Maximizing Sum Rates in Cognitive Radio Networks: Convex Relaxation and Global Optimization Algorithms

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 3, MARCH 2014 667

Maximizing Sum Rates in Cognitive RadioNetworks: Convex Relaxation and Global

Optimization AlgorithmsLiang Zheng and Chee Wei Tan, Senior Member, IEEE

Abstract—A key challenge in wireless cognitive radio networksis to maximize the total throughput also known as the sum ratesof all the users while avoiding the interference of unlicensed bandsecondary users from overwhelming the licensed band primaryusers. We study the weighted sum rate maximization problemwith both power budget and interference temperature constraintsin a cognitive radio network. This problem is nonconvex andgenerally hard to solve. We propose a reformulation-relaxationtechnique that leverages nonnegative matrix theory to first obtaina relaxed problem with nonnegative matrix spectral radiusconstraints. A useful upper bound on the sum rates is thenobtained by solving a convex optimization problem over a closedbounded convex set. It also enables the sum-rate optimality to bequantified analytically through the spectrum of specially-craftednonnegative matrices. Furthermore, we obtain polynomial-timeverifiable sufficient conditions that can identify polynomial-timesolvable problem instances, which can be solved by a fixed-pointalgorithm. As a by-product, an interesting optimality equivalencebetween the nonconvex sum rate problem and the convex max-min rate problem is established. In the general case, we propose aglobal optimization algorithm by utilizing our convex relaxationand branch-and-bound to compute an ε-optimal solution. Ourtechnique exploits the nonnegativity of the physical quantities,e.g., channel parameters, powers and rates, that enables key toolsin nonnegative matrix theory such as the (linear and nonlinear)Perron-Frobenius theorem, quasi-invertibility, Friedland-Karlininequalities to be employed naturally. Numerical results arepresented to show that our proposed algorithms are theoreticallysound and have relatively fast convergence time even for large-scale problems.

Index Terms—Optimization, convex relaxation, cognitive radionetworks, nonnegative matrix theory.

I. INTRODUCTION

A key challenge in cognitive radio networks is to enableefficient wireless resource allocation while ensuring the

unlicensed band users (the secondary users) that share thespectrum have minimal impact on the licensed band users(the primary users). Due to the broadcast nature of thewireless medium, the data rates are affected by interference

Manuscript received November 13, 2012; revised April 3, 2013 andJune 16, 2013. The work in this paper was partially supported by grantsfrom the Research Grants Council of Hong Kong Project No. RGC CityU125212, Qualcomm Inc., the Science, Technology and Innovation Commis-sion of Shenzhen Municipality, Project No. JCYJ20120829161727318 onGreen Communications in Small-cell Mobile Networks, and Project No.JCYJ20130401145617277 on Adaptive Spectrum Access Resource Allocationin Cognitive Radio Networks.

L. Zheng and C. W. Tan are with the Department of Computer Sci-ence, City University of Hong Kong, Tat Chee Ave., Hong Kong (e-mail:[email protected], [email protected]).

Digital Object Identifier 10.1109/JSAC.2014.140324

when all the users transmit simultaneously over the samespectrum. Thus, unlicensed radio transmission in cognitiveradio networks can lead to overwhelming interference thatadversely affects the overall performance of the network[1]–[3]. In this regard, the interference temperature, whichis the total interference plus noise, is an important featurefor interference management. Interference in cognitive radionetworks can be suitably controlled by imposing interferencetemperature constraints on the secondary users (to limit theinterference temperature below a predefined threshold) so thatthe interference experienced by the primary users are cappedat a limit regardless of the number of secondary users [4].Dynamic spectrum or power allocation to maximize the totalthroughput also known as the sum rates in response to varyingquality of service demands is an important problem. In thispaper, we focus on maximizing the overall weighted sumrates with both power budget constraints and interferencetemperature constraints in cognitive radio networks.

Throughput maximization for wireless cognitive radio net-works is relatively new and challenging. The authors in [5]considered throughput maximization for secondary users overnon-overlapping channels. Throughput maximization in inter-ference channels is considerably harder, as it is a nonconvexproblem. In [6], the two-user sum rate maximization specialcase with equal weights was studied. The authors in [7] provedthe NP-hardness of the problem. An often used technique totackle the nonconvexity is the standard Lagrange dual relax-ation [7], [8]. The authors in [7] used the Lyapunov theoremin functional analysis to establish a zero duality gap propertywhen the number of frequencies grows infinitely large. In[9], Lagrangian dual relaxation was used for estimating theduality gap in the general case. The authors in [10]–[13]developed successive convex approximation using geometricprogramming that only solves the problem suboptimally.

Global optimization algorithms that can compute the globaloptimal solution, albeit with exponential-time complexity,have been proposed in the literature recently. The authorsin [14] proposed global optimization algorithms using DC(difference of convex functions) programming. In [15], [16],the problem was reformulated into a convex maximizationproblem over an unbounded convex constraint set. Based onthis reformulation, an outer approximation algorithm and abranch-and-bound method were proposed in [15] and [16]respectively. However, the unbounded constraint set is suscep-tible to numerical artifacts in algorithm design, i.e., numericalcomputation is not exact. The authors in [17] obtained suffi-

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668 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 3, MARCH 2014

cient conditions to solve the problem optimally in polynomial-time when there is only a single total power constraint, and toprovide performance bounds for the problem with individualpower constraints in [18].

However, techniques that shed light on achieving optimality(in polynomial time) or giving convex relaxation have comefew and far. Most existing algorithms have drawbacks from atheoretical viewpoint as they cannot guarantee the optimalityof the numerical solution. For example, algorithms based onLagrange duality suffer from the positive duality gap andcannot guarantee finding a global optimal solution. Also,previously proposed global optimization algorithms do notconsider interference temperature constraints, which are aunique feature of cognitive radio networks. To the best ofour knowledge, there has been neither work that character-izes the quality of suboptimal solution analytically nor a(polynomial-time) optimal algorithm. Quantifying optimalityin polynomial-time (whenever that is possible) and generatingconvex relaxation can provide new perspectives to this (stillopen) NP-hard problem, especially in the large-scale setting.

The contributions of this paper are as follows:1) We propose a reformulation-relaxation technique based

on nonnegative matrix theory that uses convex relaxationto bound the nonconvex achievable rate regions for theweighted sum rate maximization problem with powerbudget and interference temperature constraints.

2) Our technique can provide analytically sufficient condi-tions to identify problem instances that are polynomial-time optimally solvable, and a fixed-point algorithm(i.e., no parameter tuning required) is proposed to com-pute the global optimal solution.

3) In the general case, we propose a global optimizationalgorithm that leverages the convex relaxation and thebranch-and-bound method to yield successively tighterbounds to the global optimal value (quantified by ε-optimality). Our optimization software is available fordownload at [19].

Using our convex relaxation, new insights can even bedrawn, e.g., an interesting optimality equivalence between theweighted sum rate problem and the weighted max-min rateproblem (that can be solved in polynomial time). These areconsequences from exploiting the inherent nonnegativity of thephysical quantities, e.g., channel gain, powers and rates, in thesum rate problem that enables key tools in nonnegative matrixtheory and nonlinear Perron-Frobenius theory to be employednaturally and successfully.1

This paper is organized as follows. We present the systemmodel in Section II. In Section III, we introduce the weightedsum rate maximization problem (in the transmit power andinterference temperature domain) and propose an equivalentreformulation (in the achievable rate domain). In Section IV,we propose a convex relaxation technique to convexify thesum-rate reformulation and identify problem instances thatcan be solved optimally in polynomial-time. In Section V,we propose a global optimization algorithm to solve the

1This idea relates to a repertory of theoretical and algorithmic developmentsfor wireless utility maximization that offer useful insights based on thenonnegative matrix theory and the nonlinear Perron-Frobenius theory, see[15]–[18], [20]–[23] for related work.

problem in the general case. We highlight the performanceof our algorithms using numerical examples in Section VI.We conclude with a summary in Section VII. All the proofscan be found in the Appendix.

The following notation is used in our paper. Column vec-tors and matrices are denoted by boldfaced lowercase anduppercase respectively. Let ρ(F) denote the Perron-Frobeniuseigenvalue of a nonnegative matrix F, and x(F) and y(F)denote the Perron right and left eigenvectors of F associatedwith ρ(F). We let el denote the lth unit coordinate vector,I denote the identity matrix and 1 = [1, . . . , 1]�. The super-scripts (·)� denotes transpose, and ‖·‖F denotes the Frobeniusnorm. We denote x ◦ y as a Schur product of x and y, i.e.,x ◦ y = [x1y1, . . . , xLyL]

�. For a vector x = [x1, . . . , xL]

�,

diag(x) is its diagonal matrix diag(x1, . . . , xL). Let ex

denote ex = (ex1 , . . . , exL)�, and logx denote logx =(log x1, . . . , log xL). An equality involving eigenvectors istrue up to a scaling constant.

II. SYSTEM MODEL

We consider a cognitive radio network with a collection ofprimary users and secondary users transmitting simultaneouslyon a single-input-single-output (SISO) channel. There are Lusers (transmitter/receiver pairs) sharing a common frequency-flat fading channel, i.e., the total number of primary users andsecondary users is L. Assume that there are S primary usersin the cognitive radio network (S < L), then the number ofsecondary users is L − S. All users are indexed by l, wherethe lth user is a primary user when l ≤ S, otherwise it isa secondary user. Each user employs a single-user decoder,i.e., treating the interference as additive Gaussian noise, andhas perfect channel state information at the receiver. Let G =[Glj ]

Ll,j=1 > 0L×L represent the channel gain, where Glj is

the channel gain from the jth transmitter to the lth receiver,and n = (n1, . . . , nL)

�> 0, where nl is the noise power

at the lth user. The vector p = (p1, . . . , pL)�

is the transmitpower vector.

The Signal-to-Interference-and-Noise Ratio (SINR) for thelth receiver is defined as the ratio of the received signal powerto the sum of interference signal power and additive noisepower. Now, the SINR of the lth user can be given in termsof p:

SINRl(p) =plGll∑

j �=l pjGlj + nl. (1)

We also define a nonnegative matrix F with entries:

Flj =

{0, if l = j

Glj/Gll, if l �= j(2)

and the vector v = ( n1

G11, n2

G22, . . . , nL

GLL)�

. Moreover, weassume that F is irreducible, i.e., each link has at least aninterferer. For brevity, the vector γ denotes the SINR for allusers. With these notations, the SINR can be expressed as:

γl =pl

(Fp+ v)l. (3)

Next, let the vector q denote the normalized interferencetemperature for all users with entries given by:

ql =

L∑j=1

pjFlj + vl, (4)

ZHENG AND TAN: MAXIMIZING SUM RATES IN COGNITIVE RADIO NETWORKS: CONVEX RELAXATION AND GLOBAL OPTIMIZATION ALGORITHMS 669

i.e., q = Fp + v. The achievable data rate rl of the lth user(assuming the Shannon capacity formula) is given by:

rl = log(1 + SINRl(p)) (5)

measured in nats per symbol. This can be expressed aserl − 1 = pl/ql for all l. Since p = diag(er − 1)q andq = Fp + v, the power and interference temperature canbe written, respectively, as p = diag(er − 1)(Fp + v) andq = F diag(er − 1)q + v. We can thus obtain a one-to-onemapping between r and p and also one between r and q,given by:

p(r) =(I+ F− diag(er)F

)−1(diag(er − 1)

)v (6)

and q(r) =(I+ F− F diag(er)

)−1v, (7)

respectively. Hence, given a feasible rate r, the correspondingpower and interference temperature can be computed by (6)and (7) respectively.

Typically, constraints in a cognitive radio network can beclassified into either resource budget constraints that limitpower at the transmitters or interference management con-straints that limit interference at the receivers. As secondaryusers are allowed to coexist with primary users [4], in-terference temperature constraints are used to manage theinterference of the secondary users from overwhelming theprimary users. The weighted power constraint set P andthe weighted interference temperature constraint set Q arerespectively given by:

P = {p | a�k p ≤ pk, k = 1, . . . ,K}, (8)

and Q = {q |b�mq ≤ qm, m = 1, . . . ,M}, (9)

where p = (p1, . . . , pK)� ∈ RK and ak ∈ R

L k = 1, . . . ,Kare respectively the upper bound and positive weight vectorsfor the weighted power constraints, and q = (q1, . . . , qM )� ∈R

M and bm ∈ RL m = 1, . . . ,M are respectively the

upper bound and positive weight vectors for the weightedinterference temperature constraints. Note that we must haveqm ≥ b�

mv for a nonempty constraint set Q. By setting theweights appropriately, we are able to guarantee prioritizedaccess of the spectrum. In particular, we can set (ak)l forl ≤ S to be relatively smaller than those for l > S and (bm)lfor l ≤ S to be relatively larger than those for l > S, i.e.,more restrictive constraints on secondary users’ power and pri-mary users’ interference temperature. As special cases, whenak = ek, we have the individual power constraints pk ≤ pkfor all k, and when ak = 1 and K = 1, we have a singletotal power constraint. Likewise, the weighted interferencetemperature constraints are general enough for modeling. Forexample, in the case of the individual interference temperatureconstraints, i.e., bm = em, we can set qm to some finite valuefor a primary user (when m ≤ S), while we can set qm toinfinity for a secondary user (when m > S).

Now, given the above power constraints, one can straight-forwardly establish (loose) upper bounds to the achievablerates in (5). In particular, since γl =

pl(Fp+ v)l

≤ plvl≤

maxk=1,...,K

pk(ak)lvl

, any r ∈ RL+ satisfies the inequalities

r ≤ r = (r1, . . . , rL)�, (10)

where rl = log

(1 + max

k=1,...,K

pk(ak)lvl

), l = 1, . . . , L.

Thus, we have r ≤ r as an implicit box constraint set, albeit aloose one, that contains the nonconvex achievable rate region.In the following, we describe how to obtain better (tighter)rate constraint set using convex relaxation.

III. PROBLEM STATEMENT AND REFORMULATION

The weighted sum rate maximization problem in a cognitiveradio network is given by:

maximizeL∑

l=1

wl log(1 + SINRl(p)

)subject to p ∈ P ,

q ∈ Q,variable: p,q,

(11)

where w = [w1, . . . , wL]�> 0 is a given probability vector,

i.e., 1�w = 1, and wl is a weight assigned to the lth link toreflect priority (a larger weight reflects a higher priority). Wedenote the optimal solution of (11) by p� = (p�1, . . . , p

�L)

�

and q� = (q�1 , . . . , q�L)

�. In general, (11) is nonconvex (dueto the nonconvex objective function) and thus is hard to solve.

We next study a reparameterization of (11) from the powerp and q to an equivalent optimization problem that onlyhas the achievable rate r and q as the explicit optimizationvariables. The advantage of the reformulation is an efficienttechnique to bound the nonconvex pareto rate region usingconvex relaxation. Let us first define the following set ofnonnegative matrices:

Bk = F+1

pkva�k , k = 1, . . . ,K, (12)

Dm =(I+

1

qm − b�mv

vb�m

)F, m = 1, . . . ,M. (13)

Note that Bk and Dm contain the problem parametersassociated with the kth power budget constraint in P and themth interference temperature constraint in Q respectively. Wenow introduce a reformulation of (11) in the following.

Theorem 1: The optimal value in (11) is equal to the optimalvalue of the following problem:

maximizeL∑

l=1

wlrl

subject to Bk diag(er)q ≤ (I+Bk)q, k = 1, . . . ,K,

Dm diag(er)q ≤ (I+Dm)q, m = 1, . . . ,M,variables: r,q.

(14)Denote the optimal r and q in (14) by r� and q� respec-tively. Using (4) and (5), we have p� in (11) given byp� = diag(er

� − 1)q�.Though (14) has a linear objective function, (14) is still

nonconvex due to the nonconvex constraint set. However,the transformation from (11) to (14) plays an instrumentalrole that facilitates a novel use of nonnegative matrix theoryfor convex relaxation and algorithm design. Fig. 1 gives anoverview of the development in the paper, particularly theconvex relaxation, the algorithms and the connections betweenkey optimization problems.


Sum Rate Maximization

in power domain (11)

(nonconvex)

Equivalent Reformulation

in rate domain (14)

(nonconvex)

Theorem 1

Optimization in (17)

(convex)

Convex Relaxation (26)

(convex)

Weighted Max-min Rate

(20)

Weighted Max-min Rate

(28)

Linear Relaxation

(27)

Global upper bound

By solving

(23) and (25)Corollary 1

Assumption 1 holds

Algorithm 3 by the

branch-and-bound

method

Global upper

bound

Global lower

bound

Algorithm 2

Lemma 3

by solving (21)

Lemma 2

Lemma 1

Algorithm 1

Normalization

The Friedland-

Karlin inequality

Nonnegative

matrix theory

Exact

polynomial-time

solution

Fig. 1. Overview of our convex relaxation approach to the sum rate maximization problem. We obtain both a polynomial-time algorithm (for special cases)and a global optimization algorithm (for the general case). The interesting reformulations of the constraint sets among problems studied in this paper areillustrated in different boxes: the ones with solid boundaries are subject to original weighted power and interference temperature constraints, the ones withbold boundaries are subject to spectral radius constraints, and the one with dotted boundary is subject to linear constraints. Notably, (14) maximizes a linearobjective function subject to a nonconvex constraint set, i,e, it is still nonconvex, but (14) enables the design of a polynomial-time algorithm (for specialcases) and a global optimization algorithm (for the general cases).

IV. CONVEX RELAXATION AND POLYNOMIAL-TIMEALGORITHMS

In this section, we first show that the constraints in (14) canbe equivalently rewritten as convex spectral radius constraintsfor those special cases in Section IV-A that enable (14) to besolved in polynomial time. For the general case, we proposea convex relaxation approach in Section IV-B and a branch-and-bound method in Section V to compute the global optimalsolution of (14), equivalently that of (11).

A. Polynomial-time Solvable Special Cases

The reformulation introduced in Theorem 1 allows us todecompose the weighted sum rate maximization problem in(11) into first optimizing r and q and then projecting thesolution (through a one-to-one mapping) back to p. Although(14) is nonconvex, the Perron-Frobenius theorem [24] enablesus to rewrite (14) into one that has the rates characterized bya set of spectral radius constraints (i.e., r is the only variable)when the following assumption is satisfied.

Assumption 1: Let Bk and Dm be given in (12) and (13)respectively, the following is satisfied:

Bk = (I+Bk)−1Bk ≥ 0, k = 1, . . . ,K, (15)

Dm = (I+Dm)−1Dm ≥ 0, m = 1, . . . ,M, (16)

i.e., Bk and Dm are irreducible nonnegative matrices for allk and m.

When there is no interference, i.e., Glj = 0, l �= j andF in (2) is an all-zero matrix, we have Bk = 1

pkva�k , which

implies that Bk = 1pk+a�

kvva�k for all k. Thus, Assumption 1

can be satisfied under certain SNR and interference conditions.

The conditions where Assumption 1 is more likely to hold areexamined in Section VI.

Corollary 1: If Assumption 1 is satisfied, then (11) can besolved by the following convex optimization problem:

maximizeL∑

l=1

wlrl

subject to log ρ(Bk diag(e

r)) ≤ 0, k = 1, . . . ,K,

log ρ(Dm diag(er)

) ≤ 0, m = 1, . . . ,M,variables: r,

(17)where Bk and Dm are given in (15) and (16) respectively.

Due to the log-convexity property of the Perron-Frobeniuseigenvalue [25], log ρ

(Bl diag(e

r))

and log ρ(Dm diag(er)

)are convex functions in r for the irreducible nonnegativematrices Bk and Dm respectively. Hence, (17) is a convexoptimization problem (implying that (11) can be solved inpolynomial time). In fact, we can obtain a closed form solutionto (11) whenever Assumption 1 holds.

Lemma 1: If Assumption 1 is satisfied, then we have

ρ(Bi diag(e

r�))= 1, p� = diag(er

� − 1)x(Bi diag(e

r�)),

and q� = x(Bi diag(e

r�)),

where i = arg maxk=1,...,K

ρ(Bk diag(e

r�)), (18)

or

ρ(Dι diag(e

r�))= 1, p� = diag(er

� − 1)x(Dι diag(e

r�)),

and q� = x(Dι diag(e

r�)),

where ι = arg maxm=1,...,M

ρ(Dm diag(er

�

)). (19)


Next, we study the weighted max-min rate fairness problemwhose solution is to provision rate fairness. How to adapt theweights in the weighted max-min rate fairness problem (a con-vex problem in general) to solve the sum rate maximization (anonconvex problem in general)? This interesting and importantconnection between these two problems will be made precisein the following. Next, instead of solving (17) directly, let usconsider the following weighted max-min rate problem withthe same constraint set in (17):

maximize minl=1,...,L

rlβl


r)) ≤ 0, k = 1, . . . ,K,

log ρ(Dm diag(er)

) ≤ 0, m = 1, . . . ,M.variables: r,

(20)where β ∈ R

L+ are weights assigned to reflect priority among

users. This means that a user requiring a higher quality-of-service has a larger weight, i.e., βl is larger. In the following,we apply nonnegative matrix theory, particularly the Perron-Frobenius theorem, to characterize the optimal solution andthe optimal value of (20).

Lemma 2: The optimal solution of (20) denoted by r� is avector with r�l /βl equal to a common value δ�, i.e., r� = δ�β,where δ� is the optimal value of (20).

We now explain how solving (20) can lead to a polynomial-time algorithm to compute the optimal solution of (17) thatrequires solving (20) as an intermediate step. In particular,with β appropriately chosen in (20), both (17) and (20) canhave the same optimal solution.

Lemma 3: Both (17) and (20) have the same optimalsolution with β =

(ξ�− (log ��)1

)/(1�ξ�−L log ��), where

ξ� and log �� are respectively the optimal solution and theoptimal value of the following optimization problem:

minimize max

{max

k=1,...,K

{ρ(Bk diag(e

ξ))}

,

maxm=1,...,M

{ρ(Dm diag(eξ)

)}}

subject toL∑

l=1

wlξl ≥ 1,

variables: ξ.

(21)

Remark 1: Using Lemma 3, an important special case willnow be explained. Solving (20) with β = 1 is equivalent tosolving (17) with w given by w = x(Ω) ◦ y(Ω), where Ω =argΩ∈{˜Bk,˜Dm} max

{max

k=1,...,K{ρ(Bk)}, max

m=1,...,M{ρ(Dm)}}.

This interesting connection between (17) (a generally hardproblem) and (20) (a polynomial-time solvable problem) isobtained using the Friedland-Karlin inequality (cf. Theorem3.1 in [26]), which is a fundamental inequality in nonnegativematrix theory. In addition, this connection will be used inthe convergence proofs of Algorithm 1 given below and theglobal optimization algorithm in Section V.

Suppose Assumption 1 holds, then the following two-timescale algorithm computes the optimal solution r� of (14): theouter loop iterates to solve (21) and to update the weight β(t)used in (20), which is solved by the inner loop.

Algorithm 1 (Polynomial-time Algorithm):

Initialize ξ(0).1) Update ξ(t+ 1) as follows:

if ρ(Bit diag(e

ξ(t))) ≥ ρ

(Dιt+1 diag(e

ξ(t)))

ξl(t+ 1) = max{ξl(t) + wl

−(x(Bit diag(e

ξ(t))) ◦ y(Bit diag(e

ξ(t))))

l, 0

},

elseξl(t+ 1) = max

{ξl(t) + wl

−(x(Dιt diag(e

ξ(t))) ◦ y(Dιt diag(e

ξ(t))))

l, 0

}.

2) Normalize ξ(t+ 1):Compute it+1 = arg max

k=1,...,Kρ(Bk diag(e

ξ(t+1)))

and

ιt+1 = arg maxm=1,...,M

ρ(Dm diag(eξ(t+1))

).

if ρ(Bit+1 diag(e

ξ(t+1))) ≥ ρ

(Dιt+1 diag(e

ξ(t+1)))

ξ(t+ 1)← log(eξ(t+1)/ρ

(Bit+1 diag(e

ξ(t+1))))

,

elseξ(t+ 1)← log

(eξ(t+1)/ρ

(Dιt+1 diag(e

ξ(t+1))))

.

3) Update β(t+ 1) as follows:�(k + 1) = max{ρ(Bit+1 diag(e

ξ(t+1))),

ρ(Dιt+1 diag(e

ξ(t+1)))},

β(k + 1) =ξ(t+ 1)− (

log �(t+ 1))1

1�ξ(k + 1)− L log �(t+ 1).

4) Compute p(t+1) and q(t+1) by solving (20) with theweight replaced by β(t+ 1):

while ‖p(τ) − p(τ − 1)‖ ≥ ε

pl(τ + 1) =

(βl(t+ 1)

log(1 + SINRl(p(τ)))

)pl(τ),

p(τ + 1)←p(τ + 1)

max{

maxk=1,...,K

{a�k p(τ + 1)

pk

}, maxm=1,...,M

{b�mFp(τ + 1)

(qm − b�mv)

}} ,

rl(τ + 1) = log(1 + SINRl(p(τ + 1))),ql(τ + 1) = pl(τ + 1)/(erl − 1).

p(t+ 1) = p(τ), q(t+ 1) = q(τ),and r(t+ 1) = r(τ).when p(τ) converges in the inner loop, and go to

Step 1.

Lemma 4: Suppose Assumption 1 holds, starting from anyinitial point ξ(0), ξ(t) converges to the optimal solution of(21) and β(t) calculated in terms of ξ(t) is of the same scalingof the optimal solution r� of (17). In addition, r(t) convergesto the optimal solution of (17) with p(t) and q(t) being thecorresponding optimal power and interference temperature.

B. Convex Relaxation

In this section, we propose a novel convex relaxation toyield useful upper bounds to (11). Our convex relaxation stemsfrom the quasi-invertibility introduced in Section IV-A. When-ever the quasi-inverse condition holds, the convex relaxation isin fact tight, otherwise the convex relaxation provides a usefulbound to the global optimal value. In particular, we exploit thespectrum of specially-constructed nonnegative matrices (Bk

and Dm introduced below) that are obtained from Bk andDm, thereby leveraging the techniques in Section IV-A.


0 0.5 1 1.5 20

1

2

3

4Rate region

Achievable rate regionConvex relaxation

r1 (nats/symbol)

r2

(nat

s/sy

mbol

)

(a)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5Rate region

Achievable rate regionConvex relaxation

r1 (nats/symbol)

r2

(nat

s/sy

mbol

)

(b)

Fig. 2. Achievable rate region and relaxed rate region for a 2-user case. Thechannel gains are given by G11 = 0.85, G12 = 0.13, G21 = 0.14 andG22 = 0.87. The noise power for both users are 0.1W. The other problemparameters are given as follows: (a) p = (5, 5)� , a1 = (10, 1)� , a2 =(1, 1)�; (b) p = (1, 100)� , a1 = (1, 1)� , a2 = (1, 1)� .

Let us introduce the following set of nonnegative matrices(without making Assumption 1):

Bk =(I+Bk+diag(ε)

)−1(Bk−X�

k), k = 1, . . . ,K, (22)

where X�k is obtained by solving

minimize ‖Xk‖Fsubject to

(I+Bk + diag(ε)

)−1(Bk − Xk) ≥ 0,

Xk ≥ 0,

variables: Xk,

(23)

and

Dm =(I+Dm +diag(ε)

)−1(Dm − Y�

m), m = 1, . . . ,M,(24)

where Y�m is obtained by solving

minimize ‖Ym‖Fsubject to

(I+Dm + diag(ε)

)−1(Dm − Ym) ≥ 0,

Ym ≥ 0,

variables: Ym.(25)

Note that ε is a vector with each entry being a givensmall positive scalar (required in case (I+Bk) or (I+Dm)is not invertible); otherwise, ε can be an all-zero vector.Notably, (23) and (25) are convex optimization problems thatcan be solved numerically using interior-point solvers, e.g.,the cvx software package [27]. Furthermore, X�

k or Y�m

are all-zero matrices whenever Bk or Dm are nonnegative,i.e., the quasi-inverse of Bk or Dm exists (Bk = Bk andDm = Dm); otherwise X�

k and Y�m are relatively small

matrices (as compared to Bk and Dm) with most of theirentries being zeros.

We replace Bk and Dm on the righthand-side of theconstraints in (14) such that we have, respectively, (Bk −X�

k) diag(er)q ≤ (I + Bk)q and (Dm − Y�

m) diag(er)q ≤(I + Dm)q. We can then compute the nonnegative Bk andDm, and rewrite the constraints as Bk diag(e

r)q ≤ q andDm diag(er)q ≤ q. By using the Perron-Frobenius theorem,we consider the following convex optimization problem:

maximizeL∑

l=1

wlrl


r)) ≤ 0, k = 1, . . . ,K,

log ρ(Dm diag(er)

) ≤ 0, m = 1, . . . ,M,variables: r.

(26)This relaxation given by (26) convexifies the nonconvex feasi-ble region in (14) to one that is convex. Fig. 2 illustrates howthe convex relaxation obtained by (26) produces a convex setfor two different sets of parameter setting. The blue curves arethe boundaries of the original pareto rate regions, while thedashed convex curves are the boundaries of the relaxed convexsets. As observed, Fig. 2(a) illustrates that the relaxationgap between the original rate region and the relaxed rateregion can be remarkably small. Furthermore, when the (ingeneral nonconvex) original rate region turns out to be convexin shape, the relaxed rate region coincides exactly with theoriginal rate region, as illustrated in Fig. 2(b).

Observe that (26) has a problem structure (linear objectiveand spectral radius constraint set) similar to that of (17). Thisallows us to leverage Algorithm 1 to solve (26) as follows.

Algorithm 2 (Convex Relaxation Algorithm):

1) Solve (23) and (25) to obtain Xk and Ym respectively.Then compute Bk and Dm by (22) and (24) respec-tively.

2) Run Algorithm 1 with Bk and Dm replaced by Bk andDm respectively.

Although an upper bound of the optimal value of (11)can be computed by Algorithm 2 (which solves (26)), it isdesirable that this bound can be further tightened and, moreimportantly, eventually yields the global optimal solution to(11). We pursue this by leveraging the relaxation in (26) anda branch-and-bound method [28], [29] in the next section.

V. GLOBAL OPTIMIZATION ALGORITHM

In this section, we exploit the convex relaxation (26)together with the branch-and-bound method to iteratively


compute the global optimal solution of (11). Recall that wehave an implicit box constraint set in (10), and this will beused as the initial set Binit = {0 ≤ r ≤ r} in branch-and-bound and to be subdivided iteratively into smaller subsets forsearching. In branch-and-bound, searching is organized usinga binary tree, where the union of the sets represented by theleaf nodes is Binit. At each leaf node, we can obtain a lowerbound and an upper bound to (11) over the subset on thatleaf node. Suppose the gap between the upper bound and thelower bound is some positive ε, which is nonincreasing withthe search iterates. Then, branch-and-bound can locate an ε-optimal neighborhood of r�. In the following, we give thedetails of this global optimization algorithm to compute theglobal optimal solution of (14), equivalently that of (11).

By applying the Friedland-Karlin inequality (cf. Theorem3.1 in [26]) to the spectral radius constraints in (26), we canfurther relax (26) into a linear program (LP). In particular, theoptimal value of the following LP is an upper bound to (26):

maximizeL∑

l=1

wlrl

subject toL∑

l=1

(x(Bk) ◦ y

(Bk)

)lrl ≤ − log ρ(Bk),

k = 1, . . . ,K,L∑

l=1

(x(Dm) ◦ y(Dm)

)lrl ≤ − log ρ(Dm),

m = 1, . . . ,M,0 ≤ r ≤ r,

variables: r,(27)

where Bk and Dm are computed using (22) and (24) respec-tively, and x(Bk)◦y

(Bk) and x(Dm)◦y(Dm) are probability

vectors. Let r denote the optimal solution of (27).Observe that we have added an extra box constraint to (27)

which will be subdivided iteratively. As mentioned, the LP in(27) is a relaxation of (26), which in turn is a relaxation of(11). The rate iterates obtained by solving this relaxed problemmay not be feasible for (11). We propose a projection using themax-min rate fairness problem. In particular, with the weightof the objective function in (20) replaced by the optimal ratecomputed using the relaxed problem, and then solving (20)(which has the same feasible set as (11)), a projection ontothe feasible rate domain of (11) is made at each iteration (thusyielding a feasible solution at each iteration as well as a lowerbound to (11)). Replacing the weight in (20) by r subject tothe constraint set in (11), we thus have:

maximize minl=1,...,L

log(1 + SINRl(p))

rlsubject to a�k p ≤ pk, k = 1, . . . ,K,

b�mq ≤ qm, m = 1, . . . ,M,

variables: p,q,

(28)

whose optimal solution is denoted by r. Note that r is asuboptimal solution for (11). Thus,

∑Ll=1 wlrl is a lower

bound to the optimal value of (11).In summary, the computational methodology is as follows.

Let R(t) = {B1(t), . . . ,BJ(t)} denote the collection of boxsubsets Bj(t) = {rj(t) ≤ r ≤ rj(t)} for all j at tth iteration,

where J(t) = |R(t)| is the number of subsets in R(t). Binit

is the initial rectangular constraint {0 ≤ r} on the root nodeof the binary tree. In the tth iteration, we solve (27) overeach subset Bj(t) to obtain the optimal solution rj(t). Alower bound w�rj(t) is then obtained by solving (28) withthe weight replaced by rj(t), i.e., taking the maximum overall the lower bound at each leaf nodes across all the levelsin the binary tree. Next, we choose the maximum over allupper bounds as U(t) and choose the maximum over all lowerbounds as L(t). We split Bj , whose corresponding w�rj(t)is maximal, into two subsets along one of its longest edges,removing it and adding the two new subsets to R(t). Thismethod generates a nested sequence of box subsets from Binit.The details are given in the following algorithm that computesthe global optimal solution of (11).

Algorithm 3 (Global Optimization Algorithm):

Initialize R(0) = {Binit}.

1) Compute J(t) = |R(t)|.2) For each Bj(t) ∈ R(t), solving the following LP:

maximizeL∑

l=1

wlrl

subject toL∑

l=1

(x(Bk) ◦ y

(Bk)

)lrl ≤ − log ρ(Bk),

k = 1, . . . ,K,L∑

l=1

(x(Dm) ◦ y(Dm)

)lrl ≤− log ρ(Dm),

m = 1, . . . ,M,r ∈ Bj(t),

variables: r,(29)

where Bk and Dm are computed using (22) and (24)respectively, and we denote the optimal solution of (29)by rj(t).Compute the optimal solution rj(t) of (28) with β =rj(t) by the following computation which iterates as aninner loop:

while ‖p′(τ) − p′(τ − 1)‖ ≥ ε

p′l(τ + 1) =

(rjl (t)

log(1 + SINRl(p′(τ)))

)p′l(τ),

p′(τ + 1)←p′(τ + 1)

max{

maxk=1,...,K

{a�k p′(τ + 1)

pk

}, maxm=1,...,M

{b�mFp′(τ + 1)

(qm − b�mv)

}} ,

r′l(τ + 1) = log(1 + SINRl(p′(τ + 1))),

q′l(τ + 1) = p′l(τ + 1)/(er′l − 1).

rj(t) = r′(τ) when this inner loop converges.3) Update the upper bound L(t) and lower bound U(t) as

follow:J = arg max

j=1,...,J(t)w�rj(t),

L(t) = maxj=1,...,J(t)

w�rj(t),

U(t) = maxj=1,...,J(t)

w�rj(t).

4) Update the collection of box subsets:


if U(t)− L(t) ≤ εAlgorithm 3 terminates.

elseFor any w�rj(t) ≤ U(t),

R(t+ 1) = R(t) − {Bj(t)}.Split BJ (t) along one of its longest edges intoBI and BII ,R(t+ 1) =

(R(t) − {BJ (t)})⋃{BI ,BII}.Set t← t+ 1, and go to Step 1.

Theorem 2: The iterates rJ (t) in Algorithm 3 converge tothe ε-optimal solution of (14).

We add the following remarks regarding the implementationissues and the optimality of Algorithm 3.

Remark 2: The limit point of rJ (t) solves (11) by thefollowing equations in (30) and (31) (see next page).

Remark 3: In Algorithm 3, we can delete the subsets thatsatisfy w�rj(t) ≤ U(t) by eliminating them from R(t) atStep 4, since these subsets do not contain the optimal solutionand need not be considered in subsequent iterations. Thisreduces overall searching time of Algorithm 3.

Remark 4: At the tth iteration of Algorithm 3, L(t) =w�rJ (t) is the maximum over all the upper bounds at eachleaf nodes in the binary tree, which gives a global upper boundon the optimal value of (11). Practical stopping criterions forAlgorithm 3 can be U(t) − L(t) ≤ ε for a given small ε.This means that we have w�r� ≤ w�rJ (t) + ε. Therefore,whenever Algorithm 3 terminates with an error tolerancenumber ε, rJ (t) is an ε-optimal solution of (11).

In general, the worst case computational complexity ofthe branch-and-bound method is exponential. However, theaverage complexity of Algorithm 3 can be significantly lower:First, the initial relaxed set that leverages the Friedland-Karlin inequality in nonnegative matrix theory is a cleverchoice. Second, the number of iterations in branch and boundincreases with the problem size. Thus, a stopping criterion εcan be used in Algorithm 3 to balance the solution accuracyand the number of iterations. Third, Algorithm 3 updates theglobal upper and lower bounds by solving a LP. Being ableto solve a sequence of (polynomial-time solvable) LPs ineach iteration enables a reasonably accurate solution to becomputed with low complexity.

VI. NUMERICAL EXAMPLES

In this section, we evaluate the performance of our algo-rithms to solve (11). Our optimization software implemen-tation (that leverages the cvx software [27]) is availablefor download at [19]. We first evaluate the performance ofAlgorithm 1 using numerical examples in which Assumption1 holds. In the general case (when Assumption 1 does nothold), we evaluate the performance of Algorithm 3.

To illustrate the validity of Assumption 1, we find theaverage number of instances when Assumption 1 holds fora 10-user network in Table I. The percentages �{Bk ≥ 0}represents a statistical average (normalized to 100%) of whenthe nonnegative Bk for all k exist, and �{Dm ≥ 0} representsa statistical average (normalized to 100%) of when the non-negative Dm for all m exist. Roughly speaking, this indicateson average how often Assumption 1 holds. We generate each

parameter randomly in a given range for fifty thousand times,and record the number of times that Bk for all k and Dm

for all m are nonnegative. From Table I, we observe that bothBk for all k and Dm for all m are nonnegative when thecross-channel interference is relatively low. In other words,Assumption 1 is more likely to hold with not only smallerGlj(l �= j) but also larger q. In addition, a smaller numberof constraints and a smaller problem size make Assumption 1more likely to hold.

Consider the following channel gain matrix for three usersgiven by

G =

⎡⎣ 1.55 0.02 + σ 0.03 + σ

0.05 + σ 1.50 0.08 + σ0.05 + σ 0.02 + σ 1.62

⎤⎦ , (32)

where σ is a constant added to the non-diagonal entries of thechannel gain matrix (which increase the interference in thenetwork). To compute the global optimal sum rates, dependingon σ, we evaluate the performance of Algorithm 1 wheneverAssumption 1 holds, otherwise we evaluate the performanceof Algorithm 3.

We assume that there are two power constraints and aninterference temperature constraint (K = 2 and M = 1). Theweights for the power and interference temperature constraintsare: a1 = (0.27, 0.51 0.22)�, a2 = (0.32, 0.74, 0.61)� andb1 = (0.63, 0.36, 0.56)�. We also set p = (1.0, 1.5)� andq = 1.2. The noise power of each user is 1 W, and the weightfor the sum rate function is w = (0.30, 0.37, 0.24)�. First, welet σ = 0 in (32). Under this setting, B1, B2 and D1 all exist,thus (11) can be solved optimally by Algorithm 1. Fig. 3 plotsthe evolution of rates for the three users that run Algorithm 1.We set the initial rate vector to r(0) = (2.11, 2.25, 2.52)�

nats/symbol, and run Algorithm 1 for 10 iterations beforeit terminates. Fig. 3(a) shows that Algorithm 1 convergesto the optimal solution (verifying Lemma 4). The optimalrate r� is (2.75, 2.10, 2.26)� nats/symbol, which satisfiesρ(B2 diag(e

r�))

= 1 (verifying Lemma 1). We also runAlgorithm 1 with a suitably large number of primary andsecondary users. Fig. 3(b) shows the rate evolution for fiveusers out of one hundred users, demonstrating that Algorithm1 converges very fast even when the number of users is large.

Fig. 4 plots the evolution of the lower bound and upperbound to the optimal value of (11) for two users that run Al-gorithm 3. The parameter settings in Figs. 4(a) and (b) are thesame as those in Figs. 2(a) and (b) respectively. The numericalexample in Fig. 4(a) that runs Algorithm 3 converges at the76th iteration with ε = 10−3, and the numerical example inFig. 4(b) that runs Algorithm 3 converges at the 130th iterationwith ε = 10−3 (verifying Theorem 2). The optimal r� inFig. 4(a) is (0, 1.68)� nats/symbol, and the optimal r� inFig. 4(b) is (2.23, 0.04)� nats/symbol.

Next, we compare Algorithm 3 with other previously pro-posed global optimization algorithms for solving (11), i.e., thebranch-and-bound method in [16] and the outer approximationmethod in [15]. Notice that both [16] and [15] solve (11) usinga reformulation in the γ = log SINR(p) domain (which is anunbounded constraint set). The unboundedness feature intro-duces numerical artifacts in the global optimization algorithmsin [16] and [15].


TABLE ITYPICAL NUMERICAL EXAMPLE IN A 10-USER NETWORK WITH DIFFERENT PARAMETER SETTINGS. ALL THE SETTINGS USE ak ∈ [0,1], bm ∈ [0,1]

AND n = 1. THE STATISTICAL PERCENTAGES OF INSTANCES WHERE ˜Bk EXISTS AND ˜Dm EXISTS ARE BOTH RECORDED.

L G K p �{˜Bk ≥ 0} M q �{˜Dm ≥ 0}5

Glj ∈ [0.01, 0.04], l �= jGlj ∈ [1.50, 2.00], l = j

5 p ∈ [1.5, 2.0] 44.14% 5 q ∈ [2.5, 3.0] 99.19%

8Glj ∈ [0.005, 0.01], l �= jGlj ∈ [2.50, 3.00], l = j

4 p ∈ [1.0, 1.2] 72.68% 6 q ∈ [2.0, 2.2] 93.00%

10Glj ∈ [0.005, 0.01], l �= jGlj ∈ [2.50, 3.00], l = j

5 p ∈ [0.4, 0.6] 41.56% 5 q ∈ [2.0, 2.5] 37.68%

10Glj ∈ [0.005, 0.01], l �= jGlj ∈ [2.50, 3.00], l = j

5 p ∈ [1.5, 2.0] 52.63% 5 q ∈ [2.5, 3.0] 98.63%

10Glj ∈ [0.01, 0.04], l �= jGlj ∈ [1.50, 2.00], l = j

5 p ∈ [1.5, 2.0] 53.18% 5 q ∈ [2.5, 3.0] 60.76%

10Glj ∈ [0.005, 0.01], l �= jGlj ∈ [2.50, 3.00], l = j

7 p ∈ [1.5, 2.0] 40.29% 10 q ∈ [2.5, 3.0] 97.14%

p� =

(I+ F− diag

(elimt→∞ rJ (t))

F

)−1(diag

(elimt→∞ rJ (t) − 1

))v, (30)

q� =

(I+ F− F diag

(elimt→∞ rJ (t)))−1

v (31)

Let us consider σ = 0.3 in (32) and the other parametersare the same as that used in the example of Fig. 3 (under thissetting Assumption 1 does not hold). This 3-user numericalexample in Fig. 5(a) runs Algorithm 3 in this paper and theinitial phase of Algorithm 2 in [16] respectively; the gapbetween the upper and lower bounds of our Algorithm 3 is0.016 after 160 seconds, while Algorithm 2 in [16] still has agap of 0.339. We next illustrate a 100-user numerical examplewith all parameters randomly generated in Fig. 5 that (b) runsAlgorithm 3 in this paper and the initial phase of Algorithm 2in [16] respectively. As shown, the gap between the lower andupper bounds is 1.274 for Algorithm 3, while the gap is still2.214 for Algorithm 2 in [16]. We can observe from Fig. 5that the relaxation gap for Algorithm 3 outperforms that in[16].

Next, we compare the outer approximation method in[15] that generates a nested sequence of polyhedrons andrequires searching the vertices of these polyhedrons. Usinga point γ(t) at time t of running Algorithm 1 in [15],we can obtain an upper bound to the optimal value by∑L

l=1 wl log(1 + eγl(t)), and a lower bound to the optimalvalue by mapping p(γ(t)) =

(I−diag(eγ(t)))−1

diag(eγ(t))vto the boundary of the constraint set, i.e., p′(γ(t)) =

p(γ(t))

max{

maxk=1,...,K{ a�k

p(γ(t))

pk},maxm=1,...,M{bmF�p(γ(t))

(qm−b�mv)

}} , and

then computing∑L

l=1 wl log(1+p′l(γ(t))/(Fp

′(γ(t))+v)l).

In addition, we consider an equivalent formulation of (26)with the constraints replaced by Bk diag(e

r)q ≤ q, k =1, . . . ,K and Dm diag(er)q ≤ q, m = 1, . . . ,M . Using thelogarithmic transformation, i.e., q = eq and then taking thelogarithm on both sides of the inequality constraints, (26) canbe written as a geometric program (GP) in terms of r and q.This relaxation for (11) is tighter than (27), but requires a GPsolver that is more complex than a LP solver. We also evaluatethe case when (29) in Algorithm 3 is replaced by the GP thatis an equivalent formulation of (26) in the following.

We compare the running time of Algorithm 1 in [15],Algorithm 3 with the LP, and Algorithm 3 with (29) replacedby a GP on an Intel (R) Core(TM) i7-2620M CPU 2.70GHzprocessor with a memory RAM of 4.00 GB (3.89 GB usable)and a 64-bit Unix operating system. We use σ = 0.3 in(32) and set the other parameters to be the same as thatused in the examples of Fig. 3 and Fig. 5(a). This 3-usernumerical example in Fig. 6(a) runs Algorithm 1 in [15] andAlgorithm 3 in this paper with the LP and the GP respectively;Algorithm 1 in [15] converges much faster than Algorithm3 (with ε = 0.02), i.e., 2.115 seconds for Algorithm 1 in[15], 114.983 and 358.802 seconds respectively for Algorithm3 with LP and GP. When the number of users is increased,e.g., the 8-user numerical example in Fig. 6(b), Algorithm 3with the LP has a faster running time, i.e., 2387.828 secondscompared with 3949.179 seconds for Algorithm 1 in [15] (withε = 0.2). From Fig. 6, we observe that the running time ofAlgorithm 3 with the LP can be much faster than the runningtime of Algorithm 3 with the GP.

VII. CONCLUSION

We studied the weighted sum rate maximization problemwith transmit power and interference temperature constraintsin a cognitive radio network. To tackle the nonconvexity, weproposed a reformulation-relaxation technique by leveragingthe nonnegative matrix theory. The reformulation transformedthe original problem in the power and interference temper-ature domain to one with the achievable rates as explicitoptimization variables. A convex relaxation to the sum rateswas obtained by solving a convex optimization problemover a closed bounded convex set in the rate domain. Theglobal optimality can even be characterized by the spectraof specially-crafted nonnegative matrices. Polynomial-timeverifiable sufficient conditions were established to identifyproblem instances that can be solved optimally in polynomialtime. We also provided an optimality equivalence between the


0 2 4 6 8 10

0.6

0.8

1

1.2

1.4

1.6

iteration

rate

(na

ts/s

ymbo

l)

Evolution of rate

User 1User 2User 3

(a)

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

iteration

rate

(na

ts/s

ymbo

l)

Evolution of rate

User 1User 2User 3User 4User 5

(b)

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

iteration

rate

(na

ts/s

ymbo

l)

Evolution of rate


(c)

0 1000 2000 3000 4000 50000

0.2

0.4

0.6

0.8

1

iterationra

te (

nats

/sym

bol)

Evolution of rate


(d)

Fig. 3. Illustration of the convergence of Algorithm 1 for (a) 3 users, (b) 50 users, (c) 100 users , and (d) 200 users. For the numerical examples with largeproblem sizes, the problem parameters are generated randomly in a given range, and Assumption 1 holds for all of them. In (b), (c) and (d) of this figure,we show the rate evolution for only five users. We can observe from the figure that the convergence of Algorithm 1 is fast.

0 20 40 600.5

0.55

0.6

0.65

0.7

iteration

Sum

rat

e (n

ats/

sym

bol)

Lower boundUpper bound

(a)

0 20 40 60 80 100 1201.48

1.5

1.52

1.54

1.56

1.58

1.6

1.62

iteration

Sum

rat

e (n

ats/

sym

bol)

Lower boundUpper bound

(b)

Fig. 4. Illustration of the convergence of Algorithm 3. The problem parameters are set the same as those in Fig. 2, and the weights for both of the numericalexamples are set as w = (0.67, 0.33)� .

nonconvex weighted sum rate maximization and the convexweighted max-min rate problem, and exploited it for algorithmdesign. In the general case, we proposed a global optimizationalgorithm to solve the sum rate maximization using the convexrelaxation and branch-and-bound method. Numerical evalua-tions demonstrated that our proposed algorithms exhibited fastconvergence behavior even for large-scale problems.

APPENDIX

A. Proof of Theorem 1

For each lth user, by combining (3) and (5), its rate can bewritten as erl = pl/(Fp+ v)l + 1, l = 1, . . . , L, which, in

matrix form, can be given by(diag(er) − I

)(Fp + v) = p.

Substituting (4) into the above equation, it can be rewritten asdiag(er)q = p + q, whose righthand-side and lefthand-sideare then both multiplied by F+ (1/pk)va

�k to obtain:(

F+ (1/pk)va�k

)diag(er)q

=(F+ (1/pk)va

�k

)p+

(F+ (1/pk)va

�k

)q

(a)

≤ (Fp+ v) +(F+ (1/pk)va

�k

)q

(b)=

(I+ F+ (1/pk)va

�k

)q,

(33)

where (a) is due to the power constraint (1/pk)a�k p ≤ 1, and(b) is due to (4).

We first establish the following two matrix equations that


0 50 100 150

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

time (seconds)

Sum

rat

e (n

ats/

sym

bol)

Lower bound (Algo. 3 with LP)Upper bound (Algo. 3 with LP)Lower bound (Algo. in [16])Upper bound (Algo. in [16])

(a)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

time (hours)

Sum

rat

e (n

ats/

sym

bol)


(b)

Fig. 5. Comparison of the upper and lower bounds between Algorithm 3 in this paper and the global optimization algorithm proposed in [16]. (a) 3-usernumerical example. We use σ = 0.3 in (32) and set the other parameters the same as that are used in Fig. 3; (b) 100-user numerical example. The channel gainis generated randomly in Glj ∈ [0.1, 0.2], l �= j and Glj ∈ [0.6, 0.9], l = j. We assume that the number of power and interference temperature constraintsaltogether are the same as the number of user with ak ∈ [0, 1] and bm ∈ [0, 1] randomly generated. We also set p ∈ [10, 20] and q ∈ [10, 20].

0 20 40 60 80 1000.6

0.7

0.8

0.9

1

1.1

time (seconds)

Sum

rat

e (n

ats/

sym

bol)

Lower bound (Algo. 3 with LP)Upper bound (Algo. 3 with LP)Lower bound (Algo. 3 with GP)Upper bound (Algo. 3 with GP)Lower bound (Algo. in [15])Upper bound (Algo. in [15])

(a)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

time (seconds)

Sum

rat

e (n

ats/

sym

bol)


(b)

Fig. 6. Comparison of the upper and lower bounds for the global optimization algorithm proposed in [15] and Algorithm 3 with LP and GP respectively.(a) 3-user numerical example. We use σ = 0.3 in (32) and set the other parameters the same as that are used in Fig. 3; (b) 8-user numerical example.The channel gain is generated randomly in Glj ∈ [0.1, 0.2], l �= j and Glj ∈ [0.6, 0.9], l = j. We assume that the number of power and interferencetemperature constraints altogether are the same as the number of user with ak ∈ [0, 1] and bm ∈ [0, 1] randomly generated. We also set p ∈ [10, 20]and q ∈ [10, 20].

are useful for a reformulation of the interference temperatureconstraints. The first one is:

(I− 1

qmvb�

m

)−1 (c)= I+ lim

N→∞

N∑n=1

( 1

qmvb�

m

)n

= I+ limN→∞

N∑n=1

1

qm

(b�mv

qm

)n−1

vb�m

(d)= I+ lim

N→∞1

qm

(1− (b�mv/qm)N−1

1− b�mv/qm

)vb�

m

(e)= I+

(1/(qm − b�

mv))vb�

m,(34)

where (c) is due to the von Neumann’s series expansion,(d) is due to the geometric series formula, and (e) is dueto qm ≥ b�

mv for a nonempty constraint set so thatlim

N→∞(b�

mv/qm)N−1 = 0. Using (34), the second matrixequation is given as:

I+ F− 1qm

vb�m

=(I− 1

qmvb�

m

)+(I− 1

qmvb�

m

)(I+ 1

qm−b�mv

vb�m

)F

=(I− 1

qmvb�

m

)(I+ F+

(1/(qm − b�

mv))vb�

mF).

(35)

Next, substituting p = diag(er−1)q and (1/qm)b�mq ≤ 1

into (4), we have:

F(diag(er)q− q

)+ (1/qm)vb�

mq ≤ q

⇒ F diag(er)q(f)

≤(I− 1

qmvb�

m

)(I+ F+ 1

qm−b�mv

vb�mF

)q

⇒(I+ 1

qm−b�mvvb

�m

)F diag(er)q

(g)

≤(I+ F+ 1

qm−b�mv

vb�mF

)q,

(36)where (f) is due to (35) and (g) is due to (34). Thus, both (33)and (36) form the constraint set in (14). By substituting (5),we rewrite the objective function in (11) as

∑Ll=1 wlrl.

B. Proof of Corollary 1

Since the nonnegative matrix Bk exists (cf. Assumption 1),the first constraint in (14) can be rewritten as:

Bk diag(er)q ≤ (I+Bk)q

⇒ (I+Bk)−1Bk diag(e

r)q ≤ q,

⇒ Bk diag(er)q ≤ q,

(37)


where Bk and Bk are given in (12) and (15) respectively.Likewise, since the nonnegative matrix Dm exists accordingto Assumption 1, the second constraint in (14) can be rewrittenas Dm diag(er)q ≤ q, m = 1, . . . ,M , where Dm and Dm

are given in (13) and (16) respectively.We now state the following lemma:Lemma 5 (The Subinvariance Theorem [30]): Let A be a

nonnegative irreducible matrix, Λ be a positive number, and υbe a nonnegative vector satisfying Aυ ≤ Λυ. Then υ > 0 andΛ > ρ(A). Moreover, Λ = ρ(A) if and only if Aυ = Λυ.

Letting Λ = 1, A = Bk diag(er) and Λ = 1, A =

Dm diag(er) respectively in Lemma 5, the inequalities in(37) can be further written as ρ

(Bk diag(e

r)) ≤ 1 and

ρ(Dm diag(er)

) ≤ 1 respectively. This algebraic manipula-tion transforms the constraint set in (14) to one that is sorelya spectral radius constraint set, which is convex (due to thelog-convex property of the spectral radius function [25]).

C. Proof of Lemma 1

The first order derivative of log ρ(Bk diag(e

r))

and log ρ(Dm diag(er)

)at r are given by

∂ log ρ(Bk diag(e

r))/∂r = x

(Bk diag(e

r))◦y(Bk diag(e

r))

and ∂ log ρ(Dm diag(er)

)/∂r = x

(Dm diag(er)

) ◦y(Dm diag(er)

)respectively [16], [26]. Due to that

Bk diag(er) and Dm diag(er) are nonnegative matrices, their

Perron right eigenvector x(Bk diag(e

r)), x

(Dm diag(er)

)and left eigenvector y

(Bk diag(e

r)), y

(Dm diag(er)

)are then all nonnegative [30], which implies that thespectral radius functions in the constraint set of (17) aremonotonically increasing. Since the objective function in(17) is linear with positive weight and thus increasing in r,maximizing the objective function in (17) leads to at least oneof the constraints becoming tight. This also means that thecorresponding constraint in (14) becomes tight. If i indexesthe ith tight power constraint, we have ρ

(Bi diag(e

r�))= 1

in (17) and Bi diag(er�)q� = (I+Bi)q

� in (14) respectively,which implies that q� = x

(Bi diag(e

r�)). By combining

q� = x(Bi diag(e

r�))

and p = diag(er� − 1)q, we can

obtain p� = diag(er� − 1)x

(Bi diag(e

r�)).

D. Proof of Lemma 2

We first write (20) in epigraph form as:

maximize δsubject to δ ≤ rl/βl, l = 1, . . . , L,

log ρ(Bk diag(e

r)) ≤ 0, k = 1, . . . ,K,

log ρ(Dm diag(er)

) ≤ 0, m = 1, . . . ,M,variables: r, δ.

(38)Next, we form the full Lagrangian for (38) by introducing

the dual variable λ ∈ RL+ for the L inequality constraints δ ≤

rl/βl l = 1, . . . , L, μ ∈ RK+ for the K inequality constraints

log ρ(Bk diag(e

r)) ≤ 0 k = 1, . . . ,K , and ν ∈ R

M+ for

the M inequality constraints log ρ(Dm diag(er)

) ≤ 0 m =1, . . . ,M to obtain:

L(δ, r,λ,μ,ν) = δ − λ�(δ1− diag(β)−1r)

−K∑

k=1

μk log ρ(Bk diag(e

r))− M∑

m=1

νm log ρ(Dm diag(er)

).

(39)Taking the partial derivative of (39) with respect to r, wehave ∂L/∂r = diag(β)−1λ − ∑K

k=1 μk

(x(Bk diag(e

r)) ◦

y(Bk diag(e

r))) − ∑M

m=1 νm

(x(Dm diag(er)

) ◦y(Dm diag(er)

)), and, setting it to zero, we have at

optimality:

λ�=diag(β)

( K∑k=1

μ�k

(x(Bk diag(e

r�)) ◦ y(Bk diag(e

r�)))

+

M∑m=1

ν�m

(x(Dm diag(er

�

)) ◦ y(Dm diag(er

�

))))

(40)According to the Perron-Frobenius theorem, the Perron right

and left eigenvectors of an irreducible nonnegative matrix are(entry-wise) nonnegative [30]. Since the dual variables are alsononnegative, we thus deduce from (40) that λ� is a positivevector. Combining the fact that λ� > 0 with complementaryslackness, we have δ� = r�l /βl for all l.

E. Proof of Lemma 3

We first form the Lagrangian for (17) by introducingthe dual variable μ ∈ R

K+ for the K inequality constraints

log ρ(diag(er)Bk

) ≤ 0, k = 1, . . . ,K and the dualvariable ν ∈ R

M+ for the M inequality constraints

log ρ(diag(er)Dm

) ≤ 0, m = 1, . . . ,M to obtainL(r,μ,ν) =

∑Ll=1 wlrl −

∑Kk=1 μk log ρ

(Bk diag(e

r)) −∑M

m=1 νm log ρ(Dm diag(er)

). Taking the first order

derivative of the above Lagrangian with respect to r, wehave ∂L(r,μ,ν)/∂r = w − ∑K

k=1 μkx(Bk diag(e

r)) ◦

y(Dm diag(er)

) − ∑Mm=1 νmx

(Bm diag(er)

) ◦y(Dm diag(er)

), and, setting it to zero, we have at

optimality:

w =

K∑k=1

μ�kx

(Bk diag(e

r�)) ◦ y(Bk diag(e

r�))

+

M∑m=1

ν�mx(Dm diag(er

�

)) ◦ y(Bm diag(er

�

)).

(41)Combining (41) with (40), the equation λ = diag(β)w/1�wmust hold if (17) and (20) have the same optimal solution.Moreover, by taking the partial derivative of (39) with respectto δ and then setting it to zero, we have 1�λ = 1. Since w isa probability vector, we have

∑Ll=1 ξlwl ≥ 1 in the constraint

set of (21). We then write (21) in epigraph form as

minimize �


ξ)) ≤ log �, k = 1, . . . ,K,

log ρ(Dm diag(eξ)

) ≤ log �, m = 1, . . . ,M,∑Ll=1 wlξl ≥ 1,

variables: �, ξ.(42)


Next, we form the Lagrangian for (42) by introducingthe dual variable μ ∈ R

K+ for the K inequality

constraints log ρ(Bk diag(e

ξ)) ≤ log �, k = 1, . . . ,K ,

the dual variable ν ∈ RM+ for the M inequality

constraints log ρ(Dm diag(eξ)

) ≤ log �, m = 1, . . . ,M ,and the dual variable λ for the single inequalityconstraint

∑Ll=1 wlξl ≥ 1 to obtain L(�, ξ, μ, ν, λ) =

� +∑K

k=1 μk

(log ρ

(Bk diag(e

ξ)) − log �

)+∑M

m=1 νm

(log ρ

(Dm diag(eξ)

)− log �)−λ(

∑Ll=1 wlξl−1).

Taking the first order derivative of the above Lagrangianwith respective to � and ξ respectively, we have∂L(�, ξ, μ, ν, λ)/∂� = 1 − ∑K

k=1 μk/� −∑M

m=1 νm/�

and ∂L(�, ξ, μ, ν, λ)/∂ξ =∑K

k=1 μkx(Bk diag(e

ξ)) ◦

y(Dm diag(eξ)

)+

∑Mm=1 νmx

(Bm diag(eξ)

) ◦y(Dm diag(eξ)

) − λw, and, setting them to zero, wehave at optimality:

K∑k=1

μ�k +

M∑m=1

ν�m = ��, (43)

and

K∑k=1

μ�kx

(Bk diag(e

ξ�

)) ◦ y(Dm diag(eξ

�

))

+

M∑m=1

ν�mx(Bm diag(eξ

�


�

))= λ�w,

(44)which imply that λ� = ��. Combining the factthat ρ

(Bk diag(

1�e

ξ�

))

= 1�ρ

(Bk diag(e

ξ�

)),

x(Bk diag(

1�e

ξ�

))

= x(Bk diag(e

ξ�

))

andy(Bk diag(

1�e

ξ�

))

= y(Bk diag(e

ξ�

))

for any positive� with (41), (43) and (44), we can deduce that the optimalsolution r� of (17) and the optimal solution ξ� of (21) satisfyer

�

= 1�� e

ξ�

. Thus, if (17) and (20) have the same optimalsolution, β is proportional to log( 1

�� eξ�

).Next, we discuss a special case and state the following

inequality to connect (17) and (20) when β = 1. SupposeΩ ∈ {B1, . . . , BK , D1, . . . , DM}. Then, taking the logarithmon the both sides of the Friedland-Karlin inequality (cf.Theorem 3.1 in [26]), we have log ρ(Ω) +

∑Ll=1

(x(Ω) ◦

y(Ω))lrl ≤ log ρ

(Ω diag(er)

). Since log ρ

(Ω diag(er)

) ≤ 0

in the constraint set in (17), we deduce that∑L

l=1

(x(Ω) ◦

y(Ω))lrl ≤ − log ρ(Ω) ⇒ (

minl=1,...,L(x(Ω) ◦y(Ω))l/wl

)∑Ll=1 wlrl ≤ log

(1/ρ(Ω)

) ⇒ ∑Ll=1 wlrl

(h)

≤(maxl=1,...,Lwl/(x(Ω) ◦ y(Ω))l

)log

(1/ρ(Ω)

). Equality is

achieved in (h) if and only if w = x(Ω) ◦ y(Ω) andρ(Ω) = max{ρ(B1), . . . , ρ(BK), ρ(D1), . . . , ρ(DM )}.

F. Proof of Lemma 4

We first prove the convergence of the outer loop by using theFriedland-Karlin inequality (cf. Theorem 3.1 in [26]). SupposeΩ ∈ {B1, . . . , BK , D1, . . . , DM}. Taking the logarithm of

the both sides of the Friedland-Karlin inequality, we have

log ρ(Ωdiag(eξ)

)≤

L∑l=1

(x(Ω diag(eξ)

) ◦ y(Ω diag(eξ)))

lξl + log ρ

(Ω).

(45)Combining the complementary slackness with (44),

we can deduce from λ� > 0 that∑L

l=1 wlξ�l = 1

at optimality of (21). By subtracting the value 1and

∑Ll=1 wlξ

�l respectively from left and right side

of (45), we have, at optimality, the inequalitieslog ρ

(Ω diag(eξ

�

)) − 1 ≤ ∑L

l=1

(x(Ω diag(eξ

�

)) ◦

y(Ω diag(eξ

�

)))

lξ�l + log ρ

(Ω) − ∑L

l=1 wlξ�l ⇒

log ρ(Ω diag(eξ

�

)) ≤ minξ

{∑Ll=1

(x(Ω diag(eξ

�

)) ◦

y(Ω diag(eξ

�

)) − w

)lξl

}+ log ρ

(Ω)

+ 1. Since

log ρ(Bk

)+ 1 is a constant, ξ� is solved by computing

minξ

{ L∑l=1

(x(Bk diag(e

ξ�

))◦y(Bk diag(e

ξ�

))−w

)lξl

}or

minξ

{ L∑l=1

(x(Dm diag(eξ

�


�

))−w

)lξl

}.

Next, we prove the convergence of the inner loop at Step4. From Lemma 2, we have δ� = r�l /βl l = 1, . . . , L, i.e.,

1

δ�p� = f(p�), (46)

where f : RL → RL is given by:

fl(p�) =

βl

log(1 + SINRl(p�))p�l . (47)

Now, we show that p� in (46) can be constrained by amonotone norm. Let us rewrite the interference temperatureconstraints in (11) as follows. By substituting q = Fp + vinto b�

mq ≤ qm, we have b�mFp ≤ qm − b�

mv. Combiningthis with a�k p ≤ pk, we see that p� must satisfy

max{

maxk=1,...,K

{a�k p�

pk

}, maxm=1,...,M

{ b�mFp�

qm − b�mv

}}= 1,

(48)which is a monotone norm in p�. We now state the nonlinearPerron-Frobenius theory in [24].

Theorem 3 (Krause’s theorem [24]): Let ‖·‖ be a monotonenorm on R

L. For a concave mapping f : RL+ → R

L+ with

f(z) > 0 for z ≥ 0, the following statements hold. Theconditional eigenvalue problem f(z) = λz, λ ∈ R, z ≥ 0,‖z‖ = 1 has a unique solution (λ∗, z∗), where λ∗ > 0, z∗ > 0.Furthermore, limk→∞ f(z(k)) converges geometrically fast toz∗, where f(z) = f(z)/‖f(z)‖.

Let f(p) be given in (47), λ = 1/δ, and we constrain p bythe monotone norm ‖·‖ given in (48). The proof of f(p) beingconcave can be found in [21]. By Theorem 3, p� convergesto the fixed point p� = f(p�)/‖f(p�)‖.G. Proof of Theorem 2

Theorem 2 can be proved by combining the previous proofsin Section IV. The proof of convergence of rJ (t) to theoptimal solution of (14) by the branch-and-bound method canbe found in [28]. Since the maps in (6) and (7) are bijective,the limit point in (30) and (31) solves (11).


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Liang Zheng received the bachelor’s degree insoftware engineering from Sichuan University,Chengdu, China, in 2011. She is pursuing her Ph.D.degree at the City University of Hong Kong. Herresearch interests are in wireless networks, mobilecomputing, and nonlinear optimization and its appli-cations. She was a Finalist of the Microsoft ResearchAsia Fellowship in 2013.

Chee Wei Tan (M’08-SM’12) received the M.A.and Ph.D. degrees in electrical engineering fromPrinceton University, Princeton, NJ, in 2006 and2008, respectively. He is an Assistant Professor atthe City University of Hong Kong. Previously, hewas a Postdoctoral Scholar at the California Instituteof Technology (Caltech), Pasadena, CA. He was aVisiting Faculty member at Qualcomm R&D, SanDiego, CA, in 2011. His research interests are innetworks, inference in online large data analytics,and optimization theory and its applications. Dr. Tan

currently serves as an Editor for the IEEE TRANSACTIONS ON COMMUNI-CATIONS. He was the recipient of the 2008 Princeton University Wu Prizefor Excellence and was awarded the 2011 IEEE Communications SocietyAP Outstanding Young Researcher Award. He was a selected participantat the U.S. National Academy of Engineering China-America Frontiers ofEngineering Symposium in 2013.

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Maximizing Sum Rates in Cognitive Radio Networks: Convex Relaxation and Global Optimization Algorithms