Generalized Cross-Layer Designs for Generic Half-Duplex Multicarrier Wireless Networks with Frequency-Reuse
aa r X i v : . [ c s . N I] A ug Generalized Cross-Layer Designs for GenericHalf-Duplex Multicarrier Wireless Networks withFrequency-Reuse
Rozita Rashtchi,
Student Member, IEEE , Ramy H. Gohary,
Senior Member, IEEE , and Halim Yanikomeroglu,
Senior Member, IEEE
Abstract —In this paper, joint designs of data routes andresource allocations are developed for generic half-duplex multi-carrier wireless networks in which each subcarrier can be reusedby multiple links. Two instances are considered. The first instancepertains to the general case in which each subcarrier can be time-shared by multiple links, whereas the second instance pertainsto a special case in which time-sharing is not allowed and asubcarrier, once assigned to a set of links, is used by thoselinks throughout the signalling interval. Novel frameworks aredeveloped to optimize the joint design of data routes, subcarrierschedules and power allocations. These design problems arenonconvex and hence difficult to solve. To circumvent thisdifficulty, efficient techniques based on geometric programmingare developed to obtain locally optimal solutions. Numericalresults show that the designs developed in both instances yieldperformance that is superior to that of their counterparts inwhich frequency-reuse is not allowed.
Index Terms —Power control, geometric programming, mono-mial approximation, time-sharing, self-concordance.
I. I
NTRODUCTION
The prospect of having ubiquitous high data-rate wirelessservices is leveraged by the versatility and portability of thecommunication devices that will form the nodes of futurewireless networks. These devices will be able to performvarious functions including sending, receiving and/or relayingdata to other nodes. As such, it is expected that future wirelessnetworks will not possess a predetermined topology, but ratheran ad hoc one that encompasses many existing and upcomingnetwork structures including current and relay-aided cellularnetworks [1], [2].Given the stringent limitations on the spectrum availablefor wireless communications, providing high data-rate servicesbanks on sharing the spectrum by multiple users which resultsin potentially significant interference. One option to mitigateinterference is to use the Orthogonal Frequency DivisionMultiple Access (OFDMA) technique, wherein a set of or-thogonal narrow-band subcarriers are exclusively assigned to
The authors are with the Department of Systems and Computer En-gineering, Carleton University, Ottawa, Ontario, Canada, { rozita, gohary,halim } @sce.carleton.caPreliminary versions of this work were presented, in part, at the IEEE Glob.Telecommun. (Globecom) , 2013, and, in part, at the
IEEE Int. Wkshp. SignalProcessing Advances in Wireless Commun. (SPAWC) , 2014.This work is supported in part by a Discovery Grant (DG) of the NaturalSciences and Engineering Research Council (NSERC) of Canada, in part byHuawei Canada Co., Ltd., and in part by the Ontario Ministry of EconomicDevelopment and Innovations ORF-RE (Ontario Research Fund - ResearchExcellence) program. each user. This technique offers several advantages includingdesign simplicity and resilience to frequency-selective fading.In spite of these advantages, rate-effective utilization of theavailable spectrum may require the OFDMA subcarriers to beused simultaneously, rather than exclusively, by multiple users.This is especially the case when the network is composed ofessentially separated clusters. In contrast, for tightly couplednetworks exclusive usage of subcarriers can be more beneficialfrom a rate perspective [3], [4].In order for a wireless network to be able to support thereliable communication of high data rates, the scarce resourcesavailable for the network must be carefully exploited. Suchresources include the spectrum available for communication,time, and the typically low power of the wireless nodes. Properexploitation of these resources involves choosing the optimalroutes of the data flows, the optimal powers to be allocatedby the nodes to each subcarrier, and the optimal schedulingand possibly duration over which the subcarriers are assignedto various links. Although these tasks have traditionally beenperformed separately, they are interrelated and performingthem in isolation may incur a significant loss in performance.To avoid the aforementioned drawback, we devise a jointoptimization framework that incorporates data routing, sub-carrier scheduling and power allocation in the design of ageneric multicarrier network. The network is generic in thesense that it possesses an ad hoc topology and its nodes canassume multiple roles simultaneously including being sources,destinations and/or relays. This framework is centralized, inthe sense that the design is performed by a central entity thatis aware of the network parameters. Hence, this frameworkcan be seen as a benchmark for distributed and potentially lesscomprehensive designs. In this framework each subcarrier canbe reused by multiple links, and the nodes acting as relaysoperate in the half-duplex mode, i.e., a node cannot send andreceive at the same time on the same subcarrier. The objectiveof the design is to maximize a weighted-sum of the ratesinjected and reliably communicated over the network. Weightsare assumed to be known a priori , but can be adapted overtime to account for fairness issues and to maintain a desiredquality of service.Two instances of networks are considered. In the firstinstance, each subcarrier can be time-shared by multiple links,thereby resulting in continuous subcarrier scheduling vari-ables. The role of these variables is to determine the fraction oftime during which a subcarrier is used over a particular link. In contrast with the first instance, in the second one time-sharingis not allowed and a subcarrier, once assigned to a set of links,is used by those links throughout the signalling interval. Thisinstance results in binary subcarrier schedules, which using aparticular change of variables, are incorporated in the powerallocation constraints. This results in a design problem thatis significantly easier than its general counterpart consideredin the first instance. It is worth noting that the first instanceis a generalization not only of the second instance, but alsoof instances in which frequency-reuse is not permitted [5].As such, the framework considered in this instance offerssignificant performance advantages over currently availabledesigns, but at the expense of increasing dimensionality anddesign complexity. This instance provides an inherent tradeoffbetween performance and design complexity; the design com-plexity can be reduced by restricting the number of links thatcan reuse a particular subcarrier.The optimization problems arising from the joint design inboth instances are nonconvex and hence difficult to solve. Toovercome this difficulty, a logarithmic transformation is usedto cast the original problem in a form that, for all but a fewconstraints, complies with the geometric programming (GP)standard form [6]. The constraints that are not compatiblewith that form are approximated by monomial expressionsthat correspond to their first order Taylor expansion arounda given initial point [7]. Using an exponential transformation,the resulting approximation can be cast in a convex form. Arefinement of this approximation can be obtained by iterativeupdating of the initial point. In particular, we use the so-called iterative monomial approximation technique, whereinthe solution of one convex approximation is used as theinitial point in the following iteration. Under relatively mildconditions, this technique is guaranteed to yield a solutionof the Karush-Kuhn-Tucker (KKT) system corresponding tothe original problem [8]. Numerical results show that thedesigns developed in both instances yield performance thatis considerably superior to that of their counterparts in whichfrequency-reuse is not allowed.In comparison with currently available designs, the onespresented herein are the first to attempt designing data routes,subcarrier schedules and power allocation jointly when thesubcarriers are both time-shared and frequency-reused. In par-ticular, the contributions in this paper include: 1) introducingthe concept of simultaneous time-sharing and frequency-reuseof subcarriers; 2) casting the joint design of data routes,subcarrier schedules and power allocation in a frameworkthat is amenable to GP-based optimization; 3) providing asimplified approach that enables mustering a considerableportion of the gains offered by the full joint design, but with asignificantly lower complexity; and 4) developing an efficientpolynomial complexity algorithm for the special case in whichthe subcarriers can be frequency-reused but not time-shared.This work builds on the results obtained in [9] and [10]. How-ever, the exposition herein is more comprehensive and includesadditional examples, a simplified approach and complexityanalysis.The paper is organized as follows. Section II provides anoverview of currently available design techniques. Section III explains the system model and design objective. Section IVconsiders the joint design of data routes and power allocationswhen time-sharing of subcarriers is allowed. The comple-mentary instance in which time-sharing is not allowed isaddressed in Section V. The complexity of the proposedalgorithms are examined in Section VI. Numerical results areprovided in Section VII, and Section VIII concludes the paper.For completeness, the GP standard form and the monomialapproximation technique are provided in Appendix A, and thederivation of the results pertaining to complexity is providedin Appendix B. II. R
ELATED W ORK
In this section we provide an overview of the currentlyavailable techniques for routing and resource allocation inwireless networks. A plethora of techniques is available foroptimizing each aspect in isolation, but significantly fewerones consider their optimization jointly.Resource allocation in wireless networks constitutes thetask of determining the power allocated for each transmis-sion and the fraction of time over which a particular sub-carrier is assigned to that transmission. Instances in whichresource allocation techniques were developed are providedin [6], [11]–[15] for various network scenarios. For instance,power allocation techniques for single-carrier cellular systemsand ad hoc multicarrier systems were developed in [11]and [6], respectively. To enable more effective utilizationof resources, power allocations were optimized jointly withbinary-constrained subcarrier schedules. For instance, the de-signs developed in [12] and [13] rely on the premise thateach subcarrier is exclusively used by one node and thesolutions obtained therein are potentially suboptimal. Whenthe binary constraint on the subcarrier schedules is relaxedallowing the subcarriers to be time-shared by multiple nodes,the optimal power allocations can be shown to be the water-filling ones [14]; a related problem was considered in [15] fora case in which the nodes experience self-noise.Further improvement can be achieved by joint optimizationof resource allocations and routing [5], [16]–[19]. For instance,a method for obtaining jointly optimal routes and powerallocations was developed in [16] for the case in whichthe nodes were restricted to use orthogonal channels fortheir transmissions. In a complementary fashion, the case inwhich the power allocations are fixed was considered in [17].Therein, a heuristic was developed for optimizing the dataroutes and subcarrier schedules jointly.Capitalizing on the potential gains of incorporating powerallocation jointly with data routing and subcarrier scheduling,the authors considered a generic network in which the nodescan assume multiple roles at the same time and each subcarriercould be either used exclusively by one link or time-sharedby multiple links [5]. Although the designs provided in [5]offer an effective means for exploiting the resources availablefor the network, these designs restrict the subcarriers to beused exclusively by only one link at any given time instant.Such a restriction may not incur a significant performanceloss in tightly coupled networks [3], but in networks with clustered structures, this restriction can be quite harmful. Forunclustered networks, frequency-reuse may result in a substan-tial increase in the interference levels. However, if properlyexploited, frequency-reuse can yield valuable performancegains. The effect of frequency-reuse was considered in single-channel networks in [18] for the case in which the data ratesare restricted to assume discrete values, and in [19] for thecase in which the nodes use superposition coding.In the current work, we will consider the joint optimizationof power allocations, subcarrier schedules and data routes inthe design of generic multicarrier networks with frequency-reuse, and with and without time-sharing. As such, thesedesigns generalize currently available ones, and will subse-quently offer a significant improvement over their perfor-mance.A summary of this review and a comparison to our work ispresented in Table I.III. P
ROBLEM S TATEMENT
A. System Model
We consider a multicarrier wireless network of N nodes,each with one transmit and one receive antenna, and a fixedpower budget, P n , n ∈ N , { , , · · · , N } . The networkoperates over a frequency-selective broadband channel ofbandwidth W , which is partitioned into K frequency-flatnarrowband channels, each of bandwidth W = W K . Nodeare assumed to be capable of simultaneously transmitting,receiving and relaying data. This assumption is generic, inthe sense that constraining some nodes to perform a subsetof tasks can be readily incorporated in the formulations thatwill be developed hereinafter. For tractability, the nodes areassumed to always have data ready for transmission [16], andfor practical considerations, the relaying nodes are assumedto operate in a multi-hop, rather than cooperative, half-duplexmode [17].The nodes are connected with L wireless links, each com-posed of K subcarriers and the set of all links is denoted by L , { , , · · · , L } . The coefficient of the k -th subcarrier oflink ℓ connecting node n to node n ′ is denoted by the complexnumber h ( k ) nn ′ which comprises pathloss, shadowing and fading.An instance of such a network with N = 4 nodes and K = 2 subcarriers is depicted in Figure 1.In addition to their desired signals, the nodes receive a su-perposition of noise and interference due to the transmissionsof other nodes in the network. Denoting the signals transmittedto and received by node n on the k -th subcarrier by u ( k ) n and y ( k ) n , respectively, we can write y ( k ) n ′ = P n ∈N \{ n ′ } h ( k ) nn ′ u ( k ) n + v ( k ) n ′ , where ‘ \ ’ denotes the set-minus operation and v ( k ) n ′ denotes the corresponding zero-mean additive Gaussian noisewith variance N . Assuming, as before, that ℓ ∈ L is the linkconnecting node n to node n ′ , it can be seen that the signal-to-noise-plus-interference ratio (SNIR) observed by node n ′ on subcarrier k of link ℓ is given bySNIR ( ℓ, k ) = p nk | h ( k ) nn ′ | W N + P n ′′ ∈N \{ n,n ′ } p ( k ) n ′′ | h ( k ) n ′′ n ′ | , (1) PSfragreplacements s (1)4 s (2)3 s (2)4 s (1)3 s (1)2 s (2)1 s (1)1 = − s (1)2 − s (1)3 − s (1)4 s (2)2 = − s (2)1 − s (2)3 − s (2)4 Node 1 Node 2Node 3Node 4 Link 1Link 2Link 3 Link 4 Link 5Link 6Link 7 Link 8Link 9Link 10 Link 11Link 12
Fig. 1: An exemplary network with N = 4 , K = 2 and D = { , } .The objective is to maximize a weighted sum of { s ( d ) n } , i.e., the rateinjected at node n and intended for destination d . where p nk is the power allocated by node n to the k -th subcar-rier. The second term in the denominator of (1) represents theaggregate interference observed by node n ′ on subcarrier k oflink ℓ . When the nodes transmit Gaussian distributed signals,the maximum data rate that can be reliably communicated onthis subcarrier is given by W log (1 + SNIR ( ℓ, k )) .For ease of exposition, we divide both the numerator anddenominator of (1) by W N and we use g ℓk to denote thenormalized channel gain, | h ( k ) nn ′ | W N , between any two nodes n, n ′ ∈ N . B. Network Topology
The considered network can be represented by a fully-connected weighted directed graph with N vertices and L = N ( N − links. To facilitate enumeration of links, the linkfrom node n to node n ′ will be labelled by ℓ = ( N − n −
1) + n ′ − if n < n ′ and by ℓ = ( N − n −
1) + n ′ if n > n ′ . The sets of incoming and outgoing links of node n ∈ N are denoted by L − ( n ) and L + ( n ) , respectively, andthe connectivity of this graph can be captured by an incidencematrix, A = [ a nℓ ] , where a nℓ = 1 if ℓ ∈ L + ( n ) , a nℓ = − if ℓ ∈ L − ( n ) and a nℓ = 0 otherwise [16]. C. Design Objective
Let D , { , . . . , D } be the set of all destination nodes,where D ⊆ N . Let s ( d ) n be the rate of the data streaminjected into node n ∈ N and intended for destination d ∈ D . The objective of our joint design is to maxi-mize a weighted-sum of the rates injected into the network,i.e., max P d ∈D P n ∈N \{ d } w ( d ) n s ( d ) n , where { w ( d ) n } are non-negative weights satisfying D ( N − P d ∈D P n ∈N \{ d } w ( d ) n =1 . Assigning weights to the injected rates provides a conve-nient means for controlling the quality of service (QoS); ahigher weight implies a higher priority. Weights are typically TABLE I: Related Work
Platform Routing Power Allocation Scheduling Frequency-reuse ReferenceMultiuser system × X X X [3]Uplink cellular × X × X [6], [11]OFDMA × X X × [12], [13], [14], [15]Generic network X X × × [16]Mesh network X × X × [17]Generic network X X × X [18], [19]Generic network X X X × [5]Generic network X X X X
This work assigned a priori , but can be adapted to meet QoS require-ments [15]. Varying the weights enables us to determine theset of all rates that the proposed design can simultaneouslyachieve.Having described the system model, in Section IV wewill characterize the constraints that must be satisfied by theroutes, the subcarrier schedules, the data rates and the powerallocations.IV. G
ENERAL C ASE : R
OUTING AND R ESOURCE A LLOCATION WITH T IME - SHARING
We consider the case when each subcarrier can be bothreused and time-shared by multiple links. This case generalizesthe cases in which either frequency-reuse or time-sharing ofsubcarriers is not considered, e.g., [5]. After characterizingthe constraints that must be satisfied by the network variables,we will formulate the cross-layer design as an optimizationproblem. Unfortunately this problem is nonconvex and toobtain a solution of its KKT system, we will use an iterativeGP-based technique that is guaranteed to converge to such asolution.
A. System Constraints
In this section, we derive the mathematical constraints thatmust be satisfied by any feasible set of data routes, time-sharing schedules and power allocations.
1) Routing Constraints:
Let x ( d ) ℓk be the data flow intendedfor destination d ∈ D on subcarrier k ∈ K of link ℓ ∈ L . Theflows, { x ( d ) ℓk } , and the injected rates, { s ( d ) n } , are related bythe flow conservation law, which must be satisfied at eachnode. This law stipulates that the sum of flows intendedfor any destination d ∈ D at each node must be equal tozero [16]. Applying this law to the current network and usingthe incidence matrix in Section III-B, it can be seen that { x ( d ) ℓk } and { s ( d ) n } must satisfy the following constraints: X ℓ ∈L X k ∈K a nl x ( d ) ℓk = s ( d ) n , n ∈ N \ { d } , d ∈ D . (2)The flow conservation law implies that the rate of data leavingthe network at d ∈ D equals the sum of the data ratesinjected into the network and intended for this destination.Hence, we can write s ( d ) d = − P n ∈N \{ d } s ( d ) n . The injectedrates, { s ( d ) n } n = d , are non-negative, and since the network isrepresented by a directed graph, the flows, { x ( d ) ℓk } , must bealso non-negative. Hence, s ( d ) n ≥ , n ∈ N \ { d } , d ∈ D , (3) x ( d ) ℓk ≥ , ℓ ∈ L , k ∈ K , d ∈ D . (4)
2) Scheduling Constraints:
Considering both time-sharingand frequency-reuse requires introducing a set of variablesto characterize the fraction of time over which a particularsubset of links utilize the same subcarrier. To do so, let γ ( k ) ℓ ··· ℓ m be the fraction of the signalling interval during whichlinks ℓ , . . . , ℓ m ∈ L are simultaneously ‘active’ on subcarrier k ∈ K ; the remaining L − m links in L are ‘silent’ onthis subcarrier. Without loss of generality, we will write theindices in an ascending order, i.e., ℓ < · · · < ℓ m . Fornotational convenience, let Γ be the set of all the subcarriertime-sharing schedules. The cardinality of Γ is given by | Γ | = K P Li =1 (cid:0) Li (cid:1) = K (2 L − . For instance, consider anetwork with L = 3 links and K = 1 subcarriers. In this case, Γ = { γ (1)1 , γ (1)2 , γ (1)3 , γ (1)1 , , γ (1)1 , , γ (1)2 , , γ (1)1 , , } and | Γ | = 7 . Tosee the role of Γ , consider the schedules in Figure 2. In thisfigure, γ (1)1 = 0 . , γ (1)1 , = 0 . , γ (1)1 , , = 0 . , and all the otherelements in Γ are zero. PSfragreplacements ℓ (1)1 ℓ (1)2 ℓ (1)3 Signalling interval fork=1 activesilent γ (1)1 (50%) γ (1)1 , (20%) γ (1)1 , , (30%) Fig. 2: An exemplary scheduling table for a network with L = 3 , K = 1 . Note that the fact that the channels are assumed constantover the signalling interval implies that only the time-sharingschedules (i.e., entries of Γ ) affect the rate expressions, irre-spective of the particular time interval over which the subcar-riers are time-shared. In other words, horizontal displacementof the shaded blocks in Figure 2 does not affect the rateexpressions.The number of variables in Γ grows exponentially with thenumber of links, L . This renders the incorporation of Γ in thejoint optimization computationally prohibitive. In most casesthis complexity can be significantly reduced without incurringheavy performance losses. For instance, if the network istightly coupled, high interference levels render the reuse ofsubcarriers on multiple links less beneficial. In such a case,restricting the reuse of a subcarrier to a fewer links mayincur negligible deterioration in performance but reduces thenumber of variables significantly. To take advantage of thisobservation, we limit the number of links that can reuse aparticular subcarrier to I ≪ L . By performing this restriction, the number of elements in Γ is reduced from K (cid:0) L − (cid:1) to K P Ii =1 (cid:0) Li (cid:1) , which, for small I , is polynomial in L . Forinstance, if at most two links are allowed to reuse a particularsubcarrier at any given time, i.e., I = 2 , the number ofelements in Γ reduces to L ( L +1)2 . It is worth noting that lim-iting the number of simultaneous transmissions, I , inherentlyoffers a trade-off between the performance and complexity. Inparticular, as I increases, the available resources are utilizedmore efficiently. However, our simulations suggest that mostof the gain of time-sharing and frequency-reuse is accrued byonly considering I ≤ simultaneous transmissions.For feasible time-sharing schedules, the elements in Γ mustbe non-negative and, to ensure no overlapping in time, thetotal time over which any subcarrier k ∈ K is used must notexceed the length of the signalling interval. These constraintsimply that Γ ≥ , elementwise , (5) I X m =1 X ℓ ··· ℓ m ∈L γ ( k ) ℓ ...ℓ m ≤ , ∀ k ∈ K . (6)Note that summations in (6) characterize the number of links, m , that reuse a particular subcarrier k . For instance, for thecase in which at most two links reuse this subcarrier, the lefthand side (LHS) of (6) can be expressed as P ℓ ∈L γ ( k ) ℓ + P ℓ ∈L P ℓ ∈L γ ( k ) ℓ ℓ .Nodes cannot broadcast data to multiple destinations at thesame time, that is, at any time instant, node n can have atmost one active link on subcarrier k . Hence, the time-sharingschedules corresponding to multiple outgoing links of node n must be zero. This can be represented as a + nℓ a + nℓ (cid:16) γ ( k ) ℓ ℓ + I X m =3 X ℓ ··· ℓ m ∈L γ ( k ) ℓ ...ℓ m (cid:17) = 0 ,ℓ ∈ L , ℓ ∈ L \ { ℓ } , k ∈ K , (7)where a + nℓ = max { , a nℓ } , that is, a + nℓ = 1 if ℓ ∈ L + ( n ) andzero, otherwise.To enforce the half-duplex constraint, we must ensure thatno two links, ℓ ∈ L − ( n ) and ℓ ∈ L + ( n ) , can be activeon the same subcarrier k ∈ K at the same time. This impliesthat all the time-sharing schedules that correspond to ℓ and ℓ , i.e., γ ( k ) ℓ ...ℓ m , m = 2 , . . . , I , must be zero. Since all theentries in Γ are non-negative, these constraints can be writtenas a + nℓ a − nℓ (cid:16) γ ( k ) ℓ ℓ + I X m =3 X ℓ ··· ℓ m ∈L γ ( k ) ℓ ...ℓ m (cid:17) = 0 ,ℓ ∈ L , ℓ ∈ L \ { ℓ } , k ∈ K , (8)where a − nℓ = | min { , a nℓ }| , that is, a − nℓ = − if ℓ ∈ L − ( n ) and zero, otherwise. Note that (7) and (8) take effect onlywhen a + nℓ a − nℓ ′ = 0 and a + nℓ a + nℓ ′ = 0 , respectively.
3) Power Allocation Constraints:
To facilitate the design,we replace the node power variables { p nk } with link power variables { q ℓk } , which are related by the following transfor-mation: p nk = max ℓ ∈L + ( n ) q ℓk , n ∈ N , k ∈ K . (9)To gain a better understanding of the transformation in (9),we note that (7) implies that, of all the links in L + ( n ) , onlyone element in the set { q ℓk } ℓ ∈L + ( n ) , ∀ n ∈ N , k ∈ K , can assume a strictly positive value. Now, (9) indicates thatthis value is the power allocated by node n to subcarrier k .Using (9), we will formulate our design in terms of { q ℓk } instead of { p nk } . These variables must satisfy the followingnon-negativity constraints: q ℓk ≥ , ℓ ∈ L , k ∈ K . (10)In a practical network, the nodes are likely to have indi-vidual power budgets which bounds the total power used byeach node on all subcarriers. To capture this constraint, wenote that only the subcarriers scheduled to outgoing linkscontribute to the power consumption of each node. Morespecifically, if ℓ ∈ L + ( n ) , then all the time-sharing schedulesthat correspond to ℓ contribute to the power consumption atnode n . This constraint can be written as X k ∈K X ℓ ∈L + ( n ) q ℓ k (cid:16) γ ( k ) ℓ + I X m =2 X ℓ ··· ℓ m ∈L γ ( k ) ℓ ...ℓ m (cid:17) ≤ P n ,n ∈ N . (11)
4) Capacity Constraints:
To complete the characterizationof the network, we point out that the data flows and the powerallocations are coupled by the maximum aggregate rate thatcan be supported by the subcarriers of each link. In particular,the aggregate rate P d ∈D x ( d ) ℓk must not exceed the capacity ofthe k -th subcarrier of link ℓ .To characterize the capacity constraints, we note that thetransmission on link ℓ ∈ L and subcarrier k ∈ K iscomposed of two parts. The first part accounts for the fractionof time over which this transmission is interference-free,whereas the second part accounts for the fraction of time overwhich this transmission interferes with other transmissions.To characterize the second part, we identify the interfer-ing links and the fraction of time over which these linksare interfering. To do so, we note that, if subcarrier k istime-shared by links ℓ , . . . , ℓ m , then the transmissions onlinks ℓ , . . . , ℓ m interfere with the transmission on link ℓ .Hence, the SNIR expression for the transmission on link ℓ is q ℓ k g ℓ k P mi =2 q ℓik g ℓ ′ ik , where ℓ ′ i denotes the index of the linkconnecting the node at which link ℓ i originates to the nodeat which link ℓ ends. Since links ℓ , . . . , ℓ m are simultane-ously active on subcarrier k for a fraction of γ ( k ) ℓ ...ℓ m , theexpression for the data rate that can be communicated overlink ℓ is γ ( k ) ℓ ...ℓ m log (cid:18) q ℓ k g ℓ k P mi =2 q ℓik g ℓ ′ ik (cid:19) . Summing overall possible combinations of the interfering links, the capacityconstraint on the aggregate flow of link ℓ on subcarrier k canbe expressed as X d ∈D x ( d ) ℓ k ≤ γ ( k ) ℓ log (1 + q ℓ k g ℓ k )+ I X m =2 X ℓ ...ℓ m ∈L γ ( k ) ℓ ...ℓ m log q ℓ k g ℓ k P mi =2 q ℓ i k g ℓ ′ i k ! . (12) B. Problem Formulation
To ensure the feasibility of the rates generated by our design,the constraints in (2)–(12) must be satisfied. Combining theseconstraints yields the following optimization problem: max { s ( d ) n } , { x ( d ) ℓk } , { q ℓk } , Γ X d ∈D X n ∈N \{ d } w ( d ) n s ( d ) n , subject toRouting constraints in (2)–(4) , Scheduling constraints in (5)–(8) , Power allocation constraints in (10) and (11) , Capacity constraints in (12) . (13)The optimization problem in (13) is nonconvex because of thepower allocation constraint in (11) and the capacity constraintsin (12). Examining (13) reveals that this problem shares somefeatures with the GP standard form, cf. Appendix A-1. Toexploit this observation, in the next section we will perform achange of variables that will enable us to express the objectiveand all, but one set, of the constraints in a GP-compatibleform. The residual constraints that do not comply with theGP standard form are approximated using the monomialapproximation technique in Appendix A-2. Under relativelymild conditions [8], iterative application of this technique isknown to yield a solution of the KKT system correspondingto (13), see e.g., [5], [19]. C. Generalized GP-Based Algorithm
To cast (13) in a form that is amenable to monomialapproximation, we define two sets of variables, { t ( d ) n } and { r ( d ) ℓk } , which are related to { s ( d ) n } and { x ( d ) ℓk } by the followingmaps: s ( d ) n = log t ( d ) n , x ( d ) ℓk = W log r ( d ) ℓk ,n ∈ N \ { d } , d ∈ D , ℓ ∈ L , k ∈ K . (14)These maps are bijective, which renders recovering { s ( d ) n , x ( d ) ℓk } straightforward. The objective and the routingconstraints in (13) can be cast in a GP-compatibleform. In particular, the objective can be expressed as Q d ∈D Q n ∈N \{ d } (cid:16) t ( d ) n (cid:17) w ( d ) n and the routing constraints canbe expressed as Y ℓ ∈L Y k ∈K (cid:0) r ( d ) ℓk (cid:1) W a nℓ = t ( d ) n , n ∈ N \ { d } , d ∈ D , (15) r ( d ) ℓk ≥ , ℓ ∈ L , k ∈ K , d ∈ D , (16) t ( d ) n ≥ , n ∈ N \ { d } , d ∈ D . (17)The non-negativity constraints in (5) and (10) are inherentlysatisfied in the GP framework. The constraints in (6) and (11)are already in a GP-compatible form. We now consider theconstraints in (7) and (8). The right hand side (RHS) of theseconstraints are zero, which makes them incompatible with the GP framework in Appendix A-1. This problem can bealleviated by constraining their LHS to be less than an arbitrarysmall number ǫ > , i.e., a + nℓ a + nℓ (cid:16) γ ( k ) ℓ ℓ + I X m =3 X ℓ ··· ℓ m ∈L γ ( k ) ℓ ...ℓ m (cid:17) ≤ ǫ,ℓ ∈ L , ℓ ∈ L \ { ℓ } , k ∈ K , (18) a + nℓ a − nℓ (cid:16) γ ( k ) ℓ ℓ + I X m =3 X ℓ ··· ℓ m ∈L γ ( k ) ℓ ...ℓ m (cid:17) ≤ ǫ,ℓ ∈ L , ℓ ∈ L \ { ℓ } , k ∈ K . (19)The remaining constraints that are not GP-compatible arethose in (12). Invoking the change of variables in (14), for ℓ ∈ L and k ∈ K , those constraints can be expressed as Y d ∈D r ( d ) ℓ k ≤ (1 + q ℓ k g ℓ k ) γ ( k ) ℓ × I Y m =2 Y ℓ ··· ℓ m ∈L q ℓ k g ℓ k P mi =2 q ℓ i k g ℓ ′ i k ! γ ( k ) ℓ ...ℓm . (20)The RHS of (20) is amenable to the monomial approximationtechnique described in Appendix A-2 [7]. One approach touse this technique is to approximate all the terms in theRHS of (20) by one monomial. This approach is overlycomplicated, and an alternative is to approximate each termby a monomial. The product of these monomials constitutesa monomial approximation of the RHS of (20). Hence, theconstraint in (20) can be approximated with Y d ∈D r ( d ) ℓ k ≤ M (cid:16) (1 + q ℓ k g ℓ k ) γ ( k ) ℓ (cid:17) × I Y m =2 Y ℓ ··· ℓ m ∈L M (cid:18)(cid:16) q ℓ k g ℓ k P mi =2 q ℓ i k g ℓ ′ i k (cid:17) γ ( k ) ℓ ...ℓm (cid:19) , (21)where the functional M ( · ) is described in Appendix A-2. Notethat, { γ ( k ) ℓ ··· ℓ m } are variables and hence inseparable from theargument of M ( · ) .Now, the problem in (13) can be approximated by thefollowing GP: max { t ( d ) n } , { r ( d ) ℓk } , { q ℓk } , Γ Y d ∈D Y n ∈N \{ d } (cid:16) t ( d ) n (cid:17) w ( d ) n , subject toRouting constraints in (15)–(17) , Scheduling constraints in (6), (18) and (19) , Power allocation constraints in (11) , Approximate capacity constraints in (21) . (22)Note that the relaxations in (18) and (19) may resultin infeasible subcarrier time-shares that do not satisfy theconstraints in (7) and (8). To construct feasible schedules, theelements of Γ that are less than or equal to ǫ are set to zero.Using a standard exponential transformation, the GP in (22)can be readily transformed into a convex optimization problemwhich can be solved in polynomial time using interior-point TABLE II: Successive GP-based Algorithm for Solving (22)
1- Let U (0) = 0 . Set accuracy to δ > .2- Choose I and a feasible (cid:0) { q (0) ℓk } , Γ (0) (cid:1) .3- Solve the GP in (13). Denote the value of the objective by U .4- While U − U (0) ≥ δ , { q (0) ℓk } ← { q ℓk } , U (0) ← U ,Solve the GP in (13). Denote the value of the objective by U , End .5- Remove the elements in Γ that are less than ǫ .6- Use (14) to recover { s ( d ) n } and { x ( d ) ℓk } . methods (IPMs) [7]. This implies that (22) enables us toefficiently solve (13) approximately in the neighbourhood ofany initial set (cid:0) { q (0) ℓk } , Γ (0) (cid:1) .Finding the global solution for the nonconvex problemin (13) is difficult, whereas solving the approximated problemin (22) is straightforward. To exploit this fact, we incorporatethe formulation in (22) in an iterative algorithm, whereby theoutput of solving (22) for an initial point (cid:0) { q (0) ℓk } , Γ (0) (cid:1) isused as a starting point for the subsequent iteration. Thistechnique is usually referred to as the single condensationmethod, e.g., [19], [20], and under relatively mild conditions,its convergence to a solution of the KKT system correspondingto (13) is guaranteed [8]. Since the original design problemis not convex, this system has multiple local solutions andthe one to which the single condensation method convergesdepends on the initial point; some of the local solutions maybe global ones. A summary of this algorithm is described inTable II.In the next section, we will discuss a special case of thisalgorithm when time-sharing of subcarriers is not allowed.Before we do that, we now provide a brief discussion onthe implementation of this algorithm. To begin with, we notethat the algorithm in Table II is centralized, in the sense thatthe design is performed by a central entity that is aware ofthe network parameters. The signalling exchange between thenodes and the central entity, required to establish communi-cation in the considered framework, are described as follows.At the beginning of each signalling interval, the central entityprompts the nodes in the network to sequentially broadcastpilot signals of prescribed power levels. Subsequently, eachnode computes the subcarrier channel gains from all othernodes in the network. There is total of LK such gains, where L is the number of links and K is the number of subcarriers.Each node sends these gains along with its destination nodes,if any, and its priority weights to the central entity. The centralentity performs the joint optimization of the power allocations,scheduling parameters and data routes as described in Table II.It then forwards these decisions to all the nodes, possiblyover a dedicated control channel. In particular, the informationforwarded by the central entity include 1) the subcarrier indexand the time allocated to each transmission. This informationis provided by the set Γ . The cardinality of this set dependson the number of simultaneous transmissions allowed in eachsubcarrier, I . For instance, for I = 2 , | Γ | = LK ( L + 1) / ; 2)The power allocated to each transmission. This information is provided by the set { q ℓk } and the cardinality of this set is LK ;and 3) The data rates at each transmitting and receiving nodein the route of the stream intended for each destination. Thisinformation is provided in the set { x ( d ) ℓk } and the cardinalityof this set is LKD , where D is the number of intendeddestinations.V. S PECIAL C ASE : R
OUTING AND R ESOURCE A LLOCATION WITHOUT T IME - SHARING
In this section, we consider a design problem similar tothe one described in Section IV, but for the case when time-sharing of subcarriers is not allowed. This corresponds to thea special case in which the entries of Γ in Section IV-A2 arerestricted to be binary. This restriction results in a mixed inte-ger program which is generally difficult to solve. To overcomethis difficulty, we capture the effect of the scheduling variablesin the power allocation constraints. We will show that thisapproach will enable us to develop a design algorithm withpolynomial-complexity. A. System Constraints1) Routing Constraints:
These constraints are identical tothose described in (2)–(4).
2) Power Allocation Constraints:
In characterizing theseconstraints, we will use the method described in Section V-A2to denote the power allocated for transmission on subcarrier k of link ℓ by the variables { q ℓk } . These variables must satisfythe non-negativity constraints in (10) and the power budgetconstraint. These constraints, using (9), can be cast as X k ∈K X ℓ ∈L a + nℓ q ℓk ≤ P n , n ∈ N . (23)Similar to the case considered in Section IV, the nodescannot simultaneously broadcast to multiple destinations onthe same subcarrier. However, this requirement in the currentcase can be implicitly captured by the allocation of the linkpowers. In particular, for any subcarrier k ∈ K , any node n ∈ N and any two links ℓ , ℓ ∈ L + ( n ) , at least q ℓ k = 0 or q ℓ k = 0 , i.e., a + nℓ a + nℓ q ℓ k q ℓ k = 0 , ℓ , ℓ ∈ L , k ∈ K , n ∈ N . (24)This constraint is significantly less involved than the one in (7).Similarly, the half-duplex requirement can be captured byensuring that, for each node n ∈ N , if the power on subcarrier k of L + ( n ) is strictly positive, then the power allocated tothis subcarrier on all the links in L − ( n ) is zero, and viceversa. Hence, the half-duplex requirement can enforced by thefollowing constraints: a − nℓ a + nℓ q ℓ k q ℓ k = 0 , ℓ , ℓ ∈ L , k ∈ K , n ∈ N . (25)Note that these constraints are simpler than their counterpartsin (8). Also, note that (24) and (25) are trivially satisfied ifeither link ℓ or ℓ are not connected to node n .
3) Capacity Constraints:
In this case, the constraints in (12)can be readily seen to reduce to X d x ( d ) ℓ k W ≤ log (cid:16) q ℓ k g ℓ k P ℓ ∈L\{ ℓ } q ℓ k g ℓ ′ k (cid:17) . (26) B. Problem Formulation
Using the characterization described in Section V-A, thedesign problem can be cast as: max { s ( d ) n } , { x ( d ) ℓk } , { q ℓk } X d ∈D X n ∈N \{ d } w ( d ) n s ( d ) n , subject toRouting constraints in (2)–(4) , Power allocation constraints in (10), (23)–(25) , Capacity constraints in (26) . (27)The optimization problem in (27) is nonconvex becausethe RHS of (26) is the logarithm of a rational function, andtherefore not concave. The equality constraints in (24) and (25)are not affine and hence, nonconvex. In the next section,we will develop a GP-based algorithm, analogous to the onedescribed in Section IV-C, to obtain a locally optimal solution. C. Proposed GP-based Algorithm
The optimization problem in (27), although nonconvex,is amenable to the GP-based monomial approximation inAppendix A-2. To use this approximation, we use (14) totransform { s ( d ) n } and { x ( d ) ℓk } to { t ( d ) n } and { r ( d ) ℓk } , respectively.Using these new variables, the routing constraints are readilyexpressed in a GP-compatible form as described in (15)–(17).Substituting from (14) into (26) yields the following set ofequivalent constraints: (cid:16) X ℓ ∈L\{ ℓ } q ℓ k g ℓ ′ k (cid:17) Y d ∈D r ( d ) ℓ k ≤ X ℓ ∈L q ℓ k g ℓ ′ k ,k ∈ K , ℓ ∈ L , (28)which are, unfortunately not GP-compatible. Using the mono-mial approximation technique in Appendix A-2 yields thefollowing approximation of (28) in the neighbourhood of { q (0) ℓk } : (cid:16) X ℓ ∈L\{ ℓ } q ℓ k g ℓ ′ k (cid:17) Y d ∈D r ( d ) ℓ k ≤ c ℓ k Y ℓ ∈L (cid:16) q ℓ k /q (0) ℓ k (cid:17) θ ℓ k ,k ∈ K , ℓ ∈ L , (29)where { q (0) ℓk } is the initial power allocation, c ℓ k = 1 + P ℓ ∈L q (0) ℓ k g ℓ ′ k , and θ ℓ k = q (0) ℓ k g ℓ ′ k /c ℓ k .Analogous to the case considered in Section IV-C, (24)and (25) are replaced with the GP-compatible inequality con-straints. The joint design of data routes and power allocationsin (27) can be approximated with the following GP: max { t ( d ) n } , { r ( d ) ℓk } , { q ℓk } Y d ∈D Y n ∈N \{ d } (cid:16) t ( d ) n (cid:17) w ( d ) n , subject toRouting constraints in (15)–(17) , Power allocation constraints in (23)–(25) (relaxed versions) , Approximate capacity constraints in (29) . (30)A locally optimal solution of (27) can be obtained by solv-ing (30) iteratively using the single condensation methoddescribed in Section IV-C. VI. C OMPLEXITY A NALYSIS
In this section we examine the computational complexityrequired for solving the problems described in Sections IV-Cand V-C for the cases with and without time-sharing, re-spectively. The algorithms in these sections iteratively solvethe families of the optimization problems in (22) and (30).Being in a GP-compatible form, these problems can be readilyconverted into convex forms and can be efficiently solvedusing IPM-based solvers.In IPM, the objective and inequality constraints are usedto construct a log-barrier function which is minimized alonga central path using Newton’s method. The complexity ofeach Newton step grows with the cube of the number ofinequality constraints and the number of Newton steps canbe bounded if the log-barrier function is self-concordant [21],cf. Appendix B. In that case, the number of Newton stepscan be shown to grow with the square root of the number ofinequality constraints [21].Unfortunately, the log-barrier functions related to the prob-lems in (22) and (30) are not self-concordant. To circumventthis difficulty, we introduce a set of auxiliary variables andconstraints which, although redundant, enables us to constructself-concordant log-barrier functions. Using these functionsand the results in [21], we arrive at the following proposition:
Proposition 1.
The complexity of solving (22) with IPM-basedsolvers is of order O LKN + N + D ( N −
1) + 2 K I X i =1 (cid:18) Li (cid:19)! . , and the complexity of solving (30) with IPM-based solvers isof order O (cid:16) ( LK (3 L + 2) + N + D ( N − . (cid:17) . Proof:
See Appendix B.The first statement of Proposition 1 pertains to the generalcase with time-sharing and frequency-reuse. This statementshows that the complexity of solving the problem in (22)is polynomial in L for small values of I . The complexityof solving (22) can be further reduced by combining thebroadcasting constraint and the half-duplex constraint in (7)and (8), respectively. In particular, examining these constraintsreveals that they are related to the network topology and do notdepend on the channel conditions. Hence, these two constraintscan be enforced by pruning the set Γ prior to solving (13) or itsapproximated version in (22). The pruning rule is as follows:For each ℓ and ℓ ′ ∈ L , if either a + nℓ a + nℓ = 0 or a + nℓ a − nℓ = 0 ,the corresponding time-shares in (7) and (8) are removed fromthe set Γ . Unfortunately, we have not been able to obtain aclosed form of the cardinality of the resulting Γ . However, thereduction in complexity, at least for small networks, appearsto be significant. For instance, for fully connected networkswith N = 4 nodes and L = N ( N −
1) = 12 links, | Γ | isreduced from 4095 to 40.The second statement of Proposition 1 pertains to the specialcase in which time-sharing is not allowed. This statementshows that the complexity of solving (30) is polynomial in the TABLE III: Normalized Channel Gains, { g ℓk } , in Example 1 [dB]. link 1 link 2 link 3 link 4 link 5 link 6subcarrier 1 -6.1 -11.1 1.86 -12.3 -13.5 42.4subcarrier 2 -5.7 1.57 5.64 -5.98 -19.2 40.7link 7 link 8 link 9 link 10 link 11 link 12subcarrier 1 15.82 -2.43 -5.82 -6.9 35.0 3.304subcarrier 2 -3.38 -5.61 0.871 -0.6 41.2 -14.5 number of nodes, N , and the number of subcarriers, K . In par-ticular, it grows as L K . . Another case in which the designcomplexity is polynomial is the one in which the subcarriersare time-shared but not frequency-reused [5]. In that case thedesign complexity is O (( LK (4+ D )+ N + K + D ( N − . ) .Hence the special cases with either no frequency-reuse or notime-sharing have polynomial complexity.VII. S IMULATION R ESULTS
In this section we provide numerical results to evaluatethe performance of joint routing and resource allocation al-gorithms for the cases with and without time-sharing.The locations of the nodes are randomly generated andevenly distributed over a × m square. The nodes areassumed to have identical power budgets, i.e, P n = P, ∀ n ∈N , and the available frequency-selective channel is partitionedinto a set of frequency-flat Rayleigh fading channels withthe values of pathloss (PL) and shadowing components ob-tained from the non line-of-sight communication of indoorhotspot (InH) scenario in the IMT-Advanced document [22].According to [22], the PL component on link ℓ ∈ L is givenby P L = 43 . ( d ℓ ) + 11 . ( f c ) , (31)where d ℓ is the length of link ℓ in meters and f c is thecarrier frequency in Gigahertz which, in our simulations, isset to f c = 3 . GHz. The shadowing component is assumedto be log-normal distributed with a mean of 0 dB and astandard deviation of 4 dB. The Rayleigh fading componentis generated by the envelope of a zero-mean unit-variancecomplex Gaussian-distributed random variable. The availablebandwidth around each subcarrier is set to W = 200 KHzand the noise power density at receivers is set to N = − dBm/Hz.The results reported herein are obtained using the CVX package [23] with an underlying
MOSEK solver [24]. Thevalue of ǫ in (7) and (8) is set to − . Example 1: (Joint Routing and Resource Allocation withTime-sharing)
Consider an exemplary network with N = 4 nodes. In this network, nodes 3 and 4 wish to communicatewith nodes 2 and 1, respectively, over K = 2 subcarriers. Inparticular, for destination node d = 1 , the source is node n = 4 and nodes { , } are potential relays, and, for destination node d = 2 , the source is node n = 3 and nodes { , } are potentialrelays. The considered network has L = 12 directional linksand therefore the channel matrix has × elements. Thechannels are assumed to be static and their normalized gain indB, i.e.,
10 log g ℓk , is given in Table III. In this example, thepower budget of each node is set to P = 20 dBm, the numberof simultaneous transmissions is set to I = 3 and the two TABLE IV: Power Allocations (mW) in Example 1. n = 1 n = 2 n = 3 n = 4 q , = 23 q , = 56 . q , = 6 . q , = 24 . q , = 13 . q , = 0 . q , = 16 . rates, s (2)3 and s (1)4 , are assigned equal weights, i.e., w (2)3 = w (1)4 = 1 . Since in this example time-sharing is allowed, thealgorithm in Section IV-C is used to generate the data routes,time-sharing schedules and power allocations.The sum-rate yielded by the algorithm in Section IV-C is7.4 b/s/Hz. The data routes generated by this algorithm areillustrated in Figure 3. For ease of exposition, the networkin this example is split into the two sub-networks: the onein Figure 3(a) depicts the routes of the data intended fordestination d = 1 , and the one in Figure 3(b) depicts theroutes of the data intended for destination d = 2 . The completenetwork is the superposition of the two sub-networks. Forinstance, the data transmitted over link 7, connecting node3 to node 1, is 4.8 b/s/Hz, of which 2 b/s/Hz is intended fordestination d = 1 and 2.8 b/s/Hz is intended for destination d = 2 .The time-sharing schedules of the subcarriers generated bythe algorithm in Section IV-C are provided in Figure 4. It canbe seen from this figure that subcarrier k = 1 is both reusedand time-shared, whereas subcarrier k = 2 is only time-shared.Figures 3 and 4 imply that link 7, connecting node 3 to node1, and link 11, connecting node 4 to node 2, carry the dataintended for both destinations on the same subcarrier, k = 1 ,during the same time interval. The fact that our designs enforcehalf-duplex requirement can be inferred from these figures.For instance, Figure 3(a) shows that node 3 uses the samesubcarrier, k = 1 , for its transmission and reception on links7 and 12, respectively, but Figure 4 shows that transmissionand reception occur during different time intervals.The power allocations yielded by the algorithm in Sec-tion IV-C are shown in Table IV. This table shows that,because of frequency-reuse, the nodes do not necessarily usetheir total power budgets. This is due to the fact that, inthis scenario, when a node increases its transmission power,it inflicts high interference on other transmissions. This isin contrast with the situation considered in [5], whereinfrequency-reuse is not allowed and increasing the transmittedpower of a node does not affect the transmissions of the othernodes in the network. (cid:3) Example 2: (Joint Routing and Resource Allocation withoutTime-sharing)
Consider an exemplary network with N = 6 nodes. In this network, as before, nodes 3 and 4 wish tocommunicate with nodes 2 and 1, respectively, over K = 4 subcarriers. In particular, for destination node d = 1 , thesource is node n = 4 and the other nodes, i.e., { , , , } arepotential relays, and, for destination node d = 2 , the source isnode n = 3 and nodes { , , , } are potential relays.The considered network has L = 30 links and therefore thechannel matrix has × elements. For space considerations,this matrix is not provided, but since the channel gain on eachsubcarrier is dominated by the PL component, we providethe coordinates of the nodes in the × m square;calculating the PL components from these coordinates is PSfragreplacements
Node 1 Node 2Node 3Node 4 s (1)4 = 3 . b/s/Hz s (1)1 = 3 . b/s/Hz (a) PSfragreplacements k = 1 k = 2 Node 1 Node 2Node 3Node 42.8 b/s/Hz 4.3 b/s/Hz2.8 b/s/Hz1.5 b/s/Hz s (2)3 = 4 . b/s/Hz s (2)2 = 4 . b/s/Hz (b) Fig. 3: Data routes for (a) d = 1 , (b) d = 2 in Example 1. PSfragreplacements k = 1 k = 2 Link 4Link 12 Link 3Link 7Link 11 Link 10Link 9
Fig. 4: Time-sharing schedules of the subcarriers in Example 1. straightforward, cf. (31). The coordinates of the nodes are { (283 , , (191 , , (287 , , (72 , , (201 , , (86 , } .Setting the node power budgets to P = 20 dBm andassuming that both rates have equal weights, w (2)3 = w (1)4 = 1 ,the joint design algorithm in Section V-C yields a sum-rate of9.1 b/s/Hz. The data routes and power allocations obtained bythis algorithm are shown in Figure 5 and Table V, respectively.For instance, in Figure 5, subcarrier k = 1 is shown to be usedtwice and due to the half-duplex constraint, transmission andreception take place over distinct subcarriers at each node. Wewill later show the advantage of the proposed algorithm overthe algorithms in which frequency-reuse is not considered. (cid:3) Example 3: (Average Weighted-Sum Rate Comparison)
Inthis example, we use Monte Carlo simulations to evaluate theaverage performance of the joint designs with and withouttime-sharing when the channels are time-varying rather than
TABLE V: Power Allocations (mW) in Example 2. n = 1 n = 3 n = 4 n = 5 q , = 12 q , = 45 q , = 100 q , = 25 q , = 55 PSfrag replacements
Node 1 Node 2 Node 3Node 4Node 5Node 6 s (1)4 = 4 . b/s/Hz s (1)1 = 4 . b/s/Hz (a) PSfrag replacements
Node 1 Node 2 Node 3Node 4Node 5Node 6 k = 1 k = 2 k = 3 k = 4 s (2)3 = 9 . b/s/Hz s (2)2 = 9 . b/s/Hz (b) Fig. 5: Data routes for (a) d = 1 , (b) d = 2 in Example 2. static as in Examples 1 and 2. We consider a network with N = 4 nodes in which nodes 3 and 4 wish to communicatewith nodes 2 and 1, respectively, over K = 4 subcarriers. Thenumber of simultaneous transmissions is set to I = 3 and thesimulation results are averaged over 10 independent networkrealizations.The average weighted-sum rates yielded by the algorithmsin Sections V-C and IV-C for the values of P ranging from 0to 30 dBm are depicted in Figures 6(a) and 6(b) for the casesof w (2)3 = 5 w (1)4 and w (2)3 = w (1)4 , respectively. These figuresalso provide a comparison with the weighted-sum rates yieldedby the designs in which frequency-reuse is not considered [5].As can be seen from Figure 6, the weighted-sum rate yieldedby the joint design with both time-sharing and frequency-reuse outperforms the designs in which either time-sharing orfrequency-reuse is exclusively considered, but at the expenseof increased complexity. For instance, Figure 6(b) suggeststhat, at the sum-rate of 12 b/s/Hz, the proposed design withboth time-sharing and frequency-reuse yields a power advan-tage of 4 dBm over the designs in which either time-sharingor frequency-reuse is exclusively considered and a poweradvantage of 8 dBm over the design in which neither of thesetechniques is considered. This figure also suggests that, forvalues of P less than 15 dBm, the design with frequency-reusebut without time-sharing yields better performance than thedesign with time-sharing but without frequency-reuse in [5].However, for values of P higher than 15 dBm, the design withtime-sharing but without frequency-reuse performs better thanthe one with frequency-reuse but without time-sharing. Thisphenomenon can be attributed to the effect of interference. Atlow powers, the effect of interference is small and frequency-reuse performs generally better than time-sharing. In contrast,at high powers, the effect of interference is more severeand time-sharing performs generally better than frequency- PSfrag replacements w/o time-sharing, w. freq. reuse, proposedw/o time-sharing, w/o freq. reuse, [5]w. time-sharing, w/o freq. reuse, [5]w. time-sharing, w. freq. reuse, proposed P [dBm] W e i gh t e d - s u m r a t e ( b / s / H z ) w (2)3 = w (1)4 w (2)3 = 5 w (1)4 w (1)4 = 5 w (2)3 (a) PSfrag replacements w/o time-sharing, w. freq. reuse, proposedw/o time-sharing, w/o freq. reuse, [5]w. time-sharing, w/o freq. reuse, [5]w. time-sharing, w. freq. reuse, proposed P [dBm] W e i gh t e d - s u m r a t e ( b / s / H z ) w (2)3 = w (1)4 w (2)3 = 5 w (1)4 w (1)4 = 5 w (2)3 (b) Fig. 6: Average weighted-sum rate comparison for (a) w (2)3 = 5 w (1)4 ,and (b) w (2)3 = w (1)4 . reuse. As expected, the design with neither time-sharing norfrequency-reuse has inferior performance. (cid:3) Example 4: (Joint Routing and Resource Allocation: Gen-eralized Algorithm)
In this example, we evaluate the per-formance of the algorithm developed in Section IV-C. Weconsider a snapshot of a network with N = 5 nodes and L = 20 links (links with a distance more than 150 m areneglected). In this network nodes 3 and 4 wish to communicatewith nodes 2 and 1, respectively, over K = 4 subcarriers. Thenumber of simultaneous transmissions is set to I = 20 , I = 3 and I = 2 , which results in Γ with − , 190 and 110variables, respectively.The sum-rate yielded by the generalized algorithm withdifferent values of I is depicted in Figure 7. For comparison,this figure also shows the rates yielded by the special casein Section V. As can be seen from Figure 7, the algorithmwith I = 2 and 3 yields rates that are slightly less thanthe rate yielded by the algorithm with I = L , howeverwith a significantly less computational complexity. In fact,the complexity of the algorithm with I ≪ L is polynomial,whereas that of the algorithm with I = L is exponential in L .This feature renders the algorithm with I ≤ more attractivefor designing large networks with potentially rapid channelvariations. From Figure 7 it can be seen that the gap betweenthe rates yielded by different values of I decreases as thepower budget increases. This is because as power increases,interference becomes more severe, which causes the reuseof a particular subcarrier on multiple links less beneficial.It can be also seen from this figure that, most of the gainof frequency-reuse is mustered by only considering two or three simultaneous transmissions, i.e., I ≤ . This implies thatincreasing I trades complexity for performance. In particular,as I increases, the performance of the algorithm becomescloser to that of the one with I = L , but at the expense ofincreased complexity. PSfragreplacements without freq. reuse ( I = 1) limited freq. reuse ( I = 3) limited freq. reuse ( I = 2) complete freq. reuse ( I = 20) P [dBm] S u m -r a t e ( b / s / H z ) Fig. 7: Sum-rate generated by the generalized algorithm for differentvalues of I . In Figure 8(a) we investigate the convergence behaviourof the generalized algorithm. We consider an instance of anetwork in which the power budget of all nodes is set to P = 10 dBm. It can be seen from this figure that, in additionto being significantly less computationally demanding, thealgorithm with a lower value of I exhibits considerably fasterconvergence than that of the one with a higher value of I . Thisconvergence can be further ameliorated by choosing the initialpoint more carefully, for instance, by choosing this point tobe the solution yielded by algorithm in [5] for the case withno frequency-reuse.To illustrate the effect of random initialization of the gen-eralized algorithm, in Figure 8(b) the value of the objectiveto which the generalized algorithm with I = 2 convergedis shown for 80 random instances of feasible initial points, ( q (0) ℓk , Γ (0) ) ∈ [0 , P ] LK × [0 , | Γ | . It can be seen from thisfigure that although the algorithm is relatively sensitive to thechoice of the initial point, finding initial points that result in‘good’ local maxima is generally easy. (cid:3) Example 5: (Average Rate-Region Comparison)
In thisexample we provide the rate regions that can be achieved bythe algorithms in Sections V-C and IV-C, when P = 10 dBm.These regions are obtained by varying the weights ( s (1)4 , s (2)3 ) over the unit simplex, i.e., (cid:8) ( w (1)4 , w (2)3 ) | w (1)4 ≥ , w (2)3 ≥ , w (1)4 + w (2)3 = 1 (cid:9) , and are depicted in Figure 9. A compar-ison between these rate regions and the ones correspondingto the case when frequency-reuse is not considered [5] isalso provided in this figure. As can be seen from Figure 9,the rate region corresponding to the design with both time-sharing and frequency-reuse properly contains the rate regionscorresponding to the designs in which either time-sharing orfrequency-reuse is exclusively used. It can be also seen thatrestricting the number of simultaneous transmissions to be lessthan three suffices to achieve most of the frequency-reuse gainand with less computational complexity. (cid:3) VIII. C
ONCLUSION
In this paper we focused on the joint optimization ofdata routes, subcarrier schedules and power allocation in a PSfragreplacements final value (I=20)final value (I=2)each iteration (I=2)each iteration (I=20) experimentTop S u m -r a t e ( b / s / H z ) Number of iterations (a)
PSfragreplacementsfinalvalue(I=20)finalvalue(I=2)eachiteration(I=2)eachiteration(I=20)experimentTop S u m -r a t e ( b / s / H z ) Number of experiments (b)
Fig. 8: (a) Convergence behaviour and (b) performance of thegeneralized algorithm with different initial points.
PSfragreplacements w/o time-sharing, w. freq. reusew/o time-sharing, w/o freq. reuse, [5]w. time-sharing, w/o freq. reuse, [5]w. time-sharing, w. freq. reuse, I = 20 w. time-sharing, w. freq. reuse, I = 2 w. time-sharing, w. freq. reuse, I = 3 s (1)4 (b/s/Hz) s ( ) ( b / s / H z ) Fig. 9: Rate-region comparison. half-duplex multicarrier network when each subcarrier canbe reused by multiple links. The goal is to maximize aweighted-sum of the rates communicated over the network.The considered network is generic in the sense that it sub-sumes many structures including cellular and device-to-devicecommunications as special cases. We considered two instancesof this problem: 1) when each subcarrier can be time-sharedby multiple links; and 2) when time-sharing is not allowed anda subcarrier, once assigned to a set of links, will be used bythose links throughout the signalling interval. The joint designin the first instance results in superior performance but withhigh complexity. The second instance is a special case of thefirst one and can be parameterized using a significantly smallernumber of variables.The joint design problem in both instances is nonconvexand locally optimal solutions are obtained using a GP-basedmonomial approximation technique. Numerical results showthat the designs developed in both instances yield performancethat is significantly better than that of their counterparts inwhich frequency-reuse is not allowed. A
PPENDIX AT HE GP S
TANDARD F ORM AND M ONOMIAL A PPROXIMATION
1) The GP Standard Form:
For self-containment, in thisappendix we will review the standard GP form. A GP opti-mization problem can be readily transformed to an efficientlysolvable convex one. To provide the standard form of a GP,let z ∈ R n be a vector of positive entries. A monomialin z is defined to be a function of the form c Q i z α i i anda posynomial in z is defined to be a function of the form P Jj =1 c j Q ni =1 z α ij i , where c j > , { α i } and { α ij } , arearbitrary constants, j = 0 , , . . . , J , and i = 1 , . . . , n . Astandard GP [6], [7], [21] is an optimization of the form: min z f ( z ) , subject to f i ( z ) ≤ , i = 1 , . . . , m, (32) g i ( z ) = 1 , i = 1 , . . . , p, where { f i } are posynomials and { g i } are monomials.
2) Monomial Approximation:
A monomial approximationof a differentiable function h ( z ) ≥ near z (0) is given byits first order Taylor expansion in the logarithmic domain [6],[7]. Defining β i = z (0) i h ( z (0) ) ∂h∂z i (cid:12)(cid:12) z = z (0) , we have M ( h ( z )) = h ( z (0) ) Q ni =1 (cid:16) z i z (0) i (cid:17) β i , where M ( · ) is the monomial approxi-mation. This approximation will be used to provide local GPapproximations in the neighbourhood of a given initial point.A PPENDIX BP ROOF OF P ROPOSITION
Definition 1.
A function f : R n → R is said to be self-concordant if, for all x, v ∈ R n , s ∈ R such that x + sv is inthe domain of f and (cid:12)(cid:12)(cid:12) ∂ ∂s f ( x + sv ) (cid:12)(cid:12)(cid:12) ≤ ∂ ∂s f ( x + sv ) / . (cid:3) A. Proof of the Second Statement of Proposition 1
To determine the complexity of solving the problem in (30),we begin by converting this problem into a convex one. Usingstandard exponential transformations, we write t ( d ) n = exp (cid:16) ln(2) s ( d ) n (cid:17) , n ∈ N \ { d } , d ∈ D ,r ( d ) ℓk = exp (cid:16) ln(2) x ( d ) ℓk W (cid:17) , ℓ ∈ L , k ∈ K , d ∈ D ,y ℓk = exp( q ℓk ) , ℓ ∈ L , k ∈ K . (33)Substituting the variables in (30) with the ones in (33) andtaking the logarithm of the obtained objective and constraintsresult in a convex optimization which can be solved effi-ciently using the IPM technique. To use this technique, alog-barrier function is synthesized from the objective andinequality constraints. The complexity analysis of the IPM technique is simplified when the log-barrier function is self-concordant [21], cf., Definition 1. The log-barrier functioncorresponding to the convex form of (30) can be written as φ = − t X n X d w ( d ) n s ( d ) n + ψ, (34)where ψ represents the component of the log-barrier functionassociated with the inequality constraints in the convex formof (30). To examine whether φ is self-concordant, we note thatthe converted objective and the inequality constraints corre-sponding to (16), (17) and the relaxed versions of (24) and (25)are linear and therefore their corresponding components in thelog-barrier function are self-concordant [21]. Hence it remainsto consider the self-concordance for the constraints in (29)and (23). For simplicity, we write the posynomial constraintin (29) in the standard form in (32). After changing thevariables and taking the logarithm of both sides, this constraintcan be written in a general form as log (cid:0)X i exp( a i α i + b i β i + c i ) (cid:1) ≤ , (35)where { α i } , { β i } are the optimization variables and { a i } , { b i } , { c i } are constants. The component correspondingto the constraint in (35) in the log-barrier function can nowbe expressed as − log (cid:0) − log X i exp( a i α i + b i β i + c i ) (cid:1) . (36)To ensure that (36) is self-concordant, we introduce auxiliaryvariables, λ i , to bound the exponentially transformed variablesin (35). Using these new variables, the constraint in (35) canbe replaced with the following set of constraints [21]: X i λ i ≤ ,λ i ≥ ,a i α i + b i β i + c i − log λ i ≤ . (37)Now the associated log-barrier function of the constraintsin (37) can be shown to be self-concordant, cf., [21, Ex-ample 9.8]. For the constraints in (23), we follow the stepsanalogous to the ones used with the constraints in (29).In particular, by introducing new auxiliary variables, weconstruct a self-concordant log-barrier function. Using thisfunction, the complexity can be shown to be proportionalto m . , where m is the number of inequality constraints.Hence, the complexity of solving (30) can be bounded by O (cid:16) ( LK (3 L + 2) + N + D ( N − . (cid:17) , which completesthe proof of the second statement of Proposition 1. B. Proof of the First Statement of Proposition 1
The proof of the first statement of Proposition 1 followsfrom arguments similar to the one used in the proof of thesecond statement and is omitted for brevity. For the firststatement, the number of inequality constraints can be readilyverified to be LKN + N + D ( N −
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