DTIC ADA536086: Maximizing Capacity in Multi-Hop Cognitive Radio Networks Under the SINR Model

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Maximizing Capacity in Multi-Hop Cognitive
Radio Networks Under the SINR Model
Yi Shi, Member, IEEE, Y. Thomas Hou, Senior Member, IEEE, Sastry Kompella, Member, IEEE,
and Hanif D. Sherali
Abstract—Cognitive radio networks (CRNs) have the potential to utilize spectrum efficiently and are positioned to be the core
technology for the next-generation multi-hop wireless networks. An important problem for such networks is its capacity. We study
thisproblemforCRNsintheSINR(signal-to-interference-and-noise-ratio)model,whichisconsideredtobeabettercharacterization
ofinterference(butalsomoredifficulttoanalyze)thandiskgraphmodel.Themaindifficultiesofthisproblemaretwo-fold.First,SINR
isanon-convexfunctionoftransmissionpowers;anoptimizationproblemintheSINRmodelisusuallyanon-convexprogramand
NP-hardingeneral.Second,intheSINRmodel,schedulingfeasibilityandthemaximumallowedflowrateoneachlinkaredetermined
bySINRatthephysicallayer.Tomaximizecapacity,itisessentialtofollowacross-layerapproach;butjointoptimizationatphysical
(powercontrol),link(scheduling),andnetwork(flowrouting)layerswiththeSINRfunctionisinherentlydifficult.Inthispaper,wegive
amathematicalcharacterizationofthejointrelationshipamongtheselayers.Wedeviseasolutionprocedurethatprovidesa(1−ε)
optimal solution to this complex problem, where ε is the required accuracy. Our theoretical result offers a performance benchmark
foranyotheralgorithmsdevelopedforpracticalimplementation.Usingnumericalresults,wedemonstratetheefficacyofthesolution
procedureandofferquantitativeunderstandingontheinteractionofpowercontrol,scheduling,andflowroutinginaCRN.
IndexTerms—Theory,multi-hopcognitiveradionetwork,non-linearoptimization,SINRmodel,cross-layer,capacity.
(cid:70)
1 INTRODUCTION thatthereisasetof“commonchannels”availableatev-
COGNITIVE radio networks (CRNs) have great po- erynodeinthenetwork.Suchassumptionishardlytrue
for CRNs, where each node may have a different set of
tential to improve spectrum efficiency and will be
available frequency bands. These important differences
the core technology for the next-generation multi-hop
between MC-MR and CR suggest that algorithm design
wireless networks. Within such a network, each node is
for CRNs is substantially more complex than that for
equipped with a cognitive radio (CR) for wireless com-
MC-MR networks. In some sense, an MC-MR network
munications, which is capable of sensing the available
can be considered as a special case of a CRN.
frequency bands (i.e., those bands that are currently not
The capacity of a CRN is usually considered a key
used by primary users) and reconfiguring RF to switch
problem in fundamental understanding. In this paper,
to newly-selected frequency bands.
we aim to study this problem in the SINR (signal-to-
From wireless networking perspective, CR offers a
interference-and-noise-ratio) model. In this model, con-
whole new set of research problems in algorithm design
current transmissions are allowed and interference (due
andprotocolimplementation.Toappreciatesuchoppor-
to transmissions by non-intended transmitter) is treated
tunity, we compare CR with a closely related wireless
asnoise.AtransmissionissuccessfulifandonlyifSINR
technology called multi-channel multi-radio (MC-MR) [2],
at the receiver is greater than or equal to a threshold.
[20], [21]. First, MC-MR employs traditional hardware-
Moreover, the achieved transmission capacity is also a
based radio technology and thus each radio can only
function of SINR (via Shannon’s formula). The current
operate on a single channel at a time. In contrast, the
understanding is that the SINR model is better than the
radio technology in CR is software-based; a soft radio is
so-called “disk graph model” (or “protocol model” [15])
capable of switching frequency bands on a per-packet
for interference characterization.
basis. As a result, the number of concurrent frequency
Although the SINR model is thought to be more
bands utilized by a CR is typically much larger than by
realistic than the protocol model, a number of problems
MC-MR. Second, a common assumption for MC-MR is
arise when trying to carry out mathematical analysis
in it. First, SINR at a receiver not only depends on
• Y. Shi and Y.T. Hou are with the Bradley Department of Electrical thetransmissionpoweratthecorrespondingtransmitter,
and Computer Engineering, Virginia Polytechnic Institute and State
but also depends on the transmission powers at other
University,Blacksburg,VA,24061.E-mail:{yshi,thou}@vt.edu.
• S. Kompella is with Information Technology Division, U.S. transmitters.Mathematically,SINRisanon-convexfunc-
Naval Research Laboratory, Washington, DC, 20375. E-mail: tion of multiple variables. Many optimization problems
[email protected]. in the SINR model are non-convex problems and NP-
• H.D.SheraliiswiththeGradoDepartmentofIndustrialandSystemsEn-
hard. Second, since both scheduling feasibility and the
gineering,VirginiaPolytechnicInstituteandStateUniversity,Blacksburg,
VA,24061.E-mail:[email protected]. maximumallowedflowrateoneachlinkaredetermined
bySINR,anoptimalsolutiontomaximizecapacitymust
ManuscriptreceivedJune10,2009;revisedJan.3,2010andMarch27,2010;
acceptedJuly1,2010. be developed with joint consideration of network, link,
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Maximizing Capacity in Multi-Hop Cognitive Radio Networks Under the
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14. ABSTRACT
Cognitive radio networks (CRNs) have the potential to utilize spectrum efficiently and are positioned to be
the core technology for the next-generation multi-hop wireless networks. An important problem for such
networks is its capacity. We study this problem for CRNs in the SINR
(signal-to-interference-and-noise-ratio) model, which is considered to be a better characterization of
interference (but also more difficult to analyze) than disk graph model. The main difficulties of this
problem are two-fold. First, SINR is a non-convex function of transmission powers; an optimization
problem in the SINR model is usually a non-convex program and NP-hard in general. Second, in the SINR
model, scheduling feasibility and the maximum allowed flow rate on each link are determined by SINR at
the physical layer. To maximize capacity, it is essential to follow a cross-layer approach; but joint
optimization at physical (power control), link (scheduling), and network (flow routing) layers with the
SINR function is inherently difficult. In this paper, we give a mathematical characterization of the joint
relationship among these layers. We devise a solution procedure that provides a (1 ? ") optimal solution to
this complex problem, where " is the required accuracy. Our theoretical result offers a performance
benchmark for any other algorithms developed for practical implementation. Using numerical results, we
demonstrate the efficacy of the solution procedure and offer quantitative understanding on the interaction
of power control, scheduling, and flow routing in a CRN.
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and physical layers. Due to these difficulties, theoretical 2 RELATED WORK
results on CRNs in the SINR model remain limited.
There have been active research efforts on cross-layer
In this paper, we investigate the network capacity optimizationforwirelessnetworks.Manyoftheseefforts
problem for multi-hop CRNs in the SINR model. For are based on the disk graph model (see, e.g., [2], [18],
a given instance where each node has access to a set [20], [24] for wireless networks with traditional radios
of available bands (likely heterogeneous), we study the and [26], [28], [33], [34] for CRNs in particular). Under
networkcapacityproblemviaacross-layeroptimization this model, the impact of interference from neighboring
approach. In particular, we consider how to maximize nodes is solely determined by whether or not a node
the rates of a set of user communication sessions, with falls within the interference range of other transmit-
joint consideration at physical layer (via power control), ting nodes. The controversy surrounding (or arguments
link layer (via frequency band scheduling), and net- against) the protocol model is that a binary decision of
work layer (via flow routing). We give a mathematical whether or not interference exists (based on interference
characterization of these layers and formulate a mixed range)doesnotaccuratelycapturephysicallayercharac-
integer nonlinear program (MINLP) problem. For this teristics.Asaresult,theaccuracy(andvalidity)ofresults
optimizationproblem,wefirstidentifycoreoptimization developed under protocol model remains unclear.
variablesandcoreoptimizationspacebasedonthephys- In contrast, the SINR model is widely regarded as a
icalsignificanceofthevariables.Wedeviseanalgorithm better model for interference characterization. Although
based on the branch-and-bound framework to obtain a such model is preferred, there are many difficulties in
(1−ε) optimal solution. carrying out analysis with this model due to the com-
putational complexity SINR involves, particularly when
Although branch-and-bound framework is standard,
we study cross-layer optimization in a multi-hop envi-
many components within this framework are not spec-
ronment. As a result, many previous efforts were done
ified. We design the following problem-specific com-
onsingle-hopnetworks,e.g.,[3],[10],[13],[14],[17].For
ponents. (1) We propose a reformulation-linearization
multi-hop networks, various simplifications have been
technique(RLT)todevelopatightlinearrelaxationsoas
employed. For example, in [4], the authors assumed
to obtain a tight upper bound. (2) For the lower bound,
synchronized power control, where transmission power
we design a local search algorithm by analyzing and
ateachnodeinthenetworkisadjustablebutis“synchro-
removing infeasibility in the linear relaxation solution.
nized” (identical). Needless to say, such synchronization
(3) For problem partitioning, we propose to choose a
in power control cannot offer optimal network perfor-
partitionvariablebasedonitsphysicalsignificance.With
mance. There are also some efforts studying cross-layer
these problem-specific designs, the overall solution can
problems involving two layers instead of three layers
find a (1−ε) optimal solution much faster than brute-
(physical, link, and network, as we do in this paper).
force exhaustive search.
For example, in [6], Bhatia and Kodialam optimized
Our theoretical result offers a solution for a given powercontrolandrouting,butassumedsomefrequency
networkinstanceandcanserveasaperformancebench- hopping mechanism is in place for scheduling, which
mark for a multi-hop CRN. Such a result is not yet helps simplify joint consideration of scheduling. In [9],
available in the literature. If the available bands at Elbatt and Ephremides optimized joint power control
each node in the network varies on a relatively larger andscheduling,butassumedroutingwasgivenapriori.
time scale than the execution of our solution, then our In [27], Shu and Krunz studied how to maximize the
solution can be used to compute a (1−ε) optimal result total rate on all links in a CRN, with the consideration
for each time period during which the available bands ofpowercontrolandchannelassignment.Forcross-layer
ateachnodeisstatic.Iftheavailablebandsateachnode optimization in the SINR model involving three layers
in the network varies on a relatively smaller time scale, (physical, link, and network), nearly all existing efforts
thentheperformancebenchmarkcanbeobtainedoffline (e.g., [7], [8]) followed a “layer-decoupled” approach to
by our solution. One can use our result as a reference simplify analysis. Under such an approach, the final so-
performance benchmark to measure the quality of some lutionisobtainedbydeterminingalgorithm/mechanism
other proposed algorithms (possibly heuristic, despite foronelayeratatimeandthenpiecingupthemtogether
distributed) that are developed for actual deployment. instead of solving a joint optimization problem. Due to
de-coupling in the solution procedure, these approaches
The remainder of this paper is organized as follows.
are heuristic at best and cannot offer any performance
In Section 2, we review related work on cross-layer
guarantee.
optimization. Section 3 gives a mathematical character-
On another line of research, various efforts have been
ization of power control, scheduling, and routing in
made to study asymptotic behavior (or scaling laws) of
the SINR model for multi-hop CRNs. In Section 4, we
wireless networks (see, e.g., [1], [5], [15], [16], [19], [21],
reformulatetheoptimizationproblemandobtainaclean
[22], [29], [30], [31], [32]). These efforts differ from ours
and compact formulation. Section 5 analyzes the core
in this paper, which focuses on designing optimal cross-
optimization space and describes the main algorithm to
layer algorithms for a finite-sized network.
obtain (1−ε) optimal solution. In Section 6, we develop
tightupperandlowerboundsinsidethemainalgorithm.
We discuss how to interpret and apply our theoretical 3 MATHEMATICAL MODELS
result in Section 7. Section 8 presents numerical results Weconsideramulti-hopnetworkwithasetofCRnodes
and Section 9 concludes this paper. N. Each node i∈N senses its environment and finds a
3
set of available frequency bands M for the given time Consideratransmissionfromnodeitonodejonband
i
instance (i.e., those bands that are currently not used m. When there is interference from concurrent transmis-
by primary users), which may not be the same as the sions on the same band, SINR at node j, denoted as sm,
ij
available frequency bands at other nodes. We assume is
that the bandwidth of each frequency band (channel) is qm
g ijP
W. Denote M the union of all frequen(cid:83)cy bands among sm = (cid:80) ij(cid:80)Q max
allthenodes(cid:84)inthenetwork,i.e.,M= i∈N Mi.Denote ij ηW + mk∈∈NM,kk(cid:54)=i mh∈∈NM,hh(cid:54)=kgkjqQkmhPmax
M = M M , which is the set of frequency bands
ij i j (i,j ∈N,i(cid:54)=j,m∈M ), (3)
that is common between nodes i and j and thus can ij
be used for communication between these two nodes. where η is the ambient Gaussian noise density and g
ij
In the rest of this section, we present mathematical is the propagation gain from node i to node j.
characterization of each layer and obtain a formulation Note that in theory, for any small SINR, the corre-
for the capacity problem in this study. spondingcapacityisstillpositive(byShannon’scapacity
formula). But in practice, if SINR is too small, then the
achieved capacity will also be very small. In this case,
3.1 Power Control, Scheduling, and Their Relation-
such a weak link will not be very useful to carry traffic
shipintheSINRModel
flow.Thus,wemayuseathresholdtoremovesuchweak
Power control on each transmitting node at the physical links from considerations. In this regard, we introduce
layeraffectsSINRatareceivingnode.TheseSINRvalues a threshold for SINR, i.e., a transmission from node i
in turn will affect scheduling decision at link layer. That to node j on band m is considered successful if and
is, if a node is scheduled to receive, then its SINR must only if sm ≥ α. Then we have the following coupling
ij
be at least α (minimum requirement). Therefore, power relationship for scheduling (xm) and SINR (sm).
ij ij
controlandschedulingaretightlycoupledviaSINRand
cannotbemodeledseparately.Inthissection,weformu- xmij =1⇐⇒smij ≥α (i,j∈N,i(cid:54)=j,m∈Mij). (4)
late power control, scheduling, and their relationship in
3.2 RoutingandLinkCapacity
the SINR model.
Scheduling for transmission at each node in the net- WeassumethereisasetofLactiveusercommunication
work can be done either in frequency domain or time (unicast)sessionsinthenetwork.Denotes(l)andd(l)the
domain. In this paper, we consider scheduling in fre- source and destination nodes of session l ∈ L and r(l)
quencydomainintheformofassigningfrequencybands the minimum rate requirement (in b/s) of session l. In
(channels). Since time domain based formulation is sim- our study, we aim to maximize a common scaling factor
ilar to that for frequency domain, our approach can be K for all session rates. That is, we aim to determine the
extended to time domain based formulation and will maximumK suchthatarateofK·r(l)canbetransmitted
yield similar results. from s(l) to d(l) for each session l∈L in the network.
Tomaximizethecapacity,theremaystillbeconcurrent To route each data flow from its source node to its
transmissions within the same channel (and thus inter- correspondingdestinationnode,multi-hoprelayingmay
ference). Denote scheduling variables xm as follows. be necessary, due to limited transmission power at each
(cid:189) ij
node.Further,foroptimalityandflexibility,itisdesirable
1 If node i transmits data to node j on band m,
xmij = 0 otherwise. to allow flow splitting and multi-path routing. This is
becauseasingle-pathflowroutingforasessionisoverly
Due to interference, a node i can use a band m for restrictive and is unlikely to offer optimal solution.
transmitting to only a single node j or receiving from Mathematically, this can be modeled as follows. De-
only a single node k. That is, note f (l) the data rate on link i → j that is attributed
ij
to session l. If node i is the source node of session l, i.e.,
m(cid:88)∈Mk m(cid:88)∈Mj i=s(l), then
xm+ xm ≤1 (i∈N,m∈M ). (1)
ki ij i
k∈N,k(cid:54)=i j∈N,j(cid:54)=i M(cid:88)ij(cid:54)=∅
f (l)=r(l)K (l∈L,i=s(l)). (5)
ij
For power control, we assume that the transmission
j∈N,j(cid:54)=i
power at a node can be tuned to a finite number of
Ifnodeiisanintermediaterelaynodeforflowattributed
levelsbetween0andP .Tomodelthisdiscretepower
max to session l, i.e., i(cid:54)=s(l) and i(cid:54)=d(l), then
control,weintroduceanintegerparameterQthatrepre-
sents the total number of power levels to which a trans- M(cid:88)ij(cid:54)=∅ M(cid:88)ki(cid:54)=∅
mitter can be adjusted, i.e., transmission power can be f (l)= f (l)
ij ki
0, 1P , 2P ,···,P . Denote qm ∈ {0,1,2,···,Q}
Q max Q max max ij j∈N,j(cid:54)=i,s(l) k∈N,k(cid:54)=i,d(l)
the integer levels for transmission power. Clearly, when (l∈L,i∈N,i(cid:54)=s(l),d(l)). (6)
node i does not transmit data to node j on band m, qm
ij
Ifnodeiisthedestinationnodeofsessionl,i.e.,i=d(l),
is0.Thus,powercontrolandschedulingiscoupledwith
then
each other via the following relationship.
(cid:189) M(cid:88)ki(cid:54)=∅
qm ∈[1,Q] If xmij =1, (i,j∈N,i(cid:54)=j,m∈M ) (2) fki(l)=r(l)K (l∈L,i=d(l)). (7)
ij =0 otherwise. ij
k∈N,k(cid:54)=i
4
Inadditiontotheaboveflowbalanceequationsateach • The constraint described in (2) is not suitable for
node i ∈ N for session l ∈ L, the aggregated flow rates mathematical programming. We reformulate it with
oneachradiolinkcannotexceedthislink’scapacity.For the following linear constraint.
a link i→j, we have
xm ≤qm ≤Qxm (i,j∈N,i(cid:54)=j,m∈M ). (9)
ij ij ij ij
s(l)(cid:54)=(cid:88)j,d(l)(cid:54)=i (cid:88)
f (l)≤ W log (1+sm) It is easy to verify that this constraint is equivalent
ij 2 ij
to (2).
l∈L m∈Mij
A closer look at (9), in conjunction with (3) and (4),
(i,j ∈N,i(cid:54)=j,M (cid:54)=∅). (8)
ij suggeststhat(9)canbefurthersimplifiedtoremove
Thisconstraintshowsthecouplingrelationshipbetween redundancy. That is, we find that (3) and (4) yield
flow routing and SINR. xm ≤ qm and thus the left-hand-side (LHS) of (9)
ij ij
can be removed. To show this, we consider the two
cases for xm. For the case when xm = 0, xm ≤ qm
3.3 TheCapacityProblem ij ij ij ij
holds by the definition of qm. For the case when
ij
Inthispaper,westudyacapacityproblemformulti-hop xm =1,by(4),sm mustbepositive.Thenby(3),qm
ij ij ij
CRNs.Forcapacityproblem,thesimplestobjectiveisthe mustalsobepositive.Bythedefinitionofqm,qm ≥1
sum of throughput achieved by all sessions. However, ij ij
when it is positive. Therefore, qm ≥ xm. So (9) (or
such an objective may lead to biased rate allocation ij ij
(2)) can be replaced by the following constraint.
among sessions [23]. Another objective is maxmin, i.e.,
maximizing the minimum throughput among all ses- qm ≤Qxm (i,j∈N,i(cid:54)=j,m∈M ). (10)
sions.Maxminhasbeenusedinanumberofworks(e.g., ij ij ij
[15], [21]) to ensure fairness. In this work, we consider • Constraint(3)isintheformofafraction.Inamath-
how to maximize a common scaling factor K for all ematicalprogramming,productisabetterform.We
sessions under some given minimum rate requirements can re-write it as
r(l), i.e., what is the maximum factor K such that a rate
qm
of K ·r(l) can be transmitted from s(l) to d(l) for each g ijP
asemssoiorne gle∈neLrailnftohremnoeftwmoarxkm. Ninoitne tthhaetstehnisseotbhjeacttwivheeins smij = ηW +(cid:80)mk∈∈NM,kk(cid:54)=iij(cid:80)Qmh∈∈mNMa,xhh(cid:54)=kgkjqQkmhPmax
r(l) = 1 for each session l ∈ L, this objective becomes g qm
the maxmin throughput objective. On the other hand, = ηWQ +(cid:80)m∈Mkij (cid:80)ijm∈Mh g qm
when r(l) (cid:54)= 1, i.e., the minimum required rate for each Pmax k∈N,k(cid:54)=i h∈N,h(cid:54)=k kj kh
session l may be different from session to session, the (i,j ∈N,i(cid:54)=j,m∈Mij).
objective function becomes to maximize each session’s
This is equivalent to
rate proportional to its minimum rate requirement.
Putting together the objective and all the constraints
forpowercontrol,scheduling,andflowrouting,wehave ηWQsm+ m(cid:88)∈Mk m(cid:88)∈Mh g qmsm−g qm =0
the following formulation. P ij kj kh ij ij ij
max
k∈N,k(cid:54)=ih∈N,h(cid:54)=k
Max K
(i,j ∈N,i(cid:54)=j,m∈M ). (11)
ij
s.t. constraints (1)–(8)
Note that in (11), qm and sm are variables while all
kh ij
xm ∈{0,1},qm ∈{0,1,2,···,Q},tm,sm ≥0 othersymbolsareconstants.Thus,wehaveadouble
ij ij i ij
sum of nonlinear terms qmsm in (11). To reduce the
(i,j ∈N,i(cid:54)=j,m∈Mij) kh ij
number of nonlinear terms, denote
K,f (l)≥0 (l∈L,i,j∈N,i(cid:54)=j,i(cid:54)=d(l),j(cid:54)=s(l),M (cid:54)=∅).
ij ij
m(cid:88)∈Mh
Note that we assume Mi, the set of current available tmk = qkmh (k ∈N,m∈Mk). (12)
bands at node i, is given for a particular time instance.
h∈N,h(cid:54)=k
The solution to the above optimization problem will
offer the best performance for this given instance. Since Then (11) can be re-written as
M maychangeovertime,theoptimalsolutionmayalso
i
change over time. ηWQ m(cid:88)∈Mk
sm+ g tmsm−g qm =0
P ij kj k ij ij ij
max
k∈N,k(cid:54)=i
4 REFORMULATION
(i,j ∈N,i(cid:54)=j,m∈M ), (13)
ij
TheformulationinSection3.3isstillina“raw”formand
isthefirststepinsettingupourcross-layeroptimization which now only involves a single sum of nonlinear
problem. Much work needs to be done to put it into termstmk smij.Notethatbyintroducingnewvariables
a more “clean” and compact form that is amenable tm,wedecreasethenumberofnonlineartermsfrom
k
to mathematical operation. In this section, we analyze O(|N|4·|M|) in (11) to O(|N|3·|M|) in (13).
eachconstraintcarefullyandperformthisnecessaryand • Similar to (2), the constraint described in (4) is not
important reformulation step. suitable for mathematical programming. We now
5
Max K
(cid:80) (cid:80)
s.t. mk∈∈NM,kk(cid:54)=ixmki+ mj∈∈NM,jj(cid:54)=ixmij ≤1 (i∈N,m∈Mi)
qimj −Qxmij ≤0 (i,j∈N,i(cid:54)=j,m∈Mij)
(cid:80)
mj∈∈NM,jj(cid:54)=iqimj −tmi =0 (i∈N,m∈Mi)
(cid:80)
PηWmaQxsmij + mk∈∈NM,kk(cid:54)=igkjtmk smij −gijqimj =0 (i,j∈N,i(cid:54)=j,m∈Mij)
αxmij −smij ≤0 (i,j∈N,i(cid:54)=j,m∈Mij)
(cid:80)
Mj∈Nij(cid:54)=,j∅(cid:54)=ifij(l)−r(l)K=0 (l∈L,i=s(l))
(cid:80) (cid:80)
Mj∈Nij(cid:54)=,j∅(cid:54)=i,s(l)fij(l)− Mk∈kNi(cid:54)=,k∅(cid:54)=i,d(l)fki(l)=0 (l∈L,i∈N,i(cid:54)=s(l),d(l))
(cid:80) (cid:80)
sl∈(lL)(cid:54)=j,d(l)(cid:54)=ifij(l)− m∈MijWlog2(1+smij)≤0 (i,j∈N,i(cid:54)=j,Mij (cid:54)=∅)
xmij ∈{0,1},qimj ∈{0,1,2,···,Q},tmi ,smij ≥0 (i,j∈N,i(cid:54)=j,m∈Mij)
K,fij(l)≥0 (l∈L,i,j∈N,i(cid:54)=j,i(cid:54)=d(l),j(cid:54)=s(l),Mij (cid:54)=∅)
Fig.1. Problemformulation.
show that (4) can be eliminated once we have (10), 5.1 CoreVariablesandTheirOptimizationSpace
(13) and the following new constraint.
For the complex MINLP problem, its variables include
smij ≥αxmij (i,j∈N,i(cid:54)=j,m∈Mij). (14) xmij,qimj,tmi ,smij,fij(l), and K, which collectively con-
tribute a seemingly huge optimization space. However,
We now verify that (10), (13), and (14) can lead to a closer investigation of these variables show that they
thefollowingtworelationshipsdescribedin(4),i.e., are inter-dependent. In particular, we find that xm and
ij
xm =1⇒sm ≥α (i,j∈N,i(cid:54)=j,m∈M ) (15) qm variablesare“core”variablesandothervariablescan
ij ij ij ij
allbederivedbycorevariables.Asaresult,wecanfocus
sm ≥α⇒xm =1 (i,j∈N,i(cid:54)=j,m∈M ) (16)
ij ij ij ourstudyonanoptimizationspacebythecorevariables,
Firstweconsiderthecasewhenxm =1.By(14),we which is a much smaller space.
ij
have sm ≥α, i.e., (15) holds. Now consider the case We now show how to derive dependent variables
when sijm ≥ α. By (13), since tm’s are non-negative from core variables. For tmi and smij variables, they can
ij k be derived by (12) and (13), respectively, once xm and
variables and η,W,Q,Pmax,gkj,gij are all positive ij
constants, qimj must be positive. Then by (10), xmij qimj are given. For fij(l) variables and K variable, their
must be positive when qm is positive. Since xm can optimal values can be determined by a linear program
onlytake0or1byitsdefiijnition,xm mustbe1i.jThat (LP). That is, once we have the values for all xmij and
ij qm variables and we have computed the values for
is, (16) holds. ij
• Finally, we can prove that (7) is redundant once we all tmi and smij variables by (12) and (13), respectively,
have (5) and (6). Thus, we will leave (5) and (6) in the optimization problem reduces to a network flow
the formulation and remove constraint (7). problem, which is an LP.
With these careful reformulations, we now have a With this new understanding of optimization vari-
cleaner and more compact problem formulation, which ables,wecannowfocusoureffortsonthecorevariables
is shown in Fig. 1. xm and qm.
ij ij
5 SOLUTION OVERVIEW 5.2 MainAlgorithm
For the optimization problem in Fig. 1, Q,η,W,α,Pmax, Inthissection,wedescribeourmainalgorithm,whichis
gij, and r(l) are constants and K,xmij,qimj,tmi ,smij, and based on the branch-and-bound framework [25]. Those
f (l) are optimization variables. This formulation is a readers who are familiar with branch-and-bound can
ij
mixed integer non-linear program (MINLP), which is skip this section and go to Section 6, where we de-
NP-hardingeneral[11]andcannotbesolvedbyCPLEX. sign problem specific algorithms for each component of
In Section 5.1, we first analyze the intricate relationship branch-and-bound.
among the variables and identify the core variables Under branch-and-bound, we aim to provide a (1−ε)
among all the variables. We show that the dependent optimal solution, where ε is a small positive constant
variables can be derived once these core variables are reflecting our desired accuracy in the final solution. In
fixed. We call the optimization space for the core vari- case we set ε=0, an optimal solution can be obtained.
ables as the core optimization space. In Section 5.2, we We can start by developing upper and lower bounds
present the main algorithm on how to determine an (UB and LB) for the objective function (see Fig. 2(a)
optimal solution in the core optimization space. for an example). Branch-and-bound requires to develop
6
n n n
o o o
ncti UB=UB1 ncti ncti
u u u
e F e F UB=UB3 e F UB=UB2
ectiv ectiv UB2 ectiv UB4 UB5
Bounds for Obj LB=LB1 Bounds for Obj LB=LB2 LB3 Bounds for Obj LB2 LB=LB4 LB5
Original Problem 1 Problem 2 Problem 3 Problem 2 Problem 4 Problem 5
(a) Iteration1. (b) Iteration2. (c) Iteration3.
Fig.2. Illustrationofbranch-and-bound.
Initialization:
problem is updated as UB = max{UB ,UB } and the
1. Lettheinitialbestsolutionψε=∅andtheinitiallower 2 3
boundLB=−∞. lower bound is updated as LB = max{LB2,LB3}. As
2. Determineinitialvaluesetforeachcorevariable. a result, we now have a smaller gap between UB and
3. Initializetheproblemlisttoincludetheoriginalproblem, LB. Then we either have a (1−ε) optimal solution (if
denotedthisproblemasproblem1.
LB ≥(1−ε)UB) or continue to choose a problem with
4. ObtainanupperboundUB1 forproblem1.
Iteration: the maximum upper bound (Problem 3 in Fig. 2(b)) and
5. Selectproblemz thathasthemaximumUBz amongall perform partitioning for this problem. By choosing a
problemsintheproblemlist. problem with the maximum upper bound for partition,
6. UpdateupperboundUB=UBz. we can ensure that UB is decreased after each partition.
7. Findafeasiblesolutionψz withalowerboundLBz.
8. If(LBz >LB){ An important technique in branch-and-bound is that
9. Updateψε=ψz andLB=LBz. we can remove some problems from further consider-
10. IfLB≥(1−ε)UB,westopwitha(1−ε)optimal ation and thus reduce complexity. In particular, if we
11. sOotlhuetriownisψe,ε.removeallproblemsz(cid:48) withLB≥ find a problem z with LB ≥(1−ε)UBz (see problem 4
(1−ε)UBz(cid:48) fromtheproblemlist.} in Fig. 2(c)), we conclude that this problem can be re-
12. Buildtwonewproblemsz1 andz2 fromproblemz. movedfromfurtherconsiderationwithoutlossof(1−ε)
13. Removeproblemz fromtheproblemlist.
optimality.
14. ObtainUBz1 andUBz2 forproblemsz1 andz2.
Figure 3 shows the main algorithm. Since our core
15. IfLB<(1−ε)UBz1,addproblemz1 intotheproblemlist.
IfLB<(1−ε)UBz2,addproblemz2 intotheproblemlist. optimization space is finite (with finite number of core
16. Gotothenextiteration. variables xm and qm, and each core variable has a finite
ij ij
Fig.3. Mainalgorithm. integer value set), the branch-and-bound algorithm is
guaranteed to converge (even for ε = 0) [25]. Here,
branch-and-bound is much faster than brute-force ex-
problemspecificalgorithmsforfindingupperandlower
haustive search because non-improving problems are
bounds, which will be described in Section 6. After we
being removed during the process to avoid wasting
obtain the two bounds, we compare the gap between
precious cycle time in future computation. As a result,
them. If the upper and lower bounds are close to each
for all the network instances studied in Section 8, brute-
other, i.e., LB ≥ (1−ε)UB, then the feasible solution
force exhaustive search cannot find an optimal solution,
corresponding to the current lower bound LB is (1−ε)
while our algorithm can find near-optimal solutions for
optimal and we are done.
all network instances (with various network sizes and
Otherwise,weneedtofurthernarrowthegapbetween
user sessions).
UB and LB. To do this, branch-and-bound partitions
the original problem 1 into two new problems 2 and 3
(see Fig. 2(b)). This is accomplished by choosing an 6 DETERMINING BOUNDS AND PARTITIONING
appropriatecorevariablexmorqmanddividingitsvalue
ij ij PROBLEMS
set into two smaller sets. The choice of specific core
variables is important as its affects complexity. We will The main algorithm was presented in Fig. 3. Several
show how to do this in Section 6. components (i.e., determining upper and lower bounds,
After partitioning, the core optimization space is di- partitioning problems) in this main algorithm are yet to
vided into two sub-spaces for problems 2 and 3, re- be developed. These algorithms should exploit problem
spectively.WeagainobtainupperboundsUB andUB specific properties to optimize performance. In this sec-
2 3
and lower bounds LB and LB for problems 2 and 3, tion, we design algorithms for these components.
2 3
respectively. Since the optimization space of problems 2
and 3 are both smaller than that of problem 1, we
6.1 DeterminingUpperBound
can have tighter upper bounds, i.e., max{UB ,UB } ≤
2 3
UB , which in turn yield better lower bounds with To find an upper bound for a problem in branch-and-
1
max{LB ,LB } ≥ LB for our maximization problem bound (see lines 4 and 14 in Fig. 3), we propose to
2 3 1
(see Fig. 2(b)). Then the upper bound of the original construct a linear relaxation. That is to say, we linearize
7
Max K
(cid:80) (cid:80)
s.t. mk∈∈NM,kk(cid:54)=ixmki+ mj∈∈NM,jj(cid:54)=ixmij ≤1 (i∈N,m∈Mi)
qimj −Qxmij ≤0 (i,j∈N,i(cid:54)=j,m∈Mij)
(cid:80)
mj∈∈NM,jj(cid:54)=iqimj −tmi =0 (i∈N,m∈Mi)
(cid:80)
PηWmaQxsmij + mk∈∈NM,kk(cid:54)=igkjumijk−gijqimj =0 (i,j∈N,i(cid:54)=j,m∈Mij)
Linearconstraintsforumijk (i,j,k∈N,i(cid:54)=j,m∈Mij,m∈Mk)
αxmij −smij ≤0 (i,j∈N,i(cid:54)=j,m∈Mij)
(cid:80)
Mj∈Nij(cid:54)=,j∅(cid:54)=ifij(l)−r(l)K=0 (l∈L,i=s(l))
(cid:80) (cid:80)
Mj∈Nij(cid:54)=,j∅(cid:54)=i,s(l)fij(l)− Mk∈kNi(cid:54)=,k∅(cid:54)=i,d(l)fki(l)=0 (l∈L,i∈N,i(cid:54)=s(l),d(l))
(cid:80) (cid:80)
sl∈(lL)(cid:54)=j,d(l)(cid:54)=ifij(l)− m∈Mij lWn2cmij ≤0 (i,j∈N,i(cid:54)=j,Mij (cid:54)=∅)
Linearconstraintsforcmij (i,j∈N,i(cid:54)=j,m∈Mij)
tmi ,smij,cmij,umijk≥0 (i,j,k∈N,i(cid:54)=j,m∈Mij,m∈Mk)
K,fij(l)≥0 (l∈L,i,j∈N,i(cid:54)=j,i(cid:54)=d(l),j(cid:54)=s(l),Mij (cid:54)=∅)
(x,q)∈Ωz
Fig.5. Linearrelaxation.
cimj (tmk )U ·smij +(smij)L·tmk −umijk ≥(tmk )U ·(smij)L ,
III (tmk )L·smij +(smij)U ·tmk −umijk ≥(tmk )L·(smij)U ,
(tm) ·sm+(sm) ·tm−um ≤(tm) ·(sm) .
k U ij ij U k ijk k U ij U
II
cm=ln(1+sm) Forthenonlineartermlog (1+sm)= 1 ln(1+sm),we
ij ij 2 ij ln2 ij
IV propose to employ three tangential supports for ln(1+
sm), which is a convex hull linear relaxation (see Fig. 4).
ij
I Suppose that we have the bounds for sm, i.e., (sm) ≤
ij ij L
sm ≤(sm) .Weintroduceavariablecm =ln(1+sm)and
ij ij U ij ij
considerhowtogetalinearrelaxationforcm.Thecurve
sm ij
(simj)L β (simj)U ij ocofncmivje=x hlnu(l1l)+, wsmihj)erceansebgembeonutnsdI,edII,baynfdouIIrIsaergemtaenngtse(notriaal
Fig.4. Aconvexhullforcm =ln(1+sm). supports and segment IV is the chord (see Fig. 4). In
ij ij
particular, the three tangent segments are tangential at
allconstraintsinFig.1sothattherelaxedproblemcanbe points (1+(sm) ,ln(1+(sm) )), (1+β,ln(1+β)), and
ij L ij L
solvedbyanLP.Thissolutionprovidesanupperbound.
(1+(sm) ,ln(1+(sm) )), where
terNmost.eFthoartpinroFdigu.c1t,ttmkmssmimj,anwdeloagp2p(1ly+samij)noarveelnomnelitnheoadr β = [1+ij(sUmij)L]·[1+(simijj)UU]·[ln(1+(smij)U)−ln(1+(smij)L)]−1
k ij (smij)U −(smij)L
basedon Reformulation-Linearization Technique(RLT)[25].
This is done by introducing a new variable um =tmsm is the horizontal location for the point that is intersected
ijk k ij
by extending segments I and III; segment IV is the
andaddlinearconstraintsforthenewvariable.Suppose
t(mksmija)Lnd≤ssmimijj ≤are(sbmijo)uUn,dreesdpebcytiv(temkly).LTh≤ust,mkw≤e h(atvmke)U and (s1eg+m(seminjt)Uth,alnt(j1o+ins(spmijo)iUn)t)s.(T1h+e(csominj)vLe,xlnre(1gi+on(sdmije)fiLn)e)danbdy
thefoursegmentscanbedescribedbythefollowingfour
[tmk −(tmk )L]·[smij −(smij)L]≥0, linear constraints.
[tmk −(tmK)L]·[(smij)U −smij]≥0, [1+(smij)L]·cmij−smij≤[1+(smij)L]·[ln(1+(smij)L)−1]+1,
[(tm) −tm]·[sm−(sm) ]≥0, (1+β)·cmij −smij ≤(1+β)·[ln(1+β)−1]+1,
k U k ij ij L
[1+(sm) ]·cm−sm≤[1+(sm) ]·[ln(1+(sm) )−1]+1,
[(tm) −tm]·[(sm) −sm]≥0. ij U ij ij ij U ij U
k U k ij U ij [(sm) −(sm) ]·cm+[ln(1+(sm) )−ln(1+(sm) )]·sm
ij U ij L ij ij L ij U ij
Substituting umijk = tmk smij, we have the following linear ≥(smij)U ·ln(1+(smij)L)−(smij)L·ln(1+(smij)U).
constraints for um .
ijk As a result, the nonlinear ln (or log) term is relaxed into
(tm) ·sm+(sm) ·tm−um ≤(tm) ·(sm) , linear constraints.
k L ij ij L k ijk k L ij L
8
Initialization:
After relaxing all nonlinear terms for a problem, say
problemz,wehavearelaxedproblemzˆinFig.5,which 1. Setxmij =(xmij)L andqimj =(qimj)L.
2. Computetheratioλij by(17)foreachlinki→j and
is an LP. In problem zˆ, x and q are the vectors for all denoteλmin=min{λij :i,j∈N,i(cid:54)=j,Mij (cid:54)=∅}.
xm and qm variables, respectively, (xm) ,(xm) ,(qm) , Iteration:
ij ij ij L ij U ij L
and (qm) are constant bounds, and Ω = {(x,q) : 3. Supposeλij =λmin.
ij U z 4. Ifwecanincreaseqm onausedband{
(xmij)L ≤ xmij ≤ (xmij)U,(qimj)L ≤ qimj ≤ (qimj)U} is the core 5. Supposebandmijhasthelargestqimj valuesinsolution
optimization space of (x,q). LP(zˆ)amongthesebands.
A relaxed problem zˆ can be solved by an LP in 6. Increaseqimj suchthatqimj ≤(qimj)U andforanyother
polynomialtime.DenoteitssolutionasLP(zˆ).Thisgives 7. linkk→h,theirnewlyupdatedλkh>λmin.}
8. else,ifwecanincreaseqm onanavailableandunused
us an upper bound to problem z. ij
band{
9. Supposebandmhasthelargestqm valuesinsolution
ij
6.2 DeterminingLowerBound LP(zˆ)amongthesebands.
10. Increaseqimj suchthatqimj ≤(qimj)U andforanyother
To find a lower bound for problem z, it is sufficient to 11. linkk→h,theirnewlyupdatedλkh>λmin.
find a feasible solution to this problem. By feasible solu- 12. Ifqm increases,thenxm=1.}
ij ij
tion,wemeanthatitsatisfiesallconstraintsforproblem 13. elsetheiterationterminates.
zinFig.1,despitethattheobjectivevaluecorresponding
Fig.6. Pseudocodeofproposedlocalsearchalgorithm.
to the feasible solution may not be optimal (maximum).
Although any feasible solution to problem z can serve λ ·K, where K is the objective value in the relaxed
min
as a lower bound, we strive to find one feasible solution solution LP(zˆ).
that can offer a tight lower bound. We can find such In the next iteration, we aim to improve the current
a feasible solution (denoted as ψz) based on LP(zˆ) by solution. Note that if we can increase λmin, then the
searching its neighborhood, which we call local search. current solution is improved. Suppose link i → j is the
The local search algorithm begins with an initial fea- link with λ = λ . To increase λ , we try to increase
ij min ij
sible solution. Such a solution may be far away from transmission power qm on some band m under the
ij
the optimum and may not provide a tight lower bound. constraintqm ≤(qm) .BasedontheconstraintsinFig.1,
Thus,wewilliterativelyimprovethecurrentsolutionto ij ij U
wemayupdatethevaluesofothervariablestomaintain
achieve a better lower bound. Until we can no longer
feasibility. For example, by the first constraint in Fig. 1,
improve (increase) the lower bound, we are done.
we need to increase xm from 0 to 1 if qm is increased
To obtain an initial feasible solution, we set xm = ij ij
ij form 0 to a positive value. Moreover, as a consequence
(xm) forschedulingandqm =(qm) forpowercontrol.
ij L ij ij L of increased qm, the interference to other transmissions
Then we can compute SINR value sm by (3). When an ij
ij on band m is increased and thus the achieved capacities
SINR value is larger than or equal to α, the achieved for other links are decreased. Thus, qm can be success-
capacity is W log (1+sm). Otherwise (i.e., SINR < α), ij
2 ij fully increased only if for any other link k → h, its
the transmission is considered unsuccessful. Note that
updated λ will not fall below the current λ . If the
kh min
although the flow rates f (l) in the relaxed solution
ij current solution can be improved (with a larger λ ),
min
LP(zˆ) guarantee flow balance at each node, such flow then we continue to the next iteration of improvement.
rates may exceed the capacities on some links under Otherwise, the local search algorithm terminates. The
the initial xmij and qimj values. To find feasible flow rates pseudocode of our local search algorithm is given in
under current xm and qm, we compare the achievable Fig. 6.
ij ij
link capacity (under current xm and qm values) to the
ij ij
aggregated flow rates f (l) on each link i → j by
ij 6.3 PartitioningProblem
computing the ratio between the two (denoted as λ )
ij
as follows. Our proposed partitioning approach differs from that
(cid:80) in standard branch-and-bound procedure. In standard
W log (1+sm)
λij = m(cid:80)∈Ms(li)j(cid:54)=j,d(l)(cid:54)=2if (l)ij . (17) binraFnigch.-3a)nids-dboonuendbypcrhooceodsiunrge,apvaarrtiiatibolneiwngith(steheellianrege1s2t
l∈L ij
relaxationerrorandusesitsvalueintherelaxedsolution
If λij < 1 for some link i → j, then the aggregated LP(zˆ) to divide its value set into two smaller sets. The
flow rates exceed the link capacity and the link capacity reasonofthisapproach(withthelargestrelaxationerror)
constraint on i → j is violated. In this case, we need to is that such a variable is likely to lead to a larger gap
scale down the flow rates on link i → j (to satisfy link between upper and lower bounds. Thus, we should
capacity constraint) and the flow rates on all other links partition its value set such that the relaxation error will
(to maintain flow balance in the network) by a value becamesmaller. This division (on value set) also divides
λ ≤ λij. On the other hand, we want to have a λ as the optimization space for problem z into two smaller
large as possible so as to maximize the scaling factor spaces, which result in two new problems z and z ,
1 2
(our objective). Such a value is the bottleneck value λij respectively.
among all links (denoted as λmin = min{λij : i,j ∈ Such standard partitioning technique, however, does
N,i (cid:54)= j,Mij (cid:54)= ∅}). We now have a complete solution not explore any problem specific property on choosing
λmin ·fij(l), (xmij)L, (qimj)L for routing, scheduling, and partition variables. We find that if we weigh the sig-
power control, respectively. The achieved objective is nificance of each variable when choosing a partitioning