij (0) = 0, ∀j and agrees amongall users on a constant δ0 > 0 for the stepsize.

Output Optimal power allocation P∗ for all users.At each iteration t:1) Each ith user receives the term

(∑j �=i γji(t)

)involving

the dual variables from the interfering users by messagepassing and minimizes the following local Lagrangian withrespect to P̃i(t),

{P̃R

ij (t)}

jsubject to the local constraints:

Li

(P̃i(t),

{P̃R

ij (t)}

j; {γij(t)}j

)

= log

⎛⎝G−1

ii exp(−P̃i(t))

⎛⎝∑

j �=i

exp(P̃Rij (t)) + σ2

⎞⎠⎞⎠

+∑j �=i

γij P̃Rij (t) −

⎛⎝∑

j �=i

γji(t)

⎞⎠ P̃i(t). (22)

2) Each ith user estimates the effective received power fromeach of the interfering users PR

ij (t) = GijPj(t) for j �= i,updates the dual variable as

γij (t + 1) = γij (t) + (δ0/t)(P̃R

ij (t) − log GijPj(t))

,

(23)and then broadcast them by message passing to all interferingusers in the system.

The amount of message passing can be further reduced inpractice by ignoring the messages from links that are phys-ically far apart, leading to suboptimal distributed heuristics.If the system can be divided into clusters so that interference

6A small quadratic term in P̃i for each i is added to the partial Lagrangianin (21) to enforce strict convexity of (21) in P̃.

0 50 100 150 2000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4

Iteration

Dual objective function

Fig. 9. Convergence of the dual objective function through distributedalgorithm (Example 6).

0 50 100 150 200−10

−8

−6

−4

−2

0

2

Iteration

Consistency of the auxiliary variables

log(P2

)

log(P R12

/G12

)

log(P R32

/G32

)

Fig. 10. Convergence of the consistency constraints through distributedalgorithm (Example 6).

happens primarily within the clusters but not across clusters,the signaling can be significantly reduced by tailoring thedistributed approach on a cluster-basis [22].

Example 6. We apply the distributed algorithm to solvethe above power control problem for three logical links withGij = 0.2, i �= j, Gii = 1, ∀i, maximal transmit powers of6mW, 7mW and 7mW for link 1, 2 and 3 respectively. Figure 9shows the convergence of the dual objective function towardsthe globally optimal total throughput of the network. Figure10 shows the convergence of the two auxiliary variables inlink 1 and 3 towards the optimal solutions.

VI. CONCLUSIONS

Power control problems with nonlinear objective and con-straints may seem to be difficult, NP-hard problems to solvefor global optimality. However, when SIR is much larger than


0dB, GP can be used to turn these problems, with a varietyof possible combinations of objective and constraint func-tions involving data rate, delay, and outage probability, intotractable, convex formulations. Then interior point algorithmscan efficiently compute the globally optimal power allocationeven for a large network. Feasibility analysis of GP naturallylead to admission control and pricing schemes. When thehigh SIR approximation cannot be made, these power controlproblems become SP and may be solved by the heuristic ofcondensation method through a series of GPs. Distributedoptimal algorithms for GP-based power control in multihopnetworks can also be carried out through message passing of‘consistency prices’.

Several interesting research issues remain to be furtherexplored: reduction of SP solution complexity by using high-SIR approximation to obtain the initial power vector and bysolving the series of GPs only approximately (except the lastGP), combination of SP solution and distributed algorithm fordistributed power control in low SIR regime, inclusion of fad-ing and mobility to the framework, and application to optimalspectrum management in DSL broadband access systems withinterference-limited performance across the tones and amongcompeting users sharing a cable binder.

ACKNOWLEDGEMENT

We would like to acknowledge collaborations on GP-basedresource allocation with Stephen Boyd at Stanford Universityand Arak Sutivong at Qualcomm, and discussions with Wei Yuat the University of Toronto. We would also like to thank JohnPapandriopoulos for discussions on the successive approxima-tion with convex problems during his visit to Princeton.

This work has been supported by NSF Grants CNS-0417607, CNS-0427677, CCF-0440443, and CCF-0448012.

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Mung Chiang (S’00 – M’03) is an Assistant Pro-fessor of Electrical Engineering, and an affiliatedfaculty of Applied and Computational Mathematicsand of Computer Science at Princeton University.He received the B.S. (Hon.) degree in electricalengineering and in mathematics, and the M.S. andPh.D. degrees in electrical engineering from Stan-ford University, Stanford, CA, in 1999, 2000, and2003, respectively.

He conducts research in the areas of optimizationof communication systems, theoretical foundations

of network architectures, algorithms for broadband access networks, andstochastic theory of communications and networking. He has been awarded aHertz Foundation Fellowship, and received the Stanford University School ofEngineering Terman Award for Academic Excellence, the SBC Communica-tions New Technology Introduction contribution award, the National ScienceFoundation CAREER Award, the ONR Young Investigator Award, and thePrinceton University Howard B. Wentz Junior Faculty Award. He co-authoredthe IEEE Globecom Best Student Paper Award 2006 and one of his papersbecame the Fast Breaking Paper in Computer Science by ISI citation data in2006.

He is a co-editor of Springer book series on Optimization and Control ofCommunication Systems, an Editor of IEEE Transaction on Wireless Com-munications, the Lead Guest Editor of the IEEE Journal of Selected Areas inCommunications special issue on Nonlinear Optimization of CommunicationSystems, a Guest Editor of the IEEE Transactions on Information Theory andIEEE/ACM Transactions on Networking joint special issue on Networkingand Information Theory, and the Program Co-Chair of the 38th Conferenceon Information Sciences and Systems.

Chee Wei Tan received the M.A. degree in Electri-cal Engineering from Princeton University, Prince-ton, NJ in 2006. He is currently working towardthe Ph.D. degree in the Department of ElectricalEngineering at Princeton University.

He was a research associate with the AdvancedNetworking and System Research Group at theChinese University of Hong Kong in 2004. He spentthe summer of 2005 with Fraser Research Lab,Princeton, NJ. His current research interests includequeueing theory, nonlinear optimization, communi-

cation networks, wireless and broadband communications.


Daniel P. Palomar (S’99-M’03) received the Elec-trical Engineering and Ph.D. degrees (both withhonors) from the Technical University of Catalonia(UPC), Barcelona, Spain, in 1998 and 2003, respec-tively.

He is an Assistant Professor in the Department ofElectronic and Computer Engineering at the HongKong University of Science and Technology, HongKong. He has held several research appointments,namely, at King’s College London (KCL), London,UK, during 1998; Technical University of Catalonia

(UPC), Barcelona, from January 1999 to December 2003; Stanford University,Stanford, CA, from April to November 2001; Telecommunications Techno-logical Center of Catalonia (CTTC), Barcelona, from January to December2002; Royal Institute of Technology (KTH), Stockholm, Sweden, from Augustto November 2003; University of Rome “La Sapienza, Rome, Italy, fromNovember 2003 to February 2004; Princeton University, Princeton, NJ, fromMarch 2004 to July 2006. His primary research interests include information-theoretic and signal processing aspects of MIMO channels, with specialemphasis on convex optimization theory applied to communication systems.He is the lead guest editor of the IEEE Journal on Selected Areas inCommunications 2007 special issue on Optimization of MIMO Transceiversfor Realistic Communication Networks.

Dr. Palomar received a 2004/06 Fulbright Research Fellowship; the 2004Young Author Best Paper Award by the IEEE Signal Processing Society;(co-recipient of) the 2006 Best Student Paper Award at ICASSP06; the2002/03 best Ph.D. prize in Information Technologies and Communicationsby the Technical University of Catalonia (UPC); the 2002/03 Rosina Ribaltafirst prize for the Best Doctoral Thesis in Information Technologies and

Communications by the Epson Foundation; and the 2004 prize for the bestDoctoral Thesis in Advanced Mobile Communications by the VodafoneFoundation and COIT.

Daniel O’Neill (M’85) received his Ph.D. fromStanford University in Electrical Engineering andBS and MBA degrees from the University of Cal-ifornia. Prior to attending Stanford he was a Di-rector at National Semiconductor responsible forintegrated processor development and President ofClearwater Networks, a web switching startup. Hiscurrent research interests are in the areas of wirelesscrosslayer network control and management andcognitive radio, especially as it pertains to video oradverse environment applications. He is currently a

Director at Sun Microsystems and a Visiting Scholar at Stanford University.

David Julian received a B.S. degree (with honors)in Electrical Engineering from New Mexico StateUniversity in 1996, and his MS and PhD degreesin Electrical Engineering from Stanford in 1998and 2004 respectively. He is a Staff Engineer atQualcomm working in Corporate Research and De-velopment as a Systems Engineer on next generationwireless systems.

2640 ieee transactions on wireless ...chiangm/gppower.pdf2640 ieee transactions on wireless...

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