Joint Load Balancing and Interference Management for Small-CellHeterogeneous Networks with Limited Backhaul Capacity

Tam, H. H. M., Tuan, H. D., Ngo, D. T., Duong, T. Q., & Poor, H. V. (2016). Joint Load Balancing andInterference Management for Small-Cell Heterogeneous Networks with Limited Backhaul Capacity. IEEETransactions on Wireless Communications. https://doi.org/10.1109/TWC.2016.2633262

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Joint Load Balancing and Interference Managementfor Small-Cell Heterogeneous Networks with

Limited Backhaul CapacityHo H. M. Tam, Hoang D. Tuan, Duy T. Ngo, Trung Q. Duong and H. Vincent Poor

Abstract—In this paper, new strategies are devised for jointload balancing and interference management in the downlink ofa heterogeneous network, where small cells are densely deployedwithin the coverage area of a traditional macrocell. Unlikeexisting work, the limited backhaul capacity at each base station(BS) is taken into account. Here users (UEs) cannot be offloadedto any arbitrary BS, but only to ones with sufficient backhaulcapacity remaining. Jointly designed with traffic offload, transmitpower allocation mitigates the intercell interference to furthersupport the quality of service of each UE. The objective here iseither (i) to maximize the network sum rate subject to minimumthroughput requirements at individual UEs, or (ii) to maximizethe minimum UE throughput. Both formulated problems belongto the difficult class of mixed-integer nonconvex optimization.The inherently binary BS-UE association variables are stronglycoupled with the transmit power variables, making the problemseven more challenging to solve. New iterative algorithms aredeveloped based on an exact penalty method combined withsuccessive convex programming, where we deal with the binaryBS-UE association problem and the nonconvex power allocationproblem one at a time. At each iteration of the proposedalgorithms, only two simple convex problems need to be solvedin the same time scale. It is proven that the algorithms improvethe objective functions in each iteration and converge eventually.Numerical results demonstrate the efficiency of the proposedalgorithms in both traffic offloading and interference mitigation.

Index Terms—Combinatorial optimization, heterogeneous net-work, limited backhaul, load balancing, power allocation, suc-cessive convex programming

Manuscript received May 21, 2016; revised August 22, 2016; acceptedNovember 18, 2016. The associate editor coordinating the review of this paperand approving it for publication was S. Mukherjee.

H. H. M. Tam and H. D. Tuan are with the Faculty of Engineer-ing and Information Technology, University of Technology Sydney, Broad-way, NSW 2007, Australia (email: HuuMinhTam.Ho@student.uts.edu.au,Tuan.Hoang@uts.edu.au).

D. T. Ngo is with the School of Electrical Engineering and Comput-ing, The University of Newcastle, Callaghan, NSW 2308, Australia (email:duy.ngo@newcastle.edu.au).

T. Q. Duong is with the School of Electronics, Electrical Engineering andComputer Science, Queen’s University Belfast, Belfast BT7 1NN, UnitedKingdom (e-mail: trung.q.duong@qub.ac.uk).

H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu).

The work of H. H. M. Tam and H. D. Tuan was supported in partby the Australian Research Council’s Discovery Projects under ProjectDP130104617. The work of T. Q. Duong was supported in part by theU.K. Royal Academy of Engineering Research Fellowship under GrantRF1415\14\22 and by the Newton Institutional Link under Grant ID172719890.

I. INTRODUCTION

Cell densification is currently the best hope to meet theunprecedented data increase (the 1000× data challenge) inthe fifth-generation (5G) wireless networks [1]–[3]. By denselydeploying cells of different types and sizes (e.g., macro, micro,pico, femto), the resulting heterogeneous networks (HetNets)can offer a substantial growth in area spectral efficiency andfull network coverage in regions traditionally difficult to pene-trate. Another key benefit of HetNet is data offloading, wheretraffic otherwise transported via the traditional macrocell isdirected to the newly deployed small cells.

Traditionally, a user (UE) is associated with the base station(BS) that offers the maximum signal-to-interference-plus noiseratio (SINR), i.e., the max-SINR rule (see, e.g., [4]). As aresult, a ‘hotspot’ BS with advantageous link conditions and/orhigh transmit power would potentially be inundated with toomany UEs while other BSs only serve a few UEs. Range ex-pansion is a heuristic method that may help balance the trafficload among different BSs, in which the SINR is regulatedthrough a positive bias level [5], [6]. Still, it is challenging todetermine optimal bias levels for multiple cells. Other relatedBS-UE association rules proposed in the literature are based onmaximizing the estimated throughput [7] and sum logarithmicthroughput [8]–[10]. Using Lagrangian duality decomposition[11], the association rule in [12] aims at maximizing the net-work sum-rate while satisfying the Quality-of-Service (QoS)constraints. A heuristic adjustment is then proposed to keepthe total number of time slots demanded by the UEs below thatavailable at the BSs. In [13], [14], a binary relaxation methodis proposed to find the optimal association rule for the sum-rate and minimum-rate maximization objectives. However, theproposed method is limited to a two-cell network. An extensiveoverview of the state-of-the-art in user association for 5Gnetworks can be found in [15].

A common assumption in the above existing work is theavailability of ideal backhaul links with unlimited capacity.This assumption is not true for HetNets. Here, the low-powerBSs of small cells (e.g., pico and femto) connect to thecore network via low capacity backhaul (e.g., DSL links) foreconomic benefits [16]. If these small-cell BSs must serve toomany UEs, their non-ideal backhaul becomes the bottleneckin transporting the required amount of data traffic to the UEs,resulting in a potentially unacceptable level of delay/jitterat the UEs. The study of [17] proposes an optimal BS-UEassociation rule that maximizes the logarithmic utility function

2

while guaranteeing a target delay. Note that one can guaranteecertain levels of delay if the demanded throughput at the BSsis kept below their respective backhaul capacity [18]. Insteadof maximizing the network sum-rate or the minimum UE’sthroughput, [19] devises a backhaul-aware BS-UE associationrule that is based on biasing, cell size and user distribution.

Interference is a major issue in dense small-cell HetNets,wherein numerous cell boundaries with poorly defined patternsare created. Compared with traditional cellular networks, theeffects of the intercell interference are much more acute andunpredictable, especially at the cell edges [20]. Power controlis an effective way to manage the interference, assuming thatthe BS-UE links have already been established. For a givenBS-UE association, [21] devises an optimal BS transmit powerpolicy for sum-rate maximization with backhaul capacity con-straints by solving the Karush-Kuhn-Tucker (KKT) conditions.However, due to the nonconvexity of the considered problem,a KKT point may not even be locally optimal or feasible.In addition, the once presumably optimal BS-UE associationswill no longer be optimal when the new transmit power valuesare used as a result of power control. It is therefore essentialto design jointly optimal strategies for both traffic offload andinterference management.

For CDMA-based networks, joint optimization of BS-UEassociation and interference management is considered in [22]for network sum-rate maximization and in [23] for minimumUE’s SINR maximization. It is not straightforward to applythe results of [22] and [23] to networks in which a BS usesorthogonal channels to serve its UEs to eliminate intracellinterference. Different from CDMA, each UE here is onlyassigned with a fraction of the time/frequency slots dependingon the current load at its serving BS. Assuming zero intracellinterference, [24] proposes an iterative procedure for joint BS-UE association and interference management that guarantees amaximum delay not be exceeded. Yet, the convergence of theproposed heuristic method is not proven. Using game theory,[25] finds the Nash equilibrium for such joint optimizationproblem, albeit without considering QoS constraints. It iscommonly known that a Nash equilibrium may not be efficientas it could be far away from the actual optimal solution.Notably, the practical issue of imperfect backhaul links isnot considered in [24] and [25], presumably due to thenonconvexity of the backhaul capacity constraints even whenthe BS-UE association is fixed.

In this paper, we formulate new problems for joint trafficoffload and interference management in the downlink of aHetNet. Aiming to maximize the network throughput andthe minimum UE rate, our formulations accommodate bothbackhaul capacity constraints and UE QoS requirements. Theconsidered problems belong to the difficult class of mixedinteger nonconvex optimization. The binary BS-UE associa-tion variables are strongly coupled with the transmit powervariables, making the problems even more challenging tosolve.

We then develop new iterative algorithms based on alter-nating descent [26] and successive convex programming forthe formulated problems. Alternating descent allows us todecouple the original problem into two subproblems and deal

MBS 1

PBS 2 PBS 3

PBS 4

Core network

UE k

BS-UE association

Backhaul link

C2

BHC1

BHC3

BHC4

BH

Fig. 1. A small-cell HetNet with limited-capacity backhaul links. ‘MBS’,‘PBS’ and ‘PoP’ refer to macro BS, pico BS and Point of Presence,respectively.

with them one at a time. Even so, each resulting subproblemis still challenging. For a fixed power allocation, the BS-UEassociation subproblem is combinatorial. And for a fixed BS-UE association, the power allocation subproblem is highlynonconvex. We propose to deal with the binary nature ofBS-UE association by relaxation combined with a penaltymethod. We then employ successive convex programming tosolve two subproblems in the same time scale. We prove thatour proposed alternating descent algorithms converge, whereonly two simple convex problems are to be solved in eachiteration. Simulation results show that the proposed algorithmsenhance the network throughput through better load balancingand interference management.

The rest of this paper is organized as follows: Sec. IIformulates the problems of joint load balancing and inter-ference management. Sec. III proposes an alternating descentalgorithm to solve the sum-rate maximization problem. Sec. IVextends the devised solution to the case of minimum UE ratemaximization. Sec. V presents numerical results to demon-strate the performance of our proposed algorithms. Finally,Sec. VI concludes the paper.

Notation. In this paper, boldfaced symbols are used foroptimization variables whereas non-boldfaced symbols are fordeterministic terms, regardless of whether they are matrix, vec-tor or scalar. The dimensions of these symbols are interpretedfrom context, and they will be explicitly specified should therebe any potential ambiguity.

II. SYSTEM MODEL AND PROBLEM FORMULATIONS

Consider the downlink of a K-cell HetNet in which onemacrocell is overlaid with K − 1 small cells, as depicted inFig. 1. To best exploit the limited radio spectrum, universalfrequency reuse is adopted [8], [10]. Without loss of generality,we assume that macro base station (MBS) serving the macro-cell is indexed as BS 1, and the BS serving small cell k (e.g.,a micro/pico/femto BS) is indexed as BS k ∈ {2, 3, . . . ,K}.Transmitting at a much lower power than an MBS, the small-cell BSs are deployed densely in order to extend network

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coverage, increase throughput and offload data traffic from theMBS.

We assume that each BS k ∈ {1, 2, . . . ,K} connects to abackhaul link with limited capacity. The backhaul link of eachBS carries the downlink traffic for its serviced UEs from thecore network to that BS via a central access point called ‘Pointof Presence’ (PoP) [17], [27], [28]. The PoP is connected to thecore network via an optical fiber link whose capacity is muchhigher than the total capacity of all links from the PoP to allBSs. Therefore, the effect of core network-PoP link capacityis neglected in our model [28]. For simplicity, we also neglectthe traffic coming from the control plane [18].

In the considered HetNet, there are N UEs looking for theserving BSs. Similar to the BSs, the UEs are each equippedwith one antenna. A snapshot model is adopted where thechannels remain unchanged during the optimization process.This channel assumption is well-justified for networks with alow degree of mobility and/or very high throughput. A centralprocessing unit is employed to collect all the channel stateinformation and perform the underlying network optimization.

In this paper, a UE is allowed to associate with at mostone BS, but a BS can serve multiple UEs. Assume that BSk ∈ {1, 2, . . . ,K} has a full buffer and transmits with powerpk. First, consider that only one UE n ∈ {1, 2, . . . , N} isconnected to BS k. The achieved data rate in nats/s/Hz of UEn is expressed as:

rnk(p) , ln

(1 +

gnkpk∑Kj 6=k,j=1 gnjpj + σ2

), (1)

where p , (p1,p2, . . . ,pK)T , gnk is the channel gain fromBS k to UE n, and σ2 is the power of background additivewhite Gaussian noise. As seen from (1), UE n is subjected tothe intercell interference from other BS j 6= k.

Next, consider the general case of multiple UEs connectingto a BS. The BS will then divide the total available time intoa number of time slots and allocates them to its serviced UEsin a round-robin fashion [8], [10]. As such, each connectedUE will receive an equal amount of transmission time whilethere is no intracell interference. Denote xnk ∈ {0, 1} asthe BS-UE association variable, i.e., xnk = 1 if UE nis associated with BS k and xnk = 0 otherwise. Definexk , [x1k, . . . ,xNk]T and x ,

[xT1 , . . . ,x

TK

]T. If BS

k serves a total of 〈xk〉 ,∑Nn=1 xnk UEs, then each of

these UEs will be allocated 1/〈xk〉 of the total availabletime. Effectively, data rate perceived by a connected UE nis rnk(p)/〈xk〉, which will be further reduced as more andmore UEs are associated to BS k. To reflect the fact that thisrate is only possible if UE n actually connects to BS k, wedefine the effective data rate given to UE n by BS k as:

reffnk(xk,p) ,

xnkrnk(p)

〈xk〉. (2)

It follows that the sum effective data rate of cell k is

reffk (x,p) ,

N∑n=1

reffnk(p) =

N∑n=1

xnkrnk(p)

〈xk〉, (3)

which is required not to exceed the limited backhaul capacityavailable to BS k [29], [30]. The total network throughputacross all K cells is then simply

K∑k=1

reffk (x,p) =

K∑k=1

N∑n=1

xnkrnk(p)

〈xk〉. (4)

This work aims to enhance the network throughput byjoint optimization of BS-UE association and transmit powerallocation. Importantly, our design takes into account both theQoS requirement of each UE and the limited backhaul capacityat each BS. If too many UEs connect a particular BS (e.g.,due to favorable channel conditions), then (i) the perceivedrate of each UE will decrease and potentially not satisfy theminimum QoS requirement, and (ii) the sum effective rate ofthe corresponding cell may increase and potentially exceedthe backhaul capacity. With a proper BS-UE association, thetraffic load will be more balanced among different BSs andthe network crowding issue can be alleviated. With adaptivepower allocation, the intercell interference can be effectivelymanaged to further improve the throughput. Here, we willconsider the following joint design problem for sum-ratemaximization.

maxp,xnk∈{0,1}

K∑k=1

N∑n=1

xnkrnk(p)

〈xk〉(5a)

s.t.K∑k=1

xnk = 1, n = 1, . . . , N (5b)

〈xk〉 ≥ 1, k = 1, . . . ,K (5c)K∑k=1

xnkrnk(p)

〈xk〉≥ RQoS

n , n = 1, . . . , N (5d)

N∑n=1

xnkrnk(p) ≤ CBHk 〈xk〉, k = 1, . . . ,K

(5e)0 ≤ pk ≤ Pmax

k , k = 1, . . . ,K. (5f)

Constraint (5b) ensures each UE be connected with one BSonly. Constraint (5c) requires each BS serves at least one UE[8]–[10], [17]. In (5d) and (5e), RQoS

n ≥ 0 and CBHk ≥ 0

specify the minimum throughput requirement for UE n andthe backhaul capacity for cell k, respectively. Finally, (5f) capsthe maximum transmit power of each BS k.

We also consider the following problem of maximizing theminimum effective rate among all UEs:

maxp,xnk∈{0,1}

minn=1,...,N

{K∑k=1

xnkrnk(p)

〈xk〉

}s.t. (5b), (5c), (5e), (5f).

(6)

In problem (6), our aim is to support the most vulnerable UEs,e.g., those at the cell edges.

Both problems (5) and (6) belong to the difficult class ofmixed-integer programming. The strong coupling between thebinary variables x and the continuous variables p make theproblems even more challenging. State-of-the-arts in existingliterature typically apply the alternating optimization frame-work [10], [22], [23], [31], [32]. In this ‘divide-and-conquer’

4

approach, instead of dealing with x and p simultaneously,one decouples the original problems into subproblems of lowerdimensions and resolve one subproblem at a time. Still for ourproblems (5) and (6) at hand, the BS-UE association problemfor optimization in binary x scales exponentially with the num-ber of BSs and UEs. It is not practical to try all the possibleBS-UE combinations, even for networks of small-to-mediumsize. Moreover, for a given BS-UE combination, the powerallocation for optimization in p remains highly nonconvex.Specifically, problem (5) has a nonconvex objective subjectto nonconvex QoS and backhaul constraints, whereas problem(6) has a nonsmooth nonconvex objective subject also to anonconvex set.

In what follows, we will address both problems (5) and(6) by a novel alternating descent method, which aims atimproving the iterative solutions. It is noteworthy that theproposed joint user association and power control algorithms,although designed for single-antenna networks, can serve asa fundamental building block for subsequent development ofjoint user association and beamforming/precoding in multiple-antenna networks [10].

III. PROPOSED ALTERNATING DESCENT ALGORITHM FORSUM-RATE MAXIMIZATION

A. BS-UE Association for Fixed Transmit Power

Given a fixed p := p, we aim to solve problem (5) invariable x. References [13], [14] considered the simplest casewith K = 2 cells, under which the objective function (5a)∑N

n=1 xn1rn1(p)

〈x1〉+

∑Nn=1 xn2rn2(p)

〈x2〉

=

∑Nn=1 xn1rn1(p)〈x2〉+

∑Nn=1 xn2rn2(p)〈x1〉

〈x1〉〈x2〉is a fraction of linear functions in the new rank-one constrainedmatrix variable X = xxT for x =

[xT1 ,x

T2

]T. By dropping

the constraint rank(X) = 1 and relaxing binary constraintson its entries to real numbers belonging to the interval [0, 1],a bisection search is used in finding the optimal solution ofthe resultant program. It should be emphasized again that suchrelaxation only works if there are two cells in the network. Thisis because for K > 2 the objective function (5a) becomes afraction of nonlinear functions in X = xxT .

Our objective here is to devise a solution that works for ageneral network with an arbitrary number of cells. To beginwith, notice that all the constraints (5b), (5c) and (5e) arelinear in x, but not (5d). The following proposition allowsus to equivalently recast the nonconvex constraint (5d) as asystem of linear constraints on x.

Proposition 1: Under the constraint (5b), the constraint (5d)is equivalent to

(M − (M − 1)xnk) rnk(p) ≥ RQoSn 〈xk〉, (7)

n = 1, . . . , N, k = 1, . . . ,K

for a sufficiently large number M .Proof: Denote by (5d)n the constraint (5d) for n. It is

sufficient to show that each (5d)n is equivalent to the following

K constraints:

(M − (M − 1)xnk) rnk(p) ≥ RQoSn 〈xk〉, k = 1, . . . ,K. (8)

Under the constraint (5b), for each n there is kn such thatxnkn = 1 and xnk = 0,∀k ∈ {1, . . . ,K} \ {kn}. Therefore,(8) merely means that

rnkn(p) ≥ RQoSn 〈xkn〉, (9)

and

Mrnk(p) ≥ RQoSn 〈xk〉, k 6= kn. (10)

Note that (9) is (5d)n, showing the implication (8)⇒ (5d)n.On the other hand, (10) holds true for 0 < M < +∞

because its right hand side is obviously bounded while thefactor rnk(p) on the left hand side is strictly positive. Theinverse implication (5d)n ⇒ (8) thus follows, yielding theequivalence between (8) and (5d)n.

Next, we deal with the binary nature of x. For xnk ∈ {0, 1},one has xnk = x2

nk and thus 〈xk〉 = 〈x2k〉 ,

∑Nn=1 x

2nk. The

objective function (5a) is then expressed as

K∑k=1

N∑n=1

x2nkrnk(p)

〈x2k〉

. (11)

On one hand, it is straightforward to see that xnk ∈ {0, 1} isequivalent to x2

nk = xnk, xnk ∈ [0, 1]. On the other hand, itholds true that x2

nk ≤ xnk for xnk ∈ [0, 1]. Following [33],we relax binary xnk to xnk ∈ [0, 1] and introduce a penaltyterm in the objective function (11) to enforce x2

nk = xnk, thusmaking xnk binary. This leads to the following problem:

maxxk∈[0,1]N

P1(x, p) ,K∑k=1

N∑n=1

x2nkrnk(p)

〈x2k〉

+λ∑Kk=1

∑Nn=1

(x2nk − xnk

)s.t. (5b)− (5e),

(12)

where λ ≥ 0 is a constant penalty factor. Parameter λsignifies the relative importance of recovering binary val-ues for x over throughput maximization. In (12), the term∑Kk=1

∑Nn=1(xnk − x2

nk) is always nonnegative and cantherefore be used to measure the degree of satisfaction ofthe binary constraints xnk ∈ {0, 1}, ∀n, k. Without squaringsuch a term, the above penalization is exact, meaning that theconstraints x ∈ {0, 1}, ∀n, k can be satisfied by a maximizerof (12) with a finite value of λ (see, e.g., [34, Ch. 16]). Thisnice property makes such exact penalization attractive.

With Proposition 1, problem (12) becomes

maxxk∈[0,1]N

P1(x, p)

s.t. (5b), (5c), (5e), (7).(13)

With an appropriate choice of λ, problems (13) and (5) areequivalent in the sense that they share the same optimalsolution [33, Sec. II]. Since problem (13) is still nonconvex,we now employ successive convex programming to solve it.

5

Proposition 2: For a given point (x(κ), p), the followingconvex problem is a global lower bound maximization for (13):

maxxk∈[0,1]N

P(κ)1 (x, p) ,

K∑k=1

N∑n=1

α(κ)nk (x, p)

+λ∑Kk=1

∑Nn=1 γ

(κ)nk (x)

s.t. (5b), (5c), (5e), (7),

(14)

where we define

α(κ)nk (x, p) ,

(x(κ)nk )2rnk(p)

〈(x(κ)k )2〉+

2x(κ)nk

(xnk − x(κ)nk

)rnk(p)

〈(x(κ)k )2〉

−(x

(κ)nk )2rnk(p)(〈(x(κ)k )2〉

)2 (〈x2k〉 − 〈(x

(κ)k )2〉

), (15a)

γ(κ)nk (x) , (x

(κ)nk )2 − x(κ)nk + (2x

(κ)nk − 1)

(xnk − x(κ)nk

).

(15b)

Proof: See Appendix A, where we show that the objectiveP(κ)1 (., p) in (14) is a global lower bound of the objectiveP1(x, p) in (13), i.e.

P1(x, p) ≥ P(κ)1 (x, p), ∀x and P1(x(κ), p) = P(κ)

1 (x(κ), p).(16)

The nonconvex problem (13) can then be addressed byinstead solving its global lower bound maximization (14) in asequential manner as follows: After initializing from a feasiblepoint x(0) of problem (13), we iteratively solve problem (14)to generate a sequence {x(κ)}, κ = 1, 2, . . . of feasible andimproved points toward the optimal solution of (13). Morespecifically, at iteration κ we use x(κ−1) as a feasible point tosolve (14) and obtain x(κ).

Theorem 1: Initialized from a feasible point x(0), thesequence {x(κ)} obtained by iteratively solving (14) is asequence of improved points of (13), which converges to aKKT point.

Proof: Note that x(κ) and x(κ+1) are a feasible point andthe optimal solution of (14), respectively. By using (16),

P1(x(κ+1), p) ≥ P(κ)1 (x(κ+1), p) ≥ P(κ)

1 (x(κ), p)

= P1(x(κ), p), (17)

i.e. x(κ+1) is a better point of (13) than x(κ). Since thesequence {x(κ)} is bounded, by Cauchy’s theorem there isa convergent subsequence {x(κν)} with a limit point x, i.e.

limν→+∞

[P1(x(κν), p)− P1(x, p)] = 0.

For every κ there is ν such that κν ≤ κ ≤ κν+1 so

0 = limν→+∞

[P1(x(κν), p)− P1(x, p)]

≤ limν→+∞

[P1(xκ), p)− P1(x, p)]

≤ limν→+∞

[P1(xκν+1), p)− P1(x, p)] = 0,

showing that limκ→+∞

P1(xκ), p) = P1(x, p). Then, each ac-

cumulation point x of the sequence {x(κ)} is a KKT-pointaccording to [35, Theorem 1]. The proof of Theorem 1 is thuscomplete.

B. Power Allocation for Fixed BS-UE Association

Given x := x, we proceed to solving problem (5) in thevariable p. Although the difference-of-convex iterations (DCI)approach of [36] can be applied in the absence of backhaulconstraints (5e), the required log-determinant optimization iscomputationally expensive even for commercialized convexsolvers. This drawback is particularly severe in HetNets whichconsist of a large number of densely deployed BSs and UEs.Reference [21] proposes solving the KKT conditions, followedby applying the gradient descent method to update Lagrangianmultipliers in order to satisfy (5e). Nevertheless, a solutionderived from the KKT conditions of a nonconvex problemmay not be locally optimal or even feasible.

Our aim here is to devise an efficient and optimal powerallocation solution. Using Proposition 1 and simple algebraicmanipulations, (5d) is expressed as the following linear con-straints:

gnkpk ≥[exp

(RQoSn 〈xk〉

M − (M − 1)xnk

)− 1

]× K∑

j 6=k,j=1

gnjpj + σ2

, n = 1, . . . , N, k = 1, . . . ,K.

(18)

Problem (5) is then reduced to

maxpP1(x,p) ,

K∑k=1

N∑n=1

(xnk)2

〈x2k〉rnk(p) (19a)

s.t.N∑n=1

xnkrnk(p) ≤ CBHk 〈xk〉, k = 1, . . . ,K (19b)

(5f), (18).

Because (19a) and (19b) are still nonconvex in p, we insteadconsider their convex bounds as given in the following result.

Proposition 3: The rate function rnk(p) in (1) admits

rnk(p) ≤θ(κ)nk (p)

, rnk(p(κ)) +1

K∑j 6=k,j=1

gnjp(κ)j + σ2

×K∑

j 6=k,j=1

(gnjp(κ)j )2

(1

gnjpj− 1

gnjp(κ)j

)

+1

K∑j=1

gnjp(κ)j + σ2

K∑j=1

gnj(pj − p(κ)j ) (20)

6

as its upper bound, and

rnk(p) ≥β(κ)nk (p)

, rnk(p(κ))− 1K∑j=1

gnjp(κ)j + σ2

×K∑j=1

(gnjp(κ)j )2

(1

gnjpj− 1

gnjp(κ)j

)

− 1K∑

j 6=k,j=1

gnjp(κ)j + σ2

K∑j 6=k,j=1

gnj(pj − p(κ)j )

(21)

as its lower bound.Proof: See Appendix B.

With the bounds in (20) and (21), we now address thenonconvex problem (19) by successive convex programming.Specifically, after initializing a feasible point p(0) of problem(19), we iteratively solve the following global lower boundmaximization of (19):

maxpP(κ)1 (x,p) ,

K∑k=1

N∑n=1

(xnk)2

〈x2k〉β(κ)nk (p) (22a)

s.t.N∑n=1

xnkθnk(p) ≤ CBHk 〈xk〉, k = 1, . . . ,K (22b)

(5f), (18)

to generate a sequence {p(κ)}, κ = 1, 2, . . . of feasible andimproved points toward the solution of (19). At iteration κ, weuse p(κ−1) as a feasible point to solve (22) and obtain p(κ).Similarly to Theorem 1, we can prove the following result.

Theorem 2: Initialized from a feasible point p(0), thesequence {p(κ)} obtained by iteratively solving (22) is asequence of improved points, which converges to a KKT pointof (19).

C. Joint Optimization of BS-UE Association and Power Allo-cation

The alternating optimization framework requires solving aseries of convex problems (14) [cf. Section III-A] followed bysolving a series of convex problems (22) [cf. Section III-B]and repeating until convergence. Realizing that solving oneinstant of (14) and (22) alone already provides a better point,we use them as an alternating descent to achieve a much fasterconvergence speed. The proposed joint optimization of BS-UEassociation and power allocation for sum-rate maximization issummarized in Algorithm 1. Our alternating descent approachgives flexibility in executing user association and power con-trol in the same time slot (as in Algorithm 1) or different timeslots (by selectively deactivating Step 3 or 4 of Algorithm 1).

At each iteration of Algorithm 1, the computational com-plexity of solving convex problems (14) and (22) is onlypolynomial in the number of variables and constraints. Tosee this, (14) can be equivalently reformulated as an opti-mization problem with a , (NK + 1) real-valued scalar

Algorithm 1 Joint BS-UE Association and Power Allocationfor Sum-Rate Maximization

1: Initialize x(0)nk := 1KN , n = 1, . . . , N, k = 1, . . . ,K and

p(0)k := Pmax

k , k = 1, . . . ,K. Set κ := 0.2: repeat3: Solve convex program (14) with p := p(κ) to find

optimal solution x?.4: Solve convex program (22) with x := x∗ to find

optimal solution p?.5: Set (x(κ+1), p(κ+1)) := (x?, p?) and κ := κ+ 1.6: until

∣∣∣P1(x(κ),p(κ))−P1(x

(κ−1),p(κ−1))P1(x(κ−1),p(κ−1))

∣∣∣ < ε

decision variables, a linear objective, b , (N + 2K + NK)linear constraints and one quadratic constraint. Similarly, (22)can be equivalently reformulated as a semidefinite programwith (2K + 1) scalar variables, a linear objective and asystem of linear matrix inequalities. The complexity requiredto solve (14) and (22) is thus O

((1 + a+ b)a2

√b+ 1

)and O

((2K + 1)2

[(K + 1)3 +NK3

]√3K +NK + 1

), re-

spectively [37].Theorem 3: Initialized from a feasible point (x(0), p(0)),

Algorithm 1 converges to a solution of problem (5) after afinite number of iterations for a given error tolerance ε > 0.

Proof: The BS-UE association problem (13) and thepower allocation problem (19) have the same objective func-tion P1(x,p). From (17), (20) and (21), we have the followingrelations:

P1(x(κ+1), p(κ+1)) ≥ P(κ)1 (x(κ+1), p(κ+1))

≥ P(κ)1 (x(κ+1), p(κ)) = P1(x(κ+1), p(κ))

≥ P(κ)1 (x(κ+1), p(κ)) ≥ P(κ)

1 (x(κ), p(κ))

= P1(x(κ), p(κ)). (23)

It means that alternatingly solving their respective convexminorants (14) and (22) always improves P1(x,p) in eachiteration. As such, once initialized from a feasible point(x(0), p(0)) that satisfies (5b), (5c) and (5f), Algorithm 1generates a sequence {(x(κ), p(κ))} of feasible and improvedpoints which eventually converges to a solution (x, p) of (5).Note that x is a KKT point of (13) for p = p while p is aKKT of (19) for x = x. Under the stopping criterion∣∣∣∣P1(x(κ+1), p(κ+1))− P1(x(κ), p(κ))

P1(x(κ), p(κ))

∣∣∣∣ < ε, (24)

Algorithm 1 terminates after a finite number of iterations fora given ε > 0.

IV. PROPOSED ALTERNATING DESCENT ALGORITHM FORMINIMUM UE RATE MAXIMIZATION

The above developed Algorithm 1 is readily extendable tosolve the max-min problem (6). In this case, we consider the

7

following objective function.

P2(x,p) ,

minn=1,...,N

{K∑k=1

x2nkrnk(p)

〈x2k〉

}+ λ

K∑k=1

N∑n=1

(x2nk − xnk

),

(25)

where xnk ∈ [0, 1], n = 1, . . . , N, k = 1, . . . ,K and λ ≥ 0is a constant penalty factor. The BS-UE association problemfor a fixed power allocation p := p is now

maxxk∈[0,1]N

P2(x, p)

s.t. (5b), (5c), (5e).(26)

Although the constraint set of (26) is convex, its objective isnonsmooth and nonconvex. Similar to Proposition 2, it can beshown that the following convex problem is a global lowerbound maximization of (26):

maxxk∈[0,1]N

P2(x, p) , minn=1,...,N

{K∑k=1

α(κ)nk (x, p)

}+λ∑Kk=1

∑Nn=1 wnkγ

(κ)nk (x),

s.t. (5b), (5c), (5e),

(27)

where α(κ)nk (x, p) and γ(κ)nk (x) have previously been defined in

(15).Next, the power allocation problem for a fixed x := x is

maxp

P2(x,p) , mink=1,...,K

{N∑n=1

x2nk〈x2k〉

rnk(p)

}s.t. (5e), (5f),

(28)

which has a nonsmooth nonconvex objective function anda nonconvex set. Similar to Proposition 3, it can be shownthat the following convex problem is a global lower boundmaximization of (28):

maxp

P(κ)2 (x,p) , min

k=1,...,K

{N∑n=1

x2nk〈x2k〉

βnk(p)

}s.t. (22b), (5f),

(29)

where βnk(p) has previously been defined in (21).To solve problem (6) in both x and p, we modify Algo-

rithm 1 as follows. In Step 3, we solve convex problem (27)instead of (14). In Step 4, we solve convex problem (29)instead of (22). And because the objective function is nowP2(x,p), the proposed algorithm for problem (6) terminateswhen

∣∣∣P2(x(κ),p(κ))−P2(x

(κ−1),p(κ−1))P2(x(κ−1),p(κ−1))

∣∣∣ < ε. We shall refer tothis modified algorithm as Algorithm 2.

In each iteration of Algorithm 2, the compu-tational complexity of solving problem (27) isO((N + a+ d)c2

√c+N

), because (27) can be equivalently

reformulated as an optimization problem with a = (NK + 1)scalar real decision variables, a linear objective, c , (N +K)linear constraints and N quadratic constraints [37].Similarly, the complexity of solving problem (29) isO([cd2(K + 1)2 + d2NK3

]√3K +NK +N

), because

(29) can be reformulated as a semidefinite program withd , (2K+1) scalar variables, a linear objective and a system

0 0.25 0.5 0.75 1 (km)0

0.25

0.5

0.75

1 (km)

Fig. 2. A three-tier network with one fixed MBS (black square), fourfixed PBSs (black diamonds), twenty random FBSs (black triangles) and 200random UEs (red circles)

of linear matrix inequalities. Finally, similar to Theorem 3,it can be proved that once initialized from a feasible point(x(0), p(0)) that satisfies (5b), (5c) and (5f), Algorithm 2converges after a finite number of iterations for a given errortolerance ε.

V. ILLUSTRATIVE EXAMPLES

Consider a three-tier HetNet where four pico BSs (PBSs)and twenty femto BSs (PBSs) are deployed within a macrocellof size 1, 000m×1, 000m. The locations of MBS and PBSs arefixed whereas those of FBSs are random, as shown in Fig. 2.We assume there are N = 200 UEs randomly distributedover the macrocell coverage area. The network topology isthen fixed during the optimization process. Without loss ofgenerality, we only consider the effect of pathloss when gen-erating the channel gains. To illustrate the impact of imperfectbackhaul links, we assume that the MBS, PBSs, and FBSs areeach equipped with a backhaul link of capacity CBH, CBH/3and CBH/10, respectively. Following [38], we choose CBH ∈{100, 150, 200,∞} Mbps where CBH = ∞ represents theideal backhaul. For simplicity, we set the required minimumUE throughput as RQoS

n = RQoS, n = 1, . . . , N . The errortolerance for the algorithms is set as ε = 10−4. Other 3GPPLTE parameters used to setup our simulations are listed inTable I [39]. Note that we divide the obtained rate results byln(2) to arrive at the unit of bps/Hz.

First, we compare the sum-rate performance of the jointdesign in Algorithm 1 to that of Algorithm 1 but with full BStransmit power (i.e., no power control). We use the heuristicBS-UE association schemes, namely, max-SINR and DCD[10] as benchmarks where full BS transmit power is alsoassumed. As both benchmark schemes assume ideal backhaul,we set CBH = ∞ here for a fair comparison. And since themax-SINR and DCD schemes do not include the minimumUE throughput constraint, we first assume RQoS = 0 in thesetwo schemes to find their BS-UE associations, followed by

8

TABLE ISIMULATION PARAMETERS USED IN ALL NUMERICAL EXAMPLES

Parameter Value

Minimum distance between MBS-UE 35m

Minimum distance between PBS/FBS-UE 10m

Path loss model for MBS-UE links 128.1 + 37.6 ln10(d), d is in km

Path loss model for PBS-UE links and FBS-UE links 140.7 + 36.7 ln10(d), d is in km

Maximum MBS transmit power 43dBm

Maximum PBS transmit power 24dBm

Maximum FBS transmit power 20dBm

Background noise power −104dBm

System bandwidth 10MHz

Frequency reuse factor 1

0 0.05 0.1 0.15 0.2 0.23200

400

600

800

1000

1200

1400

1600

1800

RQoS (Mbps)

Sum

rat

e (M

bps)

Joint opt. by Alg. 1Sole opt. by Alg. 1DCD [10]Max−SINR

Fig. 3. Sum-rate performance of Algorithm 1 under ideal backhaul links.

calculating their achieved sum-rates and minimum UE rates.Fig. 3 shows that joint design of load balancing and interfer-ence management in Algorithm 1 gives much higher networkthroughput over load balancing alone. This can be explainedby noting that the joint design has an extra dimension ofBS transmit power to optimize to further enhance the sum-rate performance. When comparing Algorithm 1 with fullBS transmit power to the max-SINR and DCD schemes, theformer offers more flexibility in setting the desired minimumRQoS. Furthermore, for the same values of RQoS that areachieved by the max-SINR and DCD schemes, the sole BS-UE design by Algorithm 1 gives a slightly better sum-rateperformance as can be observed from Fig. 3.

Next, we evaluate the effects of QoS constraints and limitedbackhaul capacity in the joint design of Algorithm 1. Fig. 4shows that as we move away from the assumption of idealbackhaul, the total throughput is gradually degraded. Thisobservation is as expected because the feasible region ofproblem (5) becomes more restricted. For each value of CBH,Fig. 4 also shows that lowering the UE throughput requirementRQoS actually increases the total throughput. However, while

0.05 0.1 0.15 0.2 0.25300

400

500

600

700

800

900

RQoS (Mbps)

Su

m r

ate

(Mb

ps)

CBH=∞CBH=200 Mbps

CBH=150 Mbps

CBH=100 Mbps

Fig. 4. Effects of QoS constraints and limited backhaul capacity on the sum-rate performance of Algorithm 1

such an throughput improvement is pronounced for the idealbackhaul, it is not much so for limited backhaul capacitycases where reducing RQoS beyond 0.2Mbps only marginallyimproves the sum-rate. Our numerical analysis reveals thatmost of the PBSs and FBSs have fully utilized their respectivelimited backhaul capacities of CBH/3 and CBH/10, leavingno room for further throughput improvement at these smallcells even if RQoS is small. Such an observation again verifiesthat backhaul capacity is in fact a bottleneck for networkperformance.

Fig. 5 further demonstrates that for a given RQoS, switchingfrom ideal backhaul (CBH = ∞) to non-ideal backhaul(CBH = 100Mbps) may limit the offloading capability of smallcells. Indeed, many UEs will be transferred from the smallcells back to the macrocell. This observation can be explainedas follows. The small cells are more easily overloaded in thenon-ideal backhaul case as their backhaul capacity is muchsmaller than that of the macrocell. To meet their backhaul lim-itations, small-cell BSs decrease their transmit power to shrinktheir cell size and serve fewer UEs with lower cell throughput.

9

0 0.5 1 (km)0

0.5

1 (km)

(a) (RQoS, CBH) = (0.05,∞) Mbps

0 0.5 1 (km)0

0.5

1 (km)

(b) (RQoS, CBH) = (0.05, 100) Mbps

0 0.5 1 (km)0

0.5

1 (km)

(c) (RQoS, CBH) = (0.25,∞) Mbps

0 0.5 1 (km)0

0.5

1 (km)

(d) (RQoS, CBH) = (0.25, 100) Mbps

Fig. 6. Changes in BS-UE associations by Algorithm 1 under various choices of UE QoS requirements and backhaul capacity

0

10

20

30

40

50

60

(RQoS, CBH) (Mbps)

Per

cen

tag

e o

f as

soci

ated

UE

s (%

)

MBSPBSFBS

(0.25, 100)(0.05, 100) (0.25, ∞)(0.05, ∞)

Fig. 5. Effects of QoS constraints and limited backhaul capacity on the loaddistribution by Algorithm 1

With 19dB-23dB higher in power budget compared to thatavailable to small-cell BSs, the MBS then increases its transmitpower (and effectively its coverage area) to take over the UEspushed out by the small cells. And with 3-10 times more

200 150 100 5010

−2

10−1

100

101

102

103

Rate: 0.05

CBH (Mbps)

Dat

a ra

te (

Mbp

s)

Rate: 0.099

Rate: 898.9

Rate: 0.461

Rate: 201

Rate: 0.05

Rate: 379.9Rate: 216.6

Algorithm 1 withRQoS=0.05 MbpsAlgorithm 2

+∞

Network sum rate

Minimum UE’s rate

Fig. 7. Fairness by Algorithm 2 and effects of limited backhaul capacity onthe minimum UE rate

backhaul capacity, the MBS is still able to accommodate theincoming traffic. This can be best observed in Fig. 6(b) [cf.Fig. 6(a)] and Fig. 6(d) [cf. Fig. 6(c)], where the MBS servesmore distant UEs for CBH = 100Mbps.

Fig. 7 demonstrates the fairness given by Algorithm 2 for

10

0 0.5 1 (km)0

0.5

1 (km)

(a) CBH =∞

0 0.5 1 (km)0

0.5

1 (km)

(b) CBH = 50 Mbps

Fig. 8. More UEs switch to MBS when backhaul capacity is limited in Algorithm 2

0 5 10 15−300

−250

−200

−150

−100

−50

0

50

100

Number of iterations

Obj

ectiv

e fu

nctio

n

λ=125

(a) Algorithm 1

0 5 10 15−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Number of iterations

Obj

ectiv

e fu

nctio

n

λ = 0.16

(b) Algorithm 2

Fig. 9. Convergence of proposed algorithms for ε = 10−4

max-min UE rates. Here, we compare against Algorithm 1 forsum-rate maximization where RQoS = 0.05Mbps is assumed.As seen from the figure, Algorithm 2 improves the minimumUE rate as the cost of reduced total throughput. Furthermore,for CBH ≥ 150 Mbps, reducing backhaul capacity does notaffect much the minimum UE rate and sum-rate performanceof Algorithm 2 because the ample backhaul capacity at eachcell can still accommodate more data traffic. However, forCBH ≤ 100 Mbps, the minimum UE rate by Algorithm 2starts to fall dramatically. And at CBH = 50Mbps, the achievedminimum UE rate drops by more than 4.5 times compared tothat in the ideal backhaul case. At this point, a significantlylarger number of UEs turns to the MBS for service as shownFig. 8. This is because the MBS still has available backhaulcapacity while the PBSs and FBSs are more likely to beoverloaded.

Finally, we examine the convergence of the proposed algo-rithms. It is sufficient to choose a large value of the penaltyfactor λ in the exact penalty method. To improve the conver-gence speed of Algorithms 1 and 2, our implementation startswith λ = 103 and fine-tunes λ through a bisection search untilthe objective functions no longer change and binary values ofx are found. For brevity, only the case of CBH = 100Mbps ispresented for illustration. Fig. 9(a) plots the convergence of the

objective function (19) by Algorithm 1 for RQoS = 0.1Mbps.Fig. 9(b) plots the convergence of the objective function (25)by Algorithm 2. In these plots, the system bandwidth isnormalized to unit to ensure the compatibility of the utilityfunction and the penalty term in (19) and (25). The numberof iterations in each plot corresponds to the presented valuesof λ. As can be seen from Fig. 9, the proposed algorithms onlyrequire at most ten iterations to converge. It is worth notingthat each iteration of our algorithms involves solving only twoeasy convex problems, each with polynomial complexity.

VI. CONCLUSIONS

In this paper, we have proposed new joint BS-UE asso-ciation and power control schemes for HetNets. Specifically,we have addressed two difficult mixed-integer optimizationproblems: (i) sum-throughput maximization under QoS con-straints and (ii) maximization of minimum UE throughput. Ourproblem formulations also include the practical constraint oflimited backhaul capacity. Our developed alternative descentalgorithms are based on an exact penalty method combinedwith successive convex programming, where we address thebinary BS-UE association problem and the nonconvex powerallocation problem separately. At each iteration, only twosimple convex problems are solved in the same time scale. Our

11

algorithms improve the objective functions in each iterationand converge eventually. Simulation results have demonstratedthe usefulness of our devised algorithms in both traffic offload-ing and interference management.

APPENDIX APROOF OF PROPOSITION 2

Firstly, notice that functions

fnk(xnk,yk) ,rnk(p)xxx2nk

yk, k = 1, . . . ,K

are jointly convex in xnk and yk. Therefore, fnk(xnk,yk)

admits its first order approximation at (x(κ)nk , y

(κ)k ) as a lower

bound [40] as follows

fnk(xnk,yk) ≥ fnk(x(κ)nk , y

(κ)k ) +

2rnk(p)x(κ)nk

(xnk − x(κ)nk

)y(κ)k

−rnk(p)(x

(κ)nk )2

(y(κ)k )2

(yk − y(κ)k

).

By replacing yk := 〈x2k〉 > 0, ∀k = 1, . . . ,K, we have

fnk(xnk, [〈x2k〉]k=1,...,K) =

rnk(p)xxx2nk〈xxx2k〉

≥ α(κ)nk (x, p).

Secondly, since x2nk − xnk is a convex quadratic function,

it also admits its first order approximation at x(κ) as a lowerbound as

x2nk − xnk ≥ (x

(κ)nk )2 − x(κ)nk +

(2x

(κ)nk − 1

)(xnk − x(κ)nk

)= γ

(κ)nk (x).

Therefore, we have

P1(x, p) ≥ P(κ)1 (x, p)

at a point (x(κ), p). This completes the proof.

APPENDIX BPROOF OF PROPOSITION 3

To prove Proposition 3, we need some preliminary resultsfirst. Let us define

gn(x1, . . . , xn) ,

(n∑i=1

x−1i

)−1. (30)

Theorem 4: For x1 > 0, x2 > 0, g2(x1, x2) is a concaveand monotonically increasing function in (x1, x2).

Proof: One has

g2(x1, x2) = x1 − x21/(x1 + x2)

which is concave in x1 > 0, x2 > 0. Moreover, as x−11 +x−12

is monotonically decreasing in x1 > 0, x2 > 0, g2(x1, x2) =1/(x−11 +x−12 ) is monotonically increasing in x1 > 0, x2 > 0.

Theorem 5: The function gn(x1, . . . , xn) =(∑n

i=1 x−1i

)−1is concave in xi > 0, i = 1, . . . , n, ∀n ≥ 1.

Proof: In Theorem 4, we have proved that g2(x1, x2)is concave in x1 > 0, x2 > 0. Assuming that

gn−1(x1, . . . , xn−1) is concave in xi > 0, i = 1, . . . , n − 1for some n, we now show that gn(x1, . . . , xn) is also concave.Firstly, one has

gn(x1, . . . , xn) =

(n∑i=1

x−1i

)−1= xn − x2n (xn + gn−1(x1, . . . , xn−1))

−1

= g2 (xn, gn−1(x1, . . . , xn−1)) .

Since g2(x, y) is a concave and monotonically increasingfunction in (x, y) and gn−1(x1, . . . , xn−1) is assumed to beconcave, we have

gn(α(x1, . . . , xn) + β(y1, . . . , yn))

= g2 (αxn + βyn, gn−1(α(x1, . . . , xn−1)

+β(y1, . . . , yn−1)))

≥ g2 (αxn + βyn, αgn−1(x1, . . . , xn−1)

+βgn−1(y1, . . . , yn−1))

≥ αg2(xn, gn−1(x1, . . . , xn−1))

+ βg2(yn, gn−1(y1, . . . , yn−1))

= αgn(x1, . . . , xn) + βgn(y1, . . . , yn)

for α ≥ 0, β ≥ 0, α + β = 1 and ∀xi > 0, yi > 0, i =1, . . . , n > 0. Thus, gn(x1, . . . , xn) is also concave in xi >0, i = 1, . . . , n, ∀n ≥ 2.

Theorem 6: The function ln(∑n

i=1 x−1i

)is convex in xi >

0, i = 1, . . . , n, ∀n ≥ 1.Proof: Let us define f(x) , ln(x). Then

ln(∑n

i=1 x−1i

)= −f(gn(x1, . . . , xn)). Since gn(x1, . . . , xn)

is a concave function in xi > 0, i = 1, . . . , n, according toTheorem 5 we have

gn(α(x1, . . . , xn) + β(y1, . . . , yn))

≥ αgn(x1, . . . , xn) + βgn(y1, . . . , yn),

with α ≥ 0, β ≥ 0, α + β = 1 and ∀xi > 0, yi >0, i = 1, . . . , n > 0. Moreover, since f(x) is a concave andmonotonically increasing function, we have

f(gn(α(x1, . . . , xn) + β(y1, . . . , yn)))

≥ f(αgn(x1, . . . , xn) + βgn(y1, . . . , yn))

≥ αf(gn(x1, . . . , xn)) + βf(gn(y1, . . . , yn)).

This shows the concavity of f(gn(x1, . . . , xn)). Therefore,−f(gn(x1, . . . , xn)) is a convex function.

We are now ready to prove Proposition 3. One can write

rnk(p) , ln

K∑j=1

gnjpj + σ2

−ln

K∑j 6=k,j=1

gnjpj + σ2

.

Since ln(x) is a concave function with x > 0, it is true that

ln

K∑j=1

gnjpj + σ2

≤ ln

K∑j=1

gnjp(κ)j + σ2

+

∑Kj=1 gnj(pj − p

(κ)j )∑K

j=1 gnjp(κ)j + σ2

, (31)

12

at some point p(κ).In addition, upon defining f(x) , ln

(∑ni=1 x

−1i

)with x =

[x1, . . . , xn], xi > 0, ∀i, one has

f(x) ≥ ln

(n∑i=1

(x(κ)i )−1

)− 1∑n

i=1(x(κ)i )−1

n∑i=1

(xi − x(κ)i

(x(κ)i )2

)at some x(κ), due to the convexity of f(x) by Theorem 6.Thus, one also has the following bound

ln

(n∑i=1

xi

)≥ ln

(n∑i=1

x(κ)i

)

− 1∑ni=1 x

(κ)i

n∑i=1

(x(κ)i )2

(1

xi− 1

x(κ)i

).

It follows that

ln

K∑j 6=k,j=1

gnjpj + σ2

≤ ln

K∑j 6=k,j=1

gnjp(κ)j + σ2

− 1∑Kj 6=k,j=1 gnjp

(κ)j + σ2

×K∑

j 6=k,j=1

(gnjp

(κ)j

)2( 1

gnjpj− 1

gnjp(κ)j

). (32)

By combining (31) and (32), (20) follows. Similarly, (21) canbe proved with minor modifications to (31) and (32).

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Ho Huu Minh Tam was born in Ho Chi Minh City,Vietnam. He received the B.S. degree in ElectricalEngineering and Telecommunications from Ho ChiMinh City University of Technology in 2012. He iscurrently working toward the Ph.D degree at the Uni-versity of Technology Sydney, Sydney, N.S.W., Aus-tralia, under the supervision of Prof. Tuan Hoang.His research interest is in optimization techniquesin signal processing for wireless communications.

Hoang Duong Tuan received the Diploma (Hons.)and Ph.D. degrees in applied mathematics fromOdessa State University, Ukraine, in 1987 and 1991,respectively. He spent nine academic years in Japanas an Assistant Professor in the Department ofElectronic-Mechanical Engineering, Nagoya Univer-sity, from 1994 to 1999, and then as an AssociateProfessor in the Department of Electrical and Com-puter Engineering, Toyota Technological Institute,Nagoya, from 1999 to 2003. He was a Professor withthe School of Electrical Engineering and Telecom-

munications, University of New South Wales, from 2003 to 2011. He iscurrently a Professor with the Centre for Health Technologies, Universityof Technology Sydney. He has been involved in research with the areasof optimization, control, signal processing, wireless communication, andbiomedical engineering for more than 20 years.

Duy Trong Ngo (S’08-M’15) received the B.Eng.(with First-class Honours and the University Medal)degree in telecommunication engineering from TheUniversity of New South Wales, Australia in 2007,the M.Sc. degree in electrical engineering (com-munication) from University of Alberta, Canada in2009, and the Ph.D. degree in electrical engineeringfrom McGill University, Canada in 2013.

Since 2013, he has been a Senior Lecturer withthe School of Electrical Engineering and Computing,The University of Newcastle, Australia, where he

currently leads the research effort in design and optimization for 5G wirelesscommunications networks.

In 2013, Dr. Ngo was awarded two prestigious Postdoctoral Fellowshipsfrom the Natural Sciences and Engineering Research Council of Canada andthe Fonds de recherche du Quebec – Nature et technologies. At The Universityof Newcastle, he received the 2015 Vice-Chancellor’s Award for Research andInnovation Excellence and the 2015 Pro Vice-Chancellor’s Award for ResearchExcellence in the Faculty of Engineering and Built Environment.

Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Sweden in2012. Since 2013, he has joined Queen’s UniversityBelfast, UK as a Lecturer (Assistant Professor). Hiscurrent research interests include physical layer se-curity, energy-harvesting communications, cognitiverelay networks. He is the author or co-author of morethan 220 technical papers published in scientificjournals (110 articles) and presented at internationalconferences (110 papers).

Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, IEEE COMMUNICATIONS LETTERS, IET COMMUNICATIONS. Hewas a Editor of WILEY TRANSACTIONS ON EMERGING TELECOMMUNI-CATIONS TECHNOLOGIES, ELECTRONICS LETTERS and has also servedas the Guest Editor of the special issue on some major journals includ-ing IEEE JOURNAL IN SELECTED AREAS ON COMMUNICATIONS, IETCOMMUNICATIONS, IEEE ACCESS, IEEE WIRELESS COMMUNICATIONSMAGAZINE, IEEE COMMUNICATIONS MAGAZINE, EURASIP JOURNALON WIRELESS COMMUNICATIONS AND NETWORKING, EURASIP JOUR-NAL ON ADVANCES SIGNAL PROCESSING. He was awarded the Best PaperAward at the IEEE Vehicular Technology Conference (VTC-Spring) in 2013,IEEE International Conference on Communications (ICC) 2014. He is therecipient of prestigious Royal Academy of Engineering Research Fellowship(2016-2021).

Vincent Poor (S72, M77, SM82, F87) received thePh.D. degree in EECS from Princeton University in1977. From 1977 until 1990, he was on the facultyof the University of Illinois at Urbana-Champaign.Since 1990 he has been on the faculty at Princeton,where he is the Michael Henry Strater UniversityProfessor of Electrical Engineering. From 2006 till2016, he served as Dean of Princetons School ofEngineering and Applied Science. Dr. Poors researchinterests are in the areas of statistical signal pro-cessing, stochastic analysis and information theory,

and their applications in wireless networks and related fields. Among hispublications in these areas is the recent book Mechanisms and Games forDynamic Spectrum Allocation (Cambridge University Press, 2014).

Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and a foreign member of the Royal Society.He is also a Fellow of the American Academy of Arts and Sciences andthe National Academy of Inventors, and of other national and internationalacademies. He received the Technical Achievement and Society Awards ofthe IEEE Signal Processing Society in 2007 and 2011, respectively. Recentrecognition of his work includes the 2014 URSI Booker Gold Medal, the2015 EURASIP Athanasios Papoulis Award, the 2016 John Fritz Medal, andhonorary doctorates from Aalborg University, Aalto University, HKUST andthe University of Edinburgh.