IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. …cheewtan/TanChiangSrikant_TSP2011.pdf · IEEE TRANSACTIONS ON SIGNAL ... DECEMBER 2011 6127 Maximizing Sum Rate and Minimizing MSE

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 12, DECEMBER 2011 6127

Maximizing Sum Rate and Minimizing MSE onMultiuser Downlink: Optimality, Fast Algorithms and

Equivalence via Max-min SINRChee Wei Tan, Member, IEEE, Mung Chiang, Senior Member, IEEE, and R. Srikant, Fellow, IEEE

Abstract—Maximizing the minimum weighted signal-to-inter-ference-and-noise ratio (SINR), minimizing the weighted summean-square error (MSE) and maximizing the weighted sum ratein a multiuser downlink system are three important performanceobjectives in nonconvex joint transceiver and power optimiza-tion, where all the users have a total power constraint. We showthat, through connections with the nonlinear Perron–Frobeniustheory, jointly optimizing power and beamformers in the max-minweighted SINR problem can be solved optimally in a distributedfashion. Then, connecting these three performance objectivesthrough the arithmetic-geometric mean inequality and nonnega-tive matrix theory, we solve the weighted sum MSE minimizationand the weighted sum rate maximization in the weak interferenceregimes using fast algorithms. In the general case, we first establishoptimality conditions to the weighted sum MSE minimization andthe weighted sum rate maximization problems and provide theirfurther connection to the max-min weighted SINR problem. Wethen propose a distributed weighted proportional SINR algorithmthat leverages our fast max-min weighted SINR algorithm to solvefor local optimal solution of the two nonconvex problems, and giveconditions under which global optimality is achieved. Numericalresults are provided to complement the analysis.

Index Terms—Beamforming, multiple-inputmultiple-output(MIMO), optimization, uplink–downlink duality, wireless commu-nication.

I. INTRODUCTION

W E consider multiuser downlink transmission on a mul-tiple-input-single-output (MISO) channel, where the

transmitter (at the base station) is equipped with an antennaarray and each user has a single receive antenna. Full channelinformation is available at both the transmitter and the receiver,and all the users share the same bandwidth under a total powerconstraint. Under this setting, the antenna array provides anextra degree of freedom, in addition to power control, to opti-mize performance, e.g., increasing the total throughput (sum

Manuscript received April 24, 2011; revised July 11, 2011; accepted Au-gust 01, 2011. Date of publication August 18, 2011; date of current versionNovember 16, 2011. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Shahram Shahbazpanahi.The material in this paper was presented in part at the IEEE InternationalSymposium for Information Theory (ISIT), South Korea, June 2009. Thisresearch has been supported in part by AFOSR FA9550-09-1-0643, ONR GrantN00014-07-1-0864, NSF CNS 0720570, NSF CNS-1011962, ARO MURIAward W911NF-08-1-0233 and City University Hong Kong project Grant7008087.

C. W. Tan is with the Department of Computer Science, City University ofHong Kong, Kowloon, Hong Kong (e-mail: [email protected]).

M. Chiang is with the Department of Electrical Engineering, Princeton Uni-versity, Princeton NJ 08544 USA (e-mail: [email protected]).

R. Srikant is with the Department of Electrical and Computer Engineering,University of Illinois, Urbana, IL 61801 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2011.2165065

rates) or the total reliability in the system. Joint optimization ofthe weighted sum rates and weighted sum mean-square error(MSE) as objectives are however challenging to solve, becausethese two problems are nonconvex. Further, the transmit beam-formers are coupled across the users, thereby making them hardto optimize in a distributed fashion.

Our approach to these two nonconvex optimization problemsbegins by first considering a joint optimization of power andtransmit beamformer for the max-min weighted signal-to-inter-ference-and-noise ratio SINR problem. While previous algo-rithms in the literature require centralized computation, e.g., it-erative computation of an eigenvalue and eigenvector of a par-ticular matrix at each step, we propose a fast distributed algo-rithm (without parameter configuration) that computes the op-timal power and transmit beamformer in the max-min weightedSINR problem with geometric convergence rate. In addition,our algorithm reuses a power control module in [1], which hasbeen used in practical CDMA cellular systems, e.g., IS-95 andCDMA2000, thus facilitating the implementation of our algo-rithm in practical systems. This is achieved by applying thenonlinear Perron–Frobenius theory in [2] and [3] and the up-link–downlink duality in [4]–[8] to optimize transmit beam-formers in the downlink.

By considering simple and low-complexity transmitter design,e.g., linear beamformer, we study the nonconvex problems of 1)minimizing the weighted sum MSE between the transmitted andestimated symbols, and 2) maximizing the weighted sum rate.The max-min SINR problem is shown to be a special case of thesetwo problems in the sense that optimal solutions are equivalentunder special cases. Previous work in the literature (e.g., [9]) onlysolve these two nonconvex problems in a centralized manner andsuboptimally. Wesolve these two nonconvexproblems optimallyunder sufficiently weak interference conditions, and develop fastalgorithms (independent of stepsize and no parameter configura-tion whatsoever) for their special cases.

We then turn to establishing the optimality conditions ofthese two nonconvex problems in the general case (any in-terference conditions). Using nonnegative matrix theory, theoptimal beamformer will be shown to be the linear minimummean-square error (LMMSE) filter. This relates to earlier workon the optimality of the LMMSE filter in the max-min SINRproblem and a related total power minimization problem [4],[7], [8]. Further theoretical and algorithmic connections be-tween the two nonconvex problems and the max-min SINRproblem are established using nonnegative matrix theory, and afast algorithm (with minimal configuration) that leverages ourfast max-min weighted SINR algorithm and the uplink–down-link duality is then proposed to solve for the local optimalsolution of these two nonconvex problems in a distributed

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6128 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 12, DECEMBER 2011

Fig. 1. Overview of the connection (solid lines) between the three optimizationproblems as revealed in the paper: i) Weighted sum MSE minimization in (25),ii) weighted sum rate maximization in (36), and iii) max-min weighted SINRin (7). The upper half of the dotted line considers power control only, while thelower half considers both power control and beamforming.

fashion. All the proposed algorithms in this paper are appli-cable to the MISO case.

This paper is organized as follows. We present the systemmodel in Section II. In Section III, we look at the max-minweighted SINR power control problem and its extension tojoint beamforming and power control, and we propose fastalgorithms to solve them. In Section IV and Section V, we lookat the MSE minimization and weighted sum rate maximizationpower control problems and solve them in the weak interferenceregime. In Section VI, we establish the optimality conditionsfor these two problems and propose a weighted proportionalSINR algorithm to solve them in the general case. We highlightthe performance of our algorithms using numerical examplesin Section VII. We conclude with a summary in Section VIII.All the proofs can be found in the Appendix.

We refer the readers to Fig. 1 for an overview of the connec-tion between the three main optimization problems in the paper.The following notation is used. Boldface uppercase letters de-note matrices, boldface lowercase letters denote column vectors,italics denote scalars, and denotes component-wise inequality between nonnegative vectors and (nonneg-ative matrices and ). The Perron–Frobenius eigenvalue of anonnegative matrix is denoted as , and the Perron (right)and left eigenvectors of associated with are denoted by

and , respectively. The super-scripts anddenote transpose and complex conjugate transpose respectively.We let denote the th unit coordinate vector and denote theidentity matrix. Let denote(Schur product). Let be the weighted maximum normof a real vector with respect to the weight , i.e.,

, . Let denote . We letdenote the th element of a function vector

.

II. SYSTEM MODEL

We consider a single cell multiuser downlink system withantennas at the base station and decentralized users,

each equipped with a single receive antenna, operating in a

frequency-flat fading channel. The downlink channel can bemodeled as a vector Gaussian broadcast channel:

(1)

where is the received signal of the th user,is the channel matrix between the base station and the th

user, is the transmitted signal vector, and ’s are thei.i.d. additive complex Gaussian noise vectors with varianceon each of its real and imaginary components.

We assume that the multiuser system adopts a linear trans-mission and reception strategy. In transmit beamforming, thebase station transmits a signal in the form of ,where is the transmit beamformer that carries theinformation signal of the th user. We assume a total powerconstraint at the transmit antennas, i.e., given inWatt (W). From (1), the received signal for the th user can beexpressed as . Next, wewrite , where is the downlink transmit power(in Watt) and is the normalized transmit beamformer, i.e.,

, of the th user. Now, the received SINR of the thuser in the downlink transmission can be given in terms ofand :

SINR

This SINR function depends only on the pairs of parameters. Thus, an equivalent form can be obtained in terms of

the normalized parameter pairs , where [10]. Let

us define the matrix with entries . In terms ofthe beamforming matrix , we also define the (cross-channelinterference) matrix with entries

ifif (2)

and

(3)

Moreover, we assume that is irreducible, i.e., each link hasat least an interferer. For brevity, we omit the dependency on

when we fix the beamformers and for the most part of thepaper. This dependency is made explicit only in Section III-B(beamforming optimization). In this paper, we shall consider theequivalent form of the SINR as

SINR

(4)

where we use and in (2) and (3), respectively.In the following, we study optimization problems having two

performance metrics that are functions of SINR , namelythe MSE at the output of a LMMSE (scalar) filter of each user[11]:

SINR(5)

TAN et al.: MAXIMIZING SUM RATE AND MINIMIZING MSE ON MULTIUSER DOWNLINK 6129

and the throughput of each user (assuming the Shannon capacityformula) [12]:

SINR (6)

III. MAX-MIN WEIGHTED SINR OPTIMIZATION

In this section, we first consider optimizing only power beforewe consider a joint optimization in power and transmit beam-formers. Let be a positive vector, where the th entry isassigned to the th link to reflect some priority. Let us considerthe following max-min weighted SINR problem:

maximizeSINR

subject to

variable (7)

Next, let us define the following nonnegative matrix:

(8)

We will extensively exploit the spectra of the product of adiagonal nonnegative matrix and (e.g., its spectral radius, itsquasi-inverse and other properties) in our problem formulation,their solution and algorithm design in this paper. We now use thespectral property of to solve (7) in the following.

A. Optimal Solution and Algorithm

By exploiting a connection between the nonlinearPerron–Frobenius theory in [2] and [3] and the algebraicstructure of (7), we can give a closed form solution to (7).1

Lemma 1: The optimal value and solution of (7) is given by

and

respectively.Note that the optimal power in Lemma 1 can also be ex-

pressed as

(9)

The following algorithm computes the solution given inLemma 1. We let index discrete time slots.

Algorithm 1: Max-min Weighted SINR

1) Update power :

SINR(10)

2) Normalize :(11)

1A closed-form solution to (7) was first obtained in [7] using a nonnegative(increased dimension) matrix totally different from �� . As such, thealgorithmic solution to (7) in [7] is different and is mainly centralized. On theother hand, our solution exploits the DPC algorithm in [1] and is distributed.

Corollary 1: Starting from any initial point , inAlgorithm 1 converges geometrically fast to the optimal solutionof (7), .

Remark 1: Interestingly, (10) in Algorithm 1 is simply theDistributed Power Control (DPC) algorithm in [1], where theth user has a virtual SINR threshold of in the downlink

transmission. The normalization at Step 2 can be computed cen-trally at the base station. In case when it cannot be expected toknow the total number of users in the system, the normaliza-tion can be made distributed using gossip algorithms to compute

at each user [13].

B. Uplink–Downlink Duality and Joint Optimization

Next, we consider the joint optimization of power andtransmit beamformer in the following max-min weighted SINRproblem:

maximizeSINR

subject to

variables: (12)

We first review the notion of uplink–downlink duality in[4]–[8]. Here, the uplink transmission uses the normalizedchannel vector and unit noise power as in Section II. Theduality theory states that, under a same total power constraintfor all the users, the achievable SINR region for a downlinktransmission with joint transmit beamforming and power con-trol optimization is equivalent to that of a reciprocal uplinktransmission with joint receive beamforming and power controloptimization. Further, the optimal receive beamforming vectorsin the uplink is also the optimal transmit beamforming vectorsin the downlink. Since the uplink problem does not have thebeamformer coupling difficulty associated with the downlink(hence easier to solve), the (dual) uplink problem can be firstused to obtain the optimal transmit beamformers in the down-link. The optimal downlink transmit power is then computedby keeping the transmit beamformers fixed.

Let the virtual uplink power be given by . Now, supposethere exists positive values (optimal max-min weighted SINR)and for all such that the virtual uplink SINR satisfies

SINR (13)

for all . Since SINR in (13) only depends on the beamformingvector , the receive beamforming optimization, with thepower fixed at , is solved by

(14)


whose solution is the LMMSE receiver given by (optimal up toa scaling factor)

(15)

Using this LMMSE receiver, the SINR constraint in (13) is al-ways met with equality, i.e., SINR by choosing

SINR

By the uplink–downlink duality, the LMMSE receiver is alsothe optimal transmit beamformer in the downlink max-minweighted SINR problem given by (12).

Now, we are ready to use Algorithm 1 to solve the joint powercontrol and beamforming problem in (12) in a fast and dis-tributed fashion as given in the following:

Algorithm 2: Max-min Weighted SINR—Joint Optimization

1) Update (virtual) uplink power :

SINR(16)

2) Normalize :

(17)

3) Update transmit beamforming matrix:

(18)

4) Update downlink power :

SINR(19)

5) Normalize :

(20)

Theorem 1: Let the optimal power and beamforming matrixin (12) be and respectively. Then, starting from any initialpoint and , in Algorithm 2 converges geometri-cally fast to

(unique up to a scaling constant).Remark 2: In Algorithm 2, the uplink power converges

geometrically fast to, which is also proportional to

.Remark 3: Note that (16) and (19) of Algorithm 2 use the

DPC algorithm in [1], where the th user has a virtual SINRthreshold of in both the (virtual) uplink and downlink trans-

mission. When the uplink and downlink channels are reciprocalto each other, in (16) is the exact uplink transmit power, andonly computing in (17) requires a global coor-dination. Similar to Algorithm 1, at Step 2 and

at Step 5 can be computed centrally at the basestation or be made distributed using gossip algorithms [13].Furthermore, in time-division duplex (TDD) system, where up-link and downlink channels are reciprocal, Step 1 can be imple-mented using local measurements in a distributed manner.

C. Nonlinear Perron–Frobenius Minimax Characterization

We first establish the following result based on the nonlinearPerron–Frobenius theory and the Friedland–Karlin inequality in[14] and [15] [see (62) in the Appendix], and then discuss howit provides further insight into the analytical solution of (7). Theresult is also useful when we consider a different reformulationof (7) in Section VI-A.

Lemma 2: Let be an irreducible nonnegativematrix, a nonnegative vector and a norm on

with a corresponding dual norm . Then,

(21)

(22)

where the optimal in (21) and (22) are both given by, and the optimal in (21) and (22) are both given by

.Furthermore, is the dual of with respect

to .2Remark 4: Lemma 2 is a general version of the Friedland-

Karlin spectral radius minimax characterization in [14] and [15].In particular, if , we obtain [15, Theorem 3.2].

Using Lemma 2 (let ,and in Lemma 2), we deduce that the optimal SINRallocation in (7) is a weighted geometric mean of the optimalSINR, where the weights are the normalized Schur product ofthe uplink power and the downlink power (i.e., the Perron andleft eigenvectors of , respectively)3:

SINR (23)

Further, both the optimal uplink and the downlink powerform dual pairs with the vector , i.e., and

are dual pairs with respect to .

2A pair �� of vectors of is said to be a dual pair with respect to � � �if �� .

3The normalization of the Schur product is done such that � �.The uplink and the downlink powers can be obtained as a sub-vector of thenormalized Perron and left eigenvectors of a �� extendedcoupling matrix, which is different from the �� matrix �� [7],[8].


Fig. 2. The uplink–downlink duality characterized through the nonlinear Perron–Frobenius theorem and the Friedland–Karlin spectral radius minimax theorem.The equality notation used in the equations denotes equality up to a scaling constant.

Finally, Fig. 2 summarizes the connection between the up-link–downlink duality with the nonlinear Perron–Frobenius the-orem and the Friedland–Karlin minimax characterization fromSection III-C.

IV. WEIGHTED SUM MSE MINIMIZATION

In this section, we fix the beamformers and study the non-convex optimization problem of minimizing the weighted sumof the MSE’s of individual data streams using two methods. Inour first method (Section IV-B), we show that it can be solvedoptimally in special cases when the problem parameters satisfycertain conditions (which will be associated with the interfer-ence and SNR level). In our second method (Section VI), forthe general case, we deduce optimality conditions and proposesuboptimal algorithm that exploits the connection with max-minweighted SINR (and its associated fast algorithms in the pre-vious section) to solve it.

A. Problem Statement

We assume that all the receivers use the LMMSE filter forestimating the received symbols of all the users. The weightedsum MSE at the output of the LMMSE receiver is given by [11]

SINR(24)

where is some positive weight assigned to the th link toreflect some priority. Without loss of generality, we assume that

is a probability vector. The weighted sum MSE minimizationproblem is given by

minimizeSINR

subject to

variables: (25)

In this section, we denote the optimal power vector to (25) by.

B. Polynomial-Time Solution for Special Cases

We can rewrite (25) as

minimize

subject to (26)

It can be shown that the total power constraint in (25) and (26)are tight at optimality (see Appendix IX-E), which we exploitto transform (26) in the variables into another optimizationproblem that can be used to solve (26) optimally. To proceedfurther, we need to introduce the notion of quasi-invertibility ofa nonnegative matrix in [16], which will be useful in solving(26) optimally.

Definition 1 (Quasi-Invertibility): A square nonnegative ma-trix is a quasi-inverse of a square nonnegative matrix if

. Furthermore, [16].For a given square nonnegative matrix , we first compute

either or . If this computed matrixis nonnegative, we say that exists (as from Definition 1, thiscomputed matrix is ). We now apply the definition ofquasi-invertibility to the matrix with in (8), andstudy the existence of , which can interestingly be associatedwith the interference and SNR regime. In the case when thetotal power is large, (high SNR regime) or when interference(off-diagonals of ) is large, it is deduced in the following that

does not exist.Lemma 3: does not exist when , where

for all and .However, when (no interference) such that

or when is sufficiently small (low SNR regime) such that, then always exists, as shown by the following

lemma.Lemma 4: For any nonnegative vector ,

when .Example 1: As a numerical example for a ten-user IEEE

802.11b network, we experiment with a total power constraint of33 mW and 1 W (the largest possible value allowed in IEEE 802.11b) and equal noise power of 1 mW. Averaging over 10 000random channel coefficient instances ( and off-di-agonal uniformly distributed between 0.01 and 0.3), thepercentage of instances where exists is 99% and 65% cor-responding to the total power constraint of 33 mW and 1 W,respectively.

In the rest of Section IV, we focus on the case when exists.We next solve (26) in the following. Let us define

(27)


Then, we can rewrite (26) in terms of the new variable as

minimize

subject to (28)

where the constraints in (28) are due to the nonnegativity of ,since, using Definition 1,

In the following, we leverage (28) to derive useful lower boundsto (25), investigate special cases, and finally solve (25) usingpolynomial-time algorithms.

C. A Lower Bound to Weighted Sum MSE Minimization

By exploiting the eigenspace of and the Friedland-Karlininequality, the following result gives a lower bound on theweighted sum MSE problem in (25).

Theorem 2: If exists,

SINR

(29)

for all feasible in (25).Equality is achieved if and only if

Thus, SINR for all . In this case,solves (25).

Interestingly, Theorem 2 shows that solving (7) withcan be seen as an approximation method to solving (25) subop-timally, but with an approximation guarantee. In particular, bytaking the logarithm of the objective function of (25), the posi-tive quantity

can be interpreted as the approximation ratio of Algorithm 1with in solving (25).

Remark 5: Based on Theorem 2, we obtain a con-nection between the min-max MSE problem, i.e.,

, and the weighted sum MSE opti-

mization in (25). By the max-min characterization of 4:

(30)

the optimal value of the min-max MSE problem is. It thus follows from Theorem 2 that (7)

with yields the equivalent solution as the min-max MSEproblem and the optimal sum MSE with

.We will establish further connections and equivalence results

between the max-min weighted SINR problem and the two non-

4The max-min characterization of the spectral radius of an irreducible non-negative matrix � is also known as the Collatz-Wielandt characterization innonnegative matrix theory, e.g., see [14], [17].

convex problems (25) and (36) in Section V-C (still requiringthe existence of ) and in Section VI-A for the general case(without requiring the existence of ).

D. Polynomial-Time Solution and Fast Algorithm

In fact, the existence of allows us to delineate cases of(25) that can be solved efficiently [through (28)] from the gen-eral problem. Although (28) is nonconvex, it is equivalent to aconvex problem by making a change of variables forall . This is allowed since . Hence, (28) is equivalent tothe problem:

minimize

subject to

variables: (31)

which is strictly convex in . Though the constraint set in (31) isunbounded, the optimal solution to (31) cannot havefor some since, at optimality, . Hence, (31) can besolved optimally by an interior-point method (e.g., see [18]),thus yielding the optimal solution to (28) after the change invariables. The following result gives the optimal solution to(25). A special case of (31) also yields a fast algorithm (Algo-rithm 3 below) that computes the optimal solution by leveragingthe nonlinear Perron–Frobenius theory in [3] instead of usingstandard convex optimization algorithms to solve (31).

Theorem 3: If exists, then the optimal solution to (25) isgiven by , where is the optimal solutionto (28). Furthermore, if (i.e., none of the constraints in(28) is tight), fulfills

(32)

for all and satisfies .Now, the righthand-side of (32) is a homogeneous and

monotone concave function (cf. proof of Corollary 2). Wecan leverage the nonlinear Perron–Frobenius theory in [3] topropose the following (step size free) algorithm that computes

in Theorem 3, and hence the optimal solution of (25).

Algorithm 3: Weighted Sum MSE Minimization

1) Update auxiliary variable :

(33)

2) Normalize :

(34)

3) Update :SINR

SINR

(35)


The following result establishes the convergence and optimalityof Algorithm 3.

Corollary 2: If exists and in (25), Algorithm 3converges geometrically fast to the unique fixed point and

in Theorem 3 from any initial point .

V. WEIGHTED SUM RATE MAXIMIZATION

In this section, we fix the beamformers and consider theweighted sum rate of all the users as the next performancemetric to be optimized. Similar to Section IV, we first show thatit can be solved optimally when , and then look at thegeneral case in Section VI. Further, we quantify a connectionof the weighted sum rate maximization and the weighted sumMSE minimization.

A. Problem Statement

The nonconvex weighted sum rate maximization problem isgiven by

maximize SINR

subject to

variables: (36)

In this section, we denote the optimal power vector to (36) by.

B. Exact Solution and Fast Algorithm

We can rewrite (36) to be equivalent to

minimize

subject to (37)

Similar to Section IV, if is the quasi-inverse of , we canrewrite (37) as

minimize

subject to (38)

where is given by (27). Similarly to (28), (38) can be solvedoptimally by first a logarithmic change in variables and thenusing convex optimization algorithms, e.g., the interior-pointmethod. The following result gives the condition under which(36) is solved optimally when exists (in the weak interfer-ence regime).

Theorem 4: If exists, then the optimal solution to (36) isgiven by , where is the optimal solutionto (38). Furthermore, if (i.e., none of the constraints in(38) is tight), fulfills

(39)

for all and satisfies .As in the previous, using the nonlinear Perron–Frobenius

theory, the following algorithm computes the optimal solutionof (36) when in (36).

Algorithm 4: Weighted Sum Rate Maximization

1) Update auxiliary variable :

(40)

2) Normalize :

(41)

3) Update :

SINRSINR

(42)

The following result establishes the convergence and optimalityof Algorithm 4.

Corollary 3: If exists and in (36), Algorithm 4converges geometrically fast to the unique fixed point and

in Theorem 4 from any initial point .Next, we connect the three optimization problems given in

(25), (36) and (7) with .

C. Connection Between Weighted Sum Rate, Sum MSE andMax-Min SINR

Applying the arithmetic-geometric mean inequality (see (64)in the Appendix) to connect (28) and (38), we deduce that theweighted sum rate maximization has the same optimal power asthe weighted sum MSE minimization when

Furthermore, from Remark 5, this optimal power is also the so-lution to the max-min weighted SINR problem (with ).

Example 2: We give a simple illustrative example for the twouser case. The channel gains are given by

and the AWGN for the firstand second user are 0.1 and 0.3 respectively. The total poweris . It can be easily checked that exists. We solve (7) with

. We then set in both(25) and (36), and find their corresponding optimal solution byAlgorithms 3 and 4. These are then plotted on the respectiveachievable MSE and rate region. Fig. 3 shows that the optimalweighted sum MSE point is the same as the weighted sum MSEpoint evaluated using the max-min SINR power control. Fig. 4shows that the optimal weighted sum rate point is the same asthe weighted sum rate point evaluated using the max-min SINRpower control.

In Section VI-B, we continue to highlight this connection be-tween (7) with and the two problems (25) and (36) with

in the general case, i.e.,without the existence condition.


Fig. 3. A two-user example with the vector � superimposed on the achievable MSE region. Its perpendicular coincides with the optimal point in the achievableregion. From a geometrical perspective, the weighted sum MSE point evaluated at the solution of the max-min SINR power control finds the largest hypercubethat supports the Pareto boundary of the achievable MSE region. When � � �� , the weighted sum MSE problem and the max-minSINR problem have the same solution.

VI. DISTRIBUTED OPTIMIZATION IN SINR

In this section, we first study the optimality conditions of (25)and (36) in the general case. We then solve the problems usinga centralized algorithm that can be made distributed using thegradient method and Algorithm 2. Conditions under which thesealgorithms solve them optimally will also be given.

A. Optimality Conditions in SINR

First, we reformulate both (36) and (25) as optimization prob-lems having a new set of variables (in the SINR domain) and aspectral radius constraint involving in (8). The new formu-lation permits us to derive optimality conditions, propose fastalgorithms and further connect the max-min SINR problem to(25) and (36). In particular, by leveraging the beamforming re-sult in Section III-B, we also address the optimal beamformerto (25) and (36).

First, let us consider the following optimization problem:

maximize

subject to

variables: (43)

Let us denote the optimal solution of (43) by . Now, the op-timal power to the weighted MSE minimization problem in (25)and the weighted sum rate maximization problem in (36) can beimplicitly obtained by solving (43) using, respectively,

weighted sum MSE (44)

and

weighted sum rate (45)

in (43). We summarize this in the following result.Lemma 5: For a feasible in (43), a feasible power in (25)

and in (36) can be computed by

(46)

This means that if we first solve (43) to obtain , the optimalpower in (25) and (36) can be obtained using (46) in Lemma 5.

We continue with a further change of variable technique in(43). For , let

for all

i.e., . Then, (43) is equivalent to

maximize

subject to

variables:

(47)

Note that the constraint set in (47) is an unbounded convex set[19] (see also [18, Sec. 4.5]). Let us denote the optimal solutionof (47) by . Note that for all .


Fig. 4. A two-user example with the vector � superimposed on the achievable rate region. Its perpendicular coincides with the optimal point in the achievableregion. From a geometrical perspective, the weighted sum rate point evaluated at the solution of the max-min SINR power control finds the largest hypercube thatis contained inside the achievable rate region. When � � �� , the weighted sum rate problem and the max-min SINR problem havethe same solution.

Suppose we know the optimal in (47). Define the first orderderivative of the objective function in (47) with respect toby . The optimality condition of (47) is given asfollows.

Theorem 5: The optimal solution of (47), , satisfies

(48)and

(49)

Furthermore, is unique.Our results in Sections IV and V established the connection

between these three problems only in the weak interference orlow signal-to-noise regime. Now, we establish the connectionfor the general case. From (48) in Theorem 5, we see that ifthe optimal SINR is equal for all , then the optimal solutionto (47) can be obtained by solving the max-min SINR problemin (12) since it means is proportional to and we have

in (48) (cf. Remark 5).Now, by extending this observation to the constraint set in

(43), we see that the uplink–downlink duality applies to theweighted sum rate maximization and weighted sum MSE mini-mization problems. In particular, the optimal in (43) is givenby the LMMSE filter in (15), where

in (15).

Further, using (21) in Lemma 2, we see that a simple proof tothe uplink–downlink duality is to observe that

(50)

where we have used both the fact that andfor any irreducible nonnegative matrices

and [17]. The first and second spectral radius expression in(50) correspond to the uplink and the downlink systems, respec-tively. Note that (50) characterizes all pareto efficient achievableSINR in both the uplink and downlink systems (since the spec-tral radius increases monotonically in the entry of a matrix [17]).

Interestingly, using the weighted geometric mean equivalenceresult in (23), (12) is equivalent to an optimization problemgiven in the form of (43) by letting

weighted SINR (51)

and its corresponding transformed problem in (47), which isconvex in , can be solved efficiently. We will now exploit thisfact and our max-min weighted SINR algorithm in Section III-Ato propose a fast algorithm that can be made distributed to solvefor a local optimal solution to (25) and (36) in the following.Conditions under which they are global optimal are also given.

B. A Weighted Proportional SINR Algorithm

We first state our weighted proportional SINR algorithm,which uses Algorithm 1 as a sub-module and adapts the weightparameter of Algorithm 1 iteratively to solve either (25) or(36).


Algorithm 5: Weighted Proportional SINR

1) Compute the weight :

(52)

2) Obtain as the optimal solution to:

maximize

subject to (53)

3) Set the output of Algorithm 1 using the input weightparameter as .5

Depending on the used in (52), Algorithm 5 can com-pute the optimal solution of (25) or (36) when the initial pointis sufficiently close to the optimal solution. This is stated in thefollowing result.

Theorem 6: For any in a sufficiently close neighborhoodof , in Algorithm 5 converges to the optimal solutionof (25) and (36) for with, respectively,

and

The convergence in Theorem 6 is proved only for initialpoints in the neighborhood of the optimal solution. However,we now state a result stronger than Theorem 6 (for a more re-laxed initialization) that also highlights the connection betweenthe max-min weighted SINR problem and the two nonconvexproblems (25) and (36).

Corollary 4: If are equal for all and, then in Algorithm 5 converges to the

optimal solution of (25) and (36) using the respectivein Theorem 6, from any initial point such that areequal for all .

Remark 6: This result is similar in flavor to the result inSection V-C, but is more general (without the existence condi-tion of ). Collorary 4 highlights a special case of (25) and (36)whose solution can be obtained by Algorithm 5 (a polynomialtime algorithm) despite the nonconvexity.

C. Distributed Optimization

Although Algorithm 5 is centralized (due to computing (53)),it can be made distributed by leveraging the output of Algo-rithm 1. In particular, we use the approximate projected gradientmethod to obtain a distributed solution to solve (53). Recall thatthe gradient of at satisfies

(54)

5The output of Algorithm 1 is obtained after it converges to within a giventolerance. Algorithm 5 is essentially a two time-scale algorithm—the powerand beamformer are updated at a much faster timescale than the variable ��.

for any feasible . In fact, is given by [15]

(55)

normalized such that . Computing the left and rightPerron eigenvector requires centralized computation in general.However, observe that it is also the normalized Schur productof the optimal uplink and downlink power when in (7)(cf. Fig. 2). Leveraging on this fact, we next apply the approx-imate projected gradient method to obtain a distributed versionof Algorithm 5.

Algorithm 6: Distributed Weighted Proportional SINR

Set the step size .1) Compute the weight :

(56)

2) Set the downlink power and uplink power output ofAlgorithm 2 using the input weight parameter ( )as and , respectively.

3) Each th user computes:

SINR

(57)

for all . Update according to Theorem 7.

Theorem 7: Let andbe the optimal solution

of Algorithm 2 when . Let, where

for all . Suppose the output of Algorithm 2 (atStep 2 of Algorithm 6) satisfies

and

for some positive and respectively. For any in asufficiently close neighborhood of and a sequence of stepsizes , , that satisfies

in Algorithm 6 converges to within a closed neighbor-hood of the optimal solution of (25) and (36) using the respective

in Theorem 6.Remark 7: We make the following remarks on Algorithm 6.

At Step 2, the computation of and is approximatelyoptimal in the sense of max-min weighted SINR as the timeto run Algorithm 2 is finite. These approximation errors carryover to Step 3, thus leading to an approximate gradient witherror. Theorem 7 states that these accumulated approximation


Fig. 5. Illustration of the convergence of Algorithm 3 and Algorithm 6 with an initial SINR vectors that are closer to, in (a), and further from, in (b), the optimalsolution.

Fig. 6. Illustration of the convergence of Algorithm 4 and Algorithm 6 with different number of inner loops executed by Algorithm 2 of (a) 150 and (b) 70.

errors however do not affect the overall convergence as long asthe number of iterations to execute Algorithm 2 (at Step 2 ofAlgorithm 6) is sufficiently long and the step size is tuned ap-propriately at each iteration. For example, in computing ,a practical stopping rule at Step 2 of Algorithm 6 is

, where index discrete time slots of Algorithm 2and is sufficiently small, or a sufficiently long fixednumber of inner loops. At Step 3, both the normalizedand can be computed centrally atthe base station or be made distributed by a gossip algorithm.

VII. NUMERICAL EXAMPLES

In this section, we fix the beamformers and consideroptimizing only powers. We evaluate the performance ofAlgorithm 3, Algorithm 4, and Algorithm 6 numerically. Wefocus on the case when exists, whereby Algorithm 3 and 4can be used. We use the following channel gain matrix:

(58)

We set the total power constraint as and thenoise power of each user be 1 W. The weight vector is givenby . It is easily verifiedthat exists. Moreover, both the optimal solution to (25) and(36) are attained at the equal SINR allocation of 0.673 for thethree users (equal to the solution of (7) with ), where

. Thus, theoptimal sum rate is 0.5144 nats/symbol.

Fig. 5 plots the evolution of the power for the three usersthat run Algorithm 4 and 6. At Step 2 of Algorithm 6, werun Algorithm 2 for 100 iterations before it terminates. Weuse a diminishing step size . InFig. 5(a) and (b), we set the initial SINR vector in Algorithm6 to (closer to the optimal solution) andSNR (further from the optimal solution) respectively.

It is observed that Algorithm 4 converges geometrically fast tothe optimal solution (verifying Corollary 3) under synchronousupdates. In the case of Algorithm 6, convergence to the optimalsolution is observed for initial vectors close to the optimalsolution (verifying Theorem 7). Interestingly, we have neverobserved a case where convergence to the optimal solutionfails even for the other initial vectors further away from theoptimal solution that we have tested. We also observe that theconvergence speed is faster when the initial SINR vector iscloser to the optimal solution.

Next, we evaluate the performance of using a constant stepsize of 0.02 in Algorithm 6, and evaluate its performanceto solve (25) for different number of inner loops executedat Step 2 of Algorithm 6. The weight vector is given by

. The optimal solution to (25) is attained at thepoint . The initial SINR vectors inAlgorithm 6 are set close to the optimal solution. Fig. 6 plots theevolution of the power for the three users that run Algorithm 3and Algorithm 6. Fig. 6(a) and (b) illustrates the cases when we


run Algorithm 2 for 120 and 60 iterations before it terminates,respectively. We observe that if the number of inner loops toexecute Algorithm 2 is smaller than a certain threshold, theiterates produced by Algorithm 6 tend towards points in theneighborhood of the optimal solution, but Algorithm 6 maynot converge to the optimal solution due to the approximategradient error. (In Fig. 6(b), less than 5% deviation from theoptimal solution is observed.)

VIII. CONCLUSION

Maximizing the minimum weighted SINR, minimizingweighted sum MSE and maximizing weighted sum rate on amultiuser downlink system are three important goals of jointtransceiver and power optimization. We established a preciseconnection between these three problems using nonnegativematrix theory, nonlinear Perron–Frobenius theory and thearithmetic-geometric mean inequality. Under sufficiently weakinterference, we showed that the weighted sum MSE minimiza-tion and the sum rate maximization problems can be solvedoptimally using fast algorithms (and without any parameterconfiguration). In the general case, we established optimalityconditions and also theoretical and algorithmic connectionsbetween the three problems. We then proposed a weightedproportional SINR algorithm that leveraged our fast max-minweighted SINR algorithm and the uplink–downlink duality tosolve for the local optimal solution of these two nonconvexproblems in a distributed manner. Numerical analysis high-lighted the robust performance of our algorithms in finding theglobal optimal solution of these nonconvex problems. All theproposed algorithms in this paper are applicable to the MISOcase. As future work, it will be interesting to extend our workto the MIMO case and go from empirical evidence to analyticunderstanding on why Algorithm 5 (centralized version) andAlgorithm 6 (distributed version) never fail to find the globaloptimal solution from any initial point.

APPENDIX

A. Proof of Lemma 1

Proof: Our proof is based on the nonlinear Perron–Frobe-nius theory [2], [3]. We let the optimal weighted max-minSINR in (7) be . A key observation is that all the SINRconstraints are tight at optimality. This implies, at optimalityof (7),

(59)

for all . Letting , (59) can be rewritten as

(60)

We first state the following conditional eigenvalue lemma:Lemma 6 (Conditional Eigenvalue [2, Corollary 13]): Letbe a nonnegative matrix and be a nonnegative vector. If

, then the conditional eigenvalue problem, , , , has a unique solution

given by and being the unique normalizedPerron eigenvector of .

Letting , ,in Lemma 6 and noting that shows that

is a fixed point of (60).

B. Proof of Corollary 1

Proof: The fixed point in Lemma 6 is also a unique fixedpoint of [2], [3]

and can be obtained by

Applying this to the system of equations in (60) shows thatAlgorithm 1 converges to the fixed point geometrically fast fromany initial point.

C. Proof of Theorem 1

Proof: We will show that Algorithm 2 produces and(optimal uplink power and receive beamformer), and this inturn produces . The uplink max-min weighted SINR problemis given by

SINR

which is equivalent to

This uplink problem can be solved by applying the up-link–downlink duality and Lemma 6: is given by (15) as afunction of , which can be solved by applying Lemma 6 to

By the uplink–downlink duality, the uplink max-min weightedSINR problem has the same optimal value (i.e., ) as thedownlink max-min weighted SINR problem. Further, isalso the optimal transmit beamformer in the downlink max-minweighted SINR problem given by (12).

Let , , inLemma 6. Hence, similarly to the proof of Lemma 1, we have

By applying the fixed point iteration in the proof of Corollary1, we obtain Steps 1–2, where is computed and is usedto compute the transmit beamforming matrix in Step3. Lastly, Steps 4–5 keep the transmit beamfomers fixed andcompute the downlink power . As and

, we have . This completes

the proof of Theorem 1.

D. Proof of Lemma 2

Proof: Let be an irreducible nonnegative matrix, anonnegative vector and a norm on with a correspondingdual norm . Then, Proposition 5 in [2] establishes that


, and the fact that

, which is the Perron eigenvector of ,is the dual of with respect to . Further, this is also theunique solution to the problem:

(61)

Next, we give further characterization of the above result byapplying the Friedland-Karlin inequality and minimax theoremin [14] and [15] on . The equality between (21) and(22) in Lemma 2 can be proved as follows. Observe that (22) isequivalent to

which can be solved optimally by first casting it in epigraphform (introduce an auxiliary variable and additional con-straints

for all ), and then using the logarithmic change of variable tech-nique on , i.e., to make the problem convex in and .Next, a partial Lagrangian from the epigraph form is consideredby relaxing the constraints

for all . Using the Lagrange duality, i.e., theKarush–Kuhn–Tucker (KKT) conditions (cf. [20]), tosolve the primal and dual problems, the optimal in (22) turnsout to be the optimal dual variable.

The next step is to apply the Friedland-Karlin inequality toupper bound the dual problem, which turns out to be tight for afeasible . The optimal and is thus given byand respectively. In fact, the optimal

in the epigraph form is related to the in (61):

Once (22) is solved completely, the equality between (21) and(22) follows from strong duality. The last step is to apply theCollatz–Wielandt characterization [cf. (30)] to relate (22) to

.The second and simpler proof is a direct application of

the Friedland–Karlin minimax theorem in [15] (see [15,Theorem 3.2]) to the matrix .

E. Total Power Constraint in (25) and (26) Tight at Optimality

Proof: Suppose at optimality. The objectivefunction in (26) can be reduced by increasing the power of all theusers proportionally such that , since SINR forall increases monotonically. But this contradicts the assump-tion, thus the constraint in (25) and (26) are tight at optimality.

F. Proof of Lemma 3

Proof: Let , where for alland . Suppose exists. Since for all , by

definition 1, , thus for all . We assume

with for all exists. By definition 1,. Thus, ,

where denotes the matrix trace. But, this cannot happenunless or , which contradicts that exists. Thisproves Lemma 3.

G. Proof of Lemma 4

Proof: Suppose that for some positivewhen . We shall show that sat-

isfies definition 1 for a unique . By definition 1,, which can be written as

using the von Neumann’s ex-pansion [17]. This leads to , whichyields . It can be checked, by definition 1, that

, thus proving Lemma 4.

H. Proof of Theorem 2

Proof: We first recall the following result in [14].Theorem 8 [14, Theorem 3.1]: For any irreducible nonnega-

tive matrix ,

(62)

for all strictly positive , where and are the Perron and lefteigenvectors of , respectively.

Furthermore, if (which implies ), then forany positive vector , (62) can be extended to

(63)

Clearly, (63) reduces to (62) when . Next, the arith-metic-geometric mean inequality states that

(64)

where and . Equality is achieved in (64)if and only if .

Suppose is the quasi-inverse of . Next, we com-pute and its eigenvectors.

Lemma 7: If exists, then has the spectral radius

with the corresponding left and Perron eigenvectors of.

Proof: We first state the following lemma.Lemma 8 (Splitting Lemma [17, Ch. 7, Theorem 5.2]): Let

with and nonsingular. Suppose that ,where . Then, if and onlyif . Letting , and

in Lemma 8, we have .

But , thus obtaining. Next, we multiply both sides of


with the Perron eigenvector of . Afterrearranging, we obtain . Thus, andhave the same Perron eigenvector . A similar proof for the lefteigenvector can also be shown.

Now, we are ready to prove (29). From the objective functionof (28), we establish the following:

(65)

where (a) is due to letting for all and in(64), (b) is due to letting in (63) since , and (c) isdue to Lemma 7. But, using (27), .Thus, we establish (29).

To prove the second part, we note that both inequalities (a)and (b) in (65) become tight if and only ifand (required only for (b)to become tight). In particular, all the users receive a commonSINR given by for all .

To establish the power vector corresponding to, we further note that equality is achieved in (29) of

Theorem 2 by the max-min SINR power allocation. More pre-cisely, is in fact the total power constrainedmax-min SINR whose associated power vector is given inLemma 1. This completes the proof of Theorem 2.

I. Proof of Theorem 3

Proof: Suppose is the quasi-inverse of .Lemma 7 shows that , thus is nonnegativeas is an M-matrix [17]. Thus, there is a unique mappingbetween all feasible in (28) and feasible in (26), and theoptimal solution to (25) is given by , whereis the optimal solution to the convex problem (31). Note that

since .Next, suppose that , i.e., the constraints in (31) are

not tight, we take the first order derivative ofwith respect to for each and set it to zero. We obtain (32)after making the change of variables back to subject to theconstraint that .

Proof of Corollary 2:Proof: Observe that the function on the right-hand side of

(32) (let ) is pos-itive, monotone, quasi-concave (see, e.g., [18]) in and homo-geneous of degree one. Therefore, it is a concave self-mapping.We first state the following key theorem in [3].

Theorem 9 (Krause’s Theorem [3]): Let be a mono-tone norm on . For a concave mapping with

for , the following statements hold. The condi-tional eigenvalue problem , , ,

has a unique solution , where , . Fur-thermore, converges geometrically fast to ,where .

We constrain by a monotone norm . By Theorem9, the convergence of the iteration

to the unique fixed point is geometrically fast, re-gardless of the initial point. Using Theorem 9 (let ), thereis a unique solution to . Therefore, we can compute(cf. Steps 1–2 of Algorithm 3). Step 3 of Algorithm 3 is ob-tained from (with a normalized suchthat is enforced at each iteration). This completesthe proof of Corollary 2.

J. Proof of Theorem 4

Proof: We make a change of variables for allin (38), and thus (38) is equivalent to the convex problem:

minimize

subject to

variables: (66)

which is strictly convex in . Hence, can be ob-tained efficiently. If , we take the first order derivative

of [the objective function of (66)] with re-spect to and set it to zero. After making a change of variablesback to the domain, we obtain (39). The optimal power vector

in (36) is then recovered from in (39).

K. Proof of Corollary 3

Proof: Similarly to the proof of Corollary 2, Corollary 3can be proved using the nonlinear Perron–Frobenius theory byobserving that the function on the right-hand side of (39) is pos-itive, monotone, homogeneous of degree one and concave in .

L. Proof of Lemma 5

Proof: It is easy to observe that for a given SINR value ,the power to achieve is given by (46) [1]. Next, we will showthat this power vector is feasible in (25) and (36), i.e., it satisfiesthe total power constraint. First, observe that isan irreducible nonnegative matrix. Now, using (46), we have

Thus, for any in (46). Usingthe Perron–Frobenius theorem [17], this implies that

.

M. Proof of Theorem 5

Proof: Theorem 5 is proved using Lagrange duality, i.e.,the KKT conditions, and the fact that the optimal dual variable is


unique as follows. We introduce a dual variable to the constraintof (47) and write the Lagrangian

Now, the gradient of with respectto is given by (unique up to a scaling constant) [15]:

, whereand are scaled such that

Using this fact, we can compute for all . In addition,we see that is unique and equals at optimality.Hence, satisfies

and

Lastly, the uniqueness of follows from the following result:Lemma 9 [21]: Let be a given ir-

reducible matrix with positive diagonal elements and positiveprobability vector, respectively. Then, there exists suchthat . Furthermore, isunique up to an addition of , being a scalar. In particular,this can be computed by solving the following convex opti-mization problem:

maximize

subject to

variables: (67)

N. Proof of Theorem 6

Proof: We use the fact that in a sufficiently close neighbor-hood of , the domain set is convex, and the objective func-tion is twice continuously differentiable. We then use asuccessive convex approximation technique to compute as-suming that the initial point is sufficiently close to . The con-vergence conditions for such a technique are given in [22] and[23]. Instead of solving (47) directly, we replace the objectivefunction of (47) in a neighborhood of a feasible point byits Taylor series (up to the first-order terms)

Assume a feasible that is close to . We then computea feasible by solving the th approximationproblem

maximize

subject to

variables:

(68)

where is the optimal solution of the th approximationproblem. This inner approximation technique converges to alocal optimal solution [22], [23]. In addition, if is suffi-ciently close to , then .

Next, we leverage Lemma 9 in [21] and Algorithm 1 to solve(68). Note that, from (51), the solution to (7) satisfies the opti-mality condition

which can be made equal to by choosing appropri-

ately. In particular, by setting , Algorithm 1 computesa feasible power :

which converges to the optimal solution of(25) and (36) for, respectively,

and

.

O. Proof of Corollary 4

Proof: If (up to a scaling constant), and, then the optimality conditions in

Theorem 5 are clearly satisfied. Any initial condition(up to a scaling constant), will satisfy (up to a scaling constant):

Thus, (up to a scaling constant) for all . This provesCorollary 4.

P. Proof of Theorem 7

Proof: Theorem 7 is proved by using the projected gra-dient method with error [20] to solve (53) and then applyinga result on the convergence of approximate gradient method in[24]. We first look at computing an approximate projected gra-dient for (53), and then show that the approximate projected gra-dient method converges to a neighborhood of the optimal solu-tion under certain conditions on the sequence of step sizes.

Define as the vector with the th entry givenby . Now, based on the gra-

dient of ,

, in (55), an approximate gradient to of(53) can be given by

Note that, at Step 2, the downlink power and uplinkpower output of Algorithm 2 using the input weight pa-


rameter are approximately optimal (in the sense ofsolving the downlink and uplink max-min weighted SINR re-spectively), because the time to compute them is finite and thiscomputation terminates within some tolerance.

Let be the vector with the th entry

Note that if and, then . How-

ever, and contain errors, as Algorithm 2 terminatesin finite time.

We now show that the gradient method converges despite ap-proximation errors made in the computation of the eigenvectors( and ) at Step 2, which spills into the gradient pro-jection computation at Step 3. We will use the following resultfrom [24] ([24, Proposition 1]; see also [20, Sec. 1.3, pp. 61]):

Theorem 10: Let be a sequence generated by thegradient method with errors

where satisfies the Lipschitz assumption, satisfies

where and are some scalars, the stepsizes satisfy

and the errors satisfy

where and are some scalars. Then eitheror else converges to a finite value and .Furthermore, every limit point of is a stationary point of

.The idea of Theorem 10 is that as long as the descent di-

rection is sufficiently aligned with the gradient and the errorsare sufficiently bounded, then the gradient method convergesto the optimal solution. Now, can be written as (let

and ):

where

In Theorem 10, we let , ,

and .It can be shown that

and

where ,and we have used the fact that

for any positive vector , , and .Assume that

and

for some positive and respectively, we can selectsuch that

in the following gradient update:

Now, the point after the above gradient update maybe infeasible with respect to the constraint set of (53). We nowproject to the feasible set by adding a constant term

(the value of the logarithmic weighted SINRevaluated at ) to obtain the following update:

SINR

To show the feasibility of , we first let. Then, we have

SINR

SINR


Hence, , i.e., is feasiblewith respect to the constraint set of (53). Using Theorem 10, weconclude that converges to the optimal solution as

.

ACKNOWLEDGMENT

The authors acknowledge helpful discussions with S. Low atCaltech, S. Friedland at UIC, and K. Tang at Cornell University.

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Chee Wei Tan (M’08) received the M.A. and Ph.D.degrees in electrical engineering from PrincetonUniversity, Princeton, NJ, in 2006 and 2008, respec-tively.

Previously, he was a Postdoctoral Scholar atthe California Institute of Technology (Caltech),Pasadena. He is currently an Assistant Professorat City University Hong Kong. He was a VisitingFaculty at Qualcomm R&D, San Diego, CA, in2011. His research interests are in wireless andbroadband communications, signal processing and

nonlinear optimization.Dr. Tan was the recipient of the 2008 Princeton Universitys Gordon Wu Prize

for Excellence and 2011 IEEE ComSoc AP Outstanding Young ResearcherAward.

Mung Chiang (M’03–SM’08) received the B.S.(Hons.), M.S., and Ph.D. degrees from StanfordUniversity, Stanford, CA, in 1999, 2000, and 2003,respectively,

He was an Assistant Professor from 2003 to 2008and an Associate Professor from 2008 to 2011 atPrinceton University, Princeton, NJ, where he iscurrently a Professor of electrical engineering and anaffiliated faculty in applied and computational math-ematics, and in computer science. His inventionsresulted in five issued patents and several technology

transfers to commercial adoption, and he founded the Princeton EDGE Lab in2009. He is currently writing an undergraduate textbook 20 Questions Aboutthe Networked Life.

Dr. Chiang received rewards for his research in networking, including theIEEE Tomiyasu Award, PECASE, TR35, ONR YIP, NSF CAREER, PrincetonWentz Faculty Award, and several best paper awards.

R. Srikant (S’90–M’91–SM’01–F’06) receivedthe B.Tech. degree from the Indian Institute ofTechnology, Madras, in 1985 and the M.S. and Ph.D.degrees from the University of Illinois in 1988 and1991, respectively, all in electrical engineering.

He was a Member of Technical Staff at AT&T BellLaboratories, Holmdel, NJ, from 1991 to 1995. He iscurrently with the University of Illinois at Urbana-Champaign, where he is the Fredric G. and ElizabethH. Nearing Professor in the Department of Electricaland Computer Engineering, and a Research Professor

in the Coordinated Science Laboratory. His research interests include communi-cation networks, stochastic processes, queueing theory, information theory, andgame theory.

He was an Associate Editor of Automatica, the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL, and the IEEE/ACM TRANSACTIONS ON NETWORKING.He has also served on the editorial boards of special issues of the IEEE J. SEL.AREAS COMMUN. and the IEEE TRANSACTIONS ON INFORMATION THEORY. Hewas the Chair of the 2002 IEEE Computer Communications Workshop in SantaFe, NM, and a Program Co-Chair of IEEE INFOCOM 2007.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. …cheewtan/TanChiangSrikant_TSP2011.pdf · IEEE TRANSACTIONS ON SIGNAL ... DECEMBER 2011 6127 Maximizing Sum Rate and Minimizing MSE

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