energy efficiency optimization for mobile ad hoc networks

Received January 28, 2016, accepted February 12, 2016, date of publication March 8, 2016, date of current version March 21, 2016.

Digital Object Identifier 10.1109/ACCESS.2016.2538269

Energy Efficiency Optimization forMobile Ad Hoc NetworksWEN-KUANG KUO1 AND SHU-HSIEN CHU21Department of Electrical Engineering, Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan 701, Taiwan2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA

Corresponding author: W.-K. Kuo ([email protected])

This work was supported by the Ministry of Science and Technology, Taiwan, under Contract 104-2221-E-006-108-.

ABSTRACT Tremendous traffic demands for ubiquitous access and emerging multimedia applicationssignificantly increase the energy consumption of battery-powered mobile devices. This trend leads tothat energy efficiency (EE) becomes an essential aspect of mobile ad hoc networks (MANETs). In thispaper, we explore EE optimization as measured in bits per Joule for MANETs based on the cross-layerdesign paradigm. We model this problem as a nonconvex mixed integer nonlinear programming (MINLP)formulation by jointly considering routing, traffic scheduling, and power control. Because the nonconvexMINLP problem is NP-hard in general, it is exceedingly difficult to globally optimize this problem.We, therefore, devise a customized branch and bound (BB) algorithm to efficiently solve this globally optimalproblem. The novelties of our proposed BB algorithm include upper and lower bounding schemes andbranching rule that are designed using the characteristics of the nonconvexMINLP problem.We demonstratethe efficiency of our proposed BB algorithm by offering numerical comparisons with a reference algorithmthat uses the relaxation manners proposed in [1]–[3]. Numerical results show that our proposed BB algorithmscheme, respectively, decreases the optimality gap 81.98% and increases the best feasible solution 32.79%compared with the reference algorithm. Furthermore, our results not only provide insights into the designof EE maximization algorithms for MANETs by employing cooperations between different layers but alsoserve as performance benchmarks for distributed protocols developed for real-world applications.

INDEX TERMS Energy efficiency, MANET, cross layer, optimization, branch and bound.

I. INTRODUCTIONA MANET is a self-organizing set of mobile devices thatcommunicate with one another across multiple hops in a dis-tributed manner. Because of the widespread use of cheaper,smaller, and more powerful portable devices, MANETshave become a promising and growing technique. Withrecent advances in information and communication tech-nology (ICT), MANETs are able to support high networkcapacity and proliferating multimedia services, such as videoon-demand, surveillance, remote education, and health mon-itoring, etc. MANET traffic produced for ubiquitous accessand multimedia applications with quality of service (QoS)requirements considerably increases energy exhaustion ofmobile devices. Energy is a scarce resource for mobiledevices, which are typically driven by batteries with limitedcapacities. Further, progress in battery technology is slowand expected to improve little in the near future [4]. Undersuch critical conditions, optimal EE design that concentrateson the most economical ways of utilizing mobile device

energy while ensuring proper network operations is an urgentrequirement for MANETs.

EE optimization of mobile communication systems hasreceived much attention in the literature. For instance, in [5],the authors optimized link-level EE of the wireless net-work under static and time-variant fading channels. In [6],the authors studied link-adaptive transmission for max-imizing the EE of the orthogonal frequency divisionmultiplexing (OFDM) system by presenting an energy-efficient water-filling power allocation algorithm. In [7], theauthors introduced channel selection and power allocationmechanisms to optimize the EE of a distributed cognitiveradio network where the transmitter directly sent data tothe receiver (i.e., a single-hop network). In [8], the authorsused game theory to develop multiuser detection and powercontrol methods to optimize EE for each user in a wirelessnetwork. In [9], the author designed a noncooperative gamewhere each user in a mobile network chooses its transmissionpower and rate to maximize the EE, while guaranteeing the

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W.-K. Kuo, S.-H. Chu: EE Optimization for MANETs

QoS requirements. In [10], the author proposed a power con-trol algorithm using noncooperative game theory to enablemultimedia transmission over a wireless network and max-imize the EE for each user. In [11]–[14], the authors dis-cussed resource allocation methods to optimize EE of cellularnetworks without routing capabilities. Although [5]–[14]examine EE optimization for different wireless communica-tion systems, they consider only physical (PHY) and linklayers but neglect network layer issues. Obviously, resultsshown in [5]–[14] are inapplicable to MANETs because oneof the most important features of MANETs is to provideroutable networking environments.

In [15], the authors presented an analytical manner forcomputing EE of the MANET by taking PHY and networklayers into account but ignored link layer issues. In [16], theauthors maximized EE for the MANET using cooperativemulti-input-single-output transmissions. They handled multi-hop routing by proposing algorithms for selecting hop dis-tance and the number of cooperating nodes around each relaynode. Nevertheless, they disregarded PHY and link layerproblems. As discussed in [4], cross-layer optimization cansubstantially enhance EE by designing resource allocationmechanisms that exploit the cooperations between differentlayers to adapt to variations of services, traffic, and envi-ronments. Although concentrating on the EE optimizationproblems in MANETs, [15], [16] ignore the consequentialinterdependencies between different layers of the entire net-work. Hence, their approaches only provision suboptimalsolutions that, in some cases, are likely to be far off from theglobal optimum. To the best of our knowledge, maximizingEE of the MANET by jointly considering the PHY, link, andnetwork layers has yet to be researched.

To fill this gap, we investigate this cross-layer optimizationproblem in this paper. We consider a set of communica-tion sessions, in which each session has its own peak andminimum sustained rate demands in a time-slotted MANET.The source of each session generates data relayed to thedestination though multihop routing. We define EE of theMANET as the total session rate divided by the aggregatepower consumption of active nodes over the scheduling timeperiod. Overall, we propose a cross-layer optimization frame-work to maximize EE by jointly computing routing path,transmission schedule, and power control corresponding tothe network, link, and PHY layers, respectively. The routingproblem involves how to choose the set of paths to routedata from the source to the destination for each session. Fortransmission schedule, the problem involves determining theset of nodes that are active in each time slot. The powercontrol problem is to specify the transmission power of eachactive node in each time slot. We formulate this problemas nonconvex MINLP problem (P1), which is NP-hard ingeneral. Challenges of globally optimizing (P1) arise from thecombinatorial property, and the nonconvexities of the linearfractional objective function and bilinear products appearingin its constraints.

To address these issues, we developed a novelBB algorithm to globally optimize (P1) by exploiting itsspecific nature. Key innovations of our proposed BB algo-rithm are as follows. First, to obtain upper bounds (UBs),we employ recent advances in piecewise linear relax-ations (PLRs) of bilinear terms and piecewise convexhulls (PCHs) of log functions to build a relaxed mixed integerlinear fractional programming (MILFP) model. Nevertheless,the MILFP is still a nonconvex MINLP and mathematicallyintractable [17], [18]. Therefore, we transform it into anequivalent mixed integer linear programming (MILP) prob-lem using the scheme proposed by [17]. This allows us toapply effective MILP techniques to solve the MILFP withglobal optimality. Second, to compute lower bounds (LBs),we keep link activations and power allocations of (P1) fixedat their optimal values from the UB problem. We then trans-form (P1) into an equivalent linear programming (LP) model.We can obtain a valid LB by combining optimal solutionof the LP problem with outcomes of link activations andpower allocations gained by solving the UB problem. Third,for problem partition, we design a branching rule that candecrease relaxation error of the UB problem to the extentpossible.

We compared our proposed BB algorithm with a refer-ence BB algorithm using the relaxation manners suggestedin [1]–[3]. Numerical results show that our algorithm sig-nificantly outperformed the reference algorithm in terms ofcomputational complexity. The contributions of the paperare: First, our theoretical outcome provides a solution todetermine the optimal EE of the MANET by exploitingthe cross-layer design principle. Second, in the real world,it still deserves to design distributed algorithms and protocols.However, to the extent of our knowledge, there is no any tech-nique that can optimize the nonconvex MINLP problem in adistributed manner. This results in that distributed algorithmsand protocols are developed using heuristic or approximationalgorithms, or even do not take EE into consideration (suchas OLSR [19], DSR [20], DSDV [21], AODV [22], andTORA [23] etc.). The common disadvantage of heuristic andapproximation algorithms is that they are unable to providethe theoretical guarantee for acquiring the global optimalsolution.Hence, the current distributed algorithms and proto-cols cannot achieve the optimal EE operation for the MANETby fully utilizing the cooperations between different layers.Moreover, it has never been studied how far the distributedalgorithms and protocols perform from the optimal EE solu-tion, and how to enhance their efficiencies. In this work,we solve these problems by providing theoretical resultsthat can furnish performance benchmark comparisons, andenable researchers to gauge the effectiveness of distributedalgorithms and protocols. Furthermore, our analyses providevaluable insights into not only the impact of routing strategy,transmission schedule, and power control on EE, but alsointo the design of novel algorithms and protocols aiming toachieve high EE for the MANET.

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The remaining paper is organized as follows.In Section 2, we describe the mathematical model forEE optimization of MANETs by jointly considering PHY,link and network layers. In Section 3, we develop a novelBB algorithm using the characteristics of the model formu-lated in Section 2. We also explain the details of the upperand lower bounding schemes and branching technique of theproposed BB algorithm. Section 4 presents the results of com-putational experiments. In these experiments, we elaboratethe computational efficiency of the proposed BB algorithm.We also discuss the impacts of power control, traffic schedul-ing and routing on the design of EE optimization protocolsand algorithms for MANETs. Finally, Section 5 concludesthe paper.

II. EE OPTIMIZATION PROBLEMWe consider a MANET comprised of one set of stationarynodes N connected by a set L of links. We consider everylink l = nt → nr to be directional, where nt and nr are thetransmitter and receiver of l, respectively.We assume channeltime is divided into time slots of equal length, and every nodeis only equipped with one transceiver. Thus, no node cansend and receive in the same time slot; further, it cannot sendto or receive from multiple nodes simultaneously. In otherwords, all nodes operate in the half-duplex mode. There areS sessions representing different end-to-end traffic demandswithin the MANET. We denote every session s as (ns, nd , rs),where ns, nd , and rs are the source node, destination node,and average transmission rate of s, respectively. To guaranteethe required QoS, we assume rs is within range [rmins , rmaxs ]for each session s, where rmins and rmaxs are two pre-specifiedQoS parameters, namely, minimum sustained rate and peakrate, respectively. Aside from being a source or destinationof a session, we assume every node can relay data for othersessions and act as a router. Our goal is to maximize EE of theentire networkwhile supportingQoS requirements of individ-ual sessions in terms of power control, traffic scheduling, androute assignment.We describe the details of ourmathematicalmodel and formulate the EE optimization problem of theMANET below.

A. MATHMATICAL MODEL FOR THEEE OPTIMIZATION PROBLEMFor every link l at every time slot t , we define binaryvariable x lt as

x lt =

{1; if link l is allowed to trasnmit at time slot t0; otherwise

(∀l ∈ L, t ∈ 0), (1)

where 0 = {1, . . .,T } and T is the total number of sched-uled time slots. We assume transmission power on link l attime slot t , i.e., plt , is continuously adjusted in given interval[0, pmax]. Following the same definition of [2] and [3],we define 0 ≤ plt ≤ pmax if link l is allowed to transmit during

time slot t and plt = 0, otherwise. We thus have the following

constraint

plt

{∈ [0, pmax], if x lt = 1= 0, if x lt = 0

(∀l ∈ L, t ∈ 0). (2)

Note that being allowed to transmit does not necessarilymean a transmission actually occurs, which is decided by theoptimization algorithm.

As every node is equipped only with a transceiver, it mustadhere to half-duplex data transmission. Thus, in every timeslot, at most only one of a node’s incoming and outgoinglinks is permitted to transmit. To characterize this constraint,we have ∑

l∈IL(i)∪OL(i)x lt ≤ 1 (∀i ∈ N , t ∈ 0), (3)

where IL (i) = {k → i|k ∈ N } and OL (i) = {i→ j|j ∈ N }are sets of links whose receiver and transmitter are both i,respectively.

The quality of a wireless link depends on the signal tointerference and noise ratio (SINR). We express SINR oflink l at time slot t as

SINRlt =hlplt

σW+∑

l′∈L,l′ 6=l h(TX l′ ,RX l )pl′t

(∀l ∈ L, t ∈ 0),

(4)

where hl is the propagation gain of link l, W is the channelbandwidth, σ is the thermal noise density, and h(TX l′ ,RX l)is the propagation gain from the transmitter of link l ′

(i.e., TX l′ ) to the receiver of link l (i.e., RX l). Using theShannon capacity theorem, we can calculate the capacity oflink l at time slot t as W log2

(1+ SINRlt

)(∀l ∈ L, t ∈ 0).

In the MANET, a source node generates data transmittedto its destination node, which forms a session. Owing to thelimited transmission power of a node, it may be necessaryto route data through multiple intermediate nodes to easetransmission over a long distance. To provide better rout-ing flexibility and reliability, we adopt flow splitting andmultipath routing. Specifically, every node can separate itsincoming traffic into subflows which are then transmittedto varied next-hop nodes. Based on the above description,we have the following flow conservation constraints. Let f ls,tdenote the flow rate of session s on link l at time slot t .If node i is the source node of session s, then∑

l∈OL(i),t∈0f ls,t = rsT (∀i ∈ N , s ∈ IS (i)), (5)

where IS(i) is the set of sessions whose source node is i. Whenboth sides of (5) are multiplied by the duration of time slot,the left hand side yields the total amount of traffic (in bits)generated by the source node and the right hand side gives theaverage session rate multiplied by the scheduling time period.Evidently, both quantities must be equal.

If node i is an intermediate node of sessions, then we have∑l∈IL(i),t∈0

f ls,t =∑

l∈OL(i),t∈0f ls,t

(∀i ∈ N , s ∈ S − (IS (i) ∪ OS (i))), (6)

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where OS(i) is the set of sessions whose destination node is i.(6) implies that for every session, the net amount of incomingand outgoing traffic must be equal at each relay node.

In [1]–[3], the authors have verified that both (5) and (6)guarantee the flow balance equation at the destination nodeof session s(∀s ∈ S). Thus, we omit flow conservationconstraints of destination nodes for all sessions.

Furthermore, at every time slot, the total amount of trafficfrom different sessions on a particular link cannot exceed thelink’s capacity. Therefore, we have the following constraint∑

s∈Sf ls,t ≤ W log2

(1+ SINRlt

)(∀l ∈ L, t ∈ 0). (7)

Finally, to support QoS requirements, we assume rs of eachsession s must satisfy

rmins ≤ rs ≤ rmaxs (∀s ∈ S) (8)

Our purpose is to maximize EE of the MANET, whichis defined as the total session rate divided by the powerconsumed by all nodes during the scheduling time period.This definition equals the amount of data transferred by allsource-destination pairs divided by the energy consumed bythe entire MANET. Thus, the objective function is shown as

EE =

∑s∈S rs∑L

l=1∑T

t=1 plt

(bits/Joule) (9)

We express the resulting EE optimization model (P) asfollows

Max

∑s∈S rs∑L

l=1∑T

t=1 plt

(P)

subject to constraints (2)-(8), x lt ∈ {0, 1},

0 ≤ plt ≤ pmax (∀l ∈ L, t ∈ 0) ; rs ≥ 0 (∀s ∈ S);

f ls,t ≥ 0 (∀l ∈ L, t ∈ 0, s ∈ S).

B. REFINEMENTS TO OUR MATHEMATICAL MODELAlthough we have successfully modeled (P), it is only a prim-itive representation and unsuitable for mathematical treat-ments. To make formulations of (P) more concise and easierto manipulate, we discuss necessary modifications and rede-velop an equivalent model that is mathematically tractable asfollows.

First, we reformulate (2) as (10), because (2) is inapplica-ble to mathematical programming.

0 ≤ plt ≤ xlt pmax (∀l ∈ L, t ∈ 0). (10)

One can easily prove that (2) and (10) are equivalent.Second, because the product form is easier to handle than

the fractional form, we rewrite (4) as follows

σWSINRlt +∑

l′∈L,l′ 6=lh(TX l′ ,RX l)p

l′t SINR

lt = hlplt

(∀l ∈ L, t ∈ 0). (11)

We notice that (11) has nonconvex bilinear products(i.e., pl

′

t SINRlt ) that cause (P) presenting amultiplicity of local

optima [24], [25]. To facilitate mathematical manipulations,

we further curtail the number of bilinear terms. We define I ltsymbolizing interference of link l at time slot t as

I lt =∑


l′t (∀l ∈ L, t ∈ 0). (12)

Next, we reformulate (11) as

σWSINRlt + Ilt SINR

lt = hlplt (∀l ∈ L, t ∈ 0). (13)

Comparing (11) with (13), we discover that the number ofbilinear products has been diminished from O(

∣∣N 4∣∣ |T |) to

O(∣∣N 2

∣∣ |T |).With the above modifications, we redevelop an equivalent

model (P1) for (P) as

Max

∑s∈S rs∑L

l=1∑T

t=1 plt

(P1)

subject to constraints (3), (5)-(8), (10), (12), (13),

x lt ∈ {0, 1}, 0 ≤ plt ≤ pmax , 0 ≤ SINR

lt ≤ hlpmax/σW ,

0 ≤ I lt ≤ pmax∑

l′∈L,l′ 6=lh(TX l′ ,RX l) (∀l ∈ L, t ∈ 0);

rs ≥ 0 (∀s ∈ S); f ls,t ≥ 0 (∀l ∈ L, t ∈ 0, s ∈ S).

By inspecting (P1), we observe that to globally optimizeit, three major challenges need to be overcome: (1) there arebinary variables with combinatorial nature, (2) the objectivefunction is a nonconvex linear fractional function [17], [18],and (3) the bilinear terms are also nonconvex. These prop-erties cause (P1) becoming a nonconvex MINLP which isNP-hard in general and thus incredibly difficult to solve.To settle this issue, we propose a novel BB algorithm forsolving (P1) with global optimality in Section 3.

III. OUR PROPOSED BB SOLUTION PROCEDUREThe BB algorithm, which consists of several components,defines a common framework to solve a wide class of opti-mization problems. The framework itself provides flexibilityin the sense that for some its key components, one can designcustomized algorithms based on the problem’s structure andproperties. Because performance of the BB process criticallyhinges on these customized algorithms’ efficiencies [26],focusing on the design of these algorithms is absolutely cru-cial. In the following subsections, we present the proposedBB procedure and problem-specific algorithms to its impor-tant elements developed by employing characteristics of (P1).

A. THE MAIN ALGORITHMThe idea underlying the BB procedure is based on the ‘‘divideand conquer’’ paradigm, which begins by considering theoriginal problem with the entire feasible region. The BB pro-cess performs lower-bounding and upper-bounding methodsto the original problem to find LB and UB of the globaloptimum. If the gap between the bounds is within tolerance ε,then we have achieved an ε-optimal solution and the pro-cedure terminates. Here, tolerance ε is a sufficiently smallpositive constant that signifies the required precision of theultimate solution. Otherwise, we partition the feasible region

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into two disjoint subregions, which together comprise thecomplete feasible region. Each of these subregions formsa new nonconvex MINLP model with the same objectivefunction and constraints as the original problem, which col-lectively is called the subproblem.

The same algorithms are applied recursively to each sub-problem to decide LB and UB of the subproblem. TheLB and UB of the subproblem provide information about(1) whether a further partition on this subproblem is nec-essary, (2) whether the subproblem can be discarded, and(3) a better feasible solution to the original problem. Theprocess itself generates a search tree where the root noderepresents the original problem. The tree search procedurecontinues until all nodes have been solved or deleted, or untilan ε-optimal solution is found.

FIGURE 1. Our proposed BB algorithm.

Fig.1 shows our proposed BB algorithm. We denote theUB, LB, and optimal solution of (P1) as UB, LB, and x∗,respectively. In step 1, we initialize values of LB and x∗,and set convergence tolerance ε. We add the root node(i.e., (P1)) to problem list W , which serves as the collectionof subproblems generated during the tree search. We assignan initial value to UB(P1), which designates the UB of P1.In step 2, we check whether W is empty. If so, we furtherexamine LB. If LBis −∞, (P1) is infeasible; otherwise weobtain optimal solution x∗.In step 3, we update UB as the maximum UBq among all

nodes q in W , where UBq is the UB of node q. We then

compute the optimality gap ε0 between UB and LB, whichis defined as

ε0 =

∞, if LB = −∞∣∣∣ UB−LBUB

∣∣∣, otherwise.

If ε0 ≤ ε, we end the procedure by achieving ε-convergence.Otherwise, we select and delete node q with UBq = UBfrom W. In step 4, we solve the UB problem of q to find

its optimal objective value. If this problem is infeasible,we proceed to step 2. Otherwise, we updateUBq and calculatethe approximation gap εq defined as

εq =

{∞, if LB = −∞∣∣∣UBq−LBUBq

∣∣∣, otherwise.

If εq ≤ ε, we go to step 2 because the best feasible solutionin q cannot be strictly better than LB above the ε toler-ance. Therefore, we exclude q from further consideration andchoose another node from W to investigate.

In step 5, we solve the LB problem of q to obtain feasiblesolution x∗q . Since x

∗q is feasible to (P1), its objective value

LBq provides a LB of (P1). We then compare values ofLB and LBq. If LBq > LB, we update LB andx∗, becauseLBq improves the best known feasible solution of (P1) so far.We also delete all nodes q′ with UBq′ ≤ LB fromW because they cannot possibly contain the global opti-mum. Subsequently, we update εq and check whetherεq ≤ ε. If yes, we proceed to step 2 for the samereason mentioned in step 4. In contrast, if LBq ≤ LBor εq > ε, we go to step 6. This is because theremay still be scope for locating a better LB in q. Hence,we need to partition q further. In step 6, we divide q intotwo child nodes q1 and q2 by employing the branching rule.In step 7, we assign UB of q1 and q2,UBq1 andUBq2, asUBq.We then add q1 and q2 to W , and go to step 3.Clearly, the success of the BB algorithm primarily relies

on how fast ε0 decreases. To expedite the convergence rateof the BB algorithm, ε0 must quickly reduce during the treesearch procedure. Because ε0 is determined by the tightnessof the LB and UB, designing efficient bounding schemesto obtain stringent bounds thus becomes critical for theBB process. To approach this issue, we propose novel upperand lower bounding methods in Sections 3.2 and 3.3, respec-tively. Further, to improve UBs through the BB process,we devise an innovative branching strategy to diminish thelargest relaxation error of the UB problem in Section 3.4.

B. UPPER BOUNDING SCHEME1) MOTIVATIONIn [1]–[3], the authors studied different cross-layer opti-mization problems in wireless communications. These prob-lems are all nonconvex MINLPs with bilinear terms andlog functions but lack fractional functions. [1]–[3] also usedBB algorithms to solve these optimization problems. To buildthe relaxation problem for the BB procedure, [1]–[3] all usedthe same linearization methods, including the reformulation-linearization technique (RLT) and the single convexhull (SCH), to approximate the bilinear term and the log func-tion, respectively. Although being helpful for creating therelaxation problem, the relaxation methods used in [1]–[3]have several major drawbacks. First, as reported by exhaus-tive computational studies presented in [24] and [25],the RLT yields weak relaxations for the bilinear term andis very slow in declining UBs of the BB procedure for

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FIGURE 2. PCH and SCH for C lt = ln(1 + SINRl

t ).

maximization problems. It thus requires exploration of manymore nodes, considerably decelerating convergence of theBB process and causing excessive computational complexity.Second, as illustrated in Fig. 2, the SCH is too loose toaccurately bound the log function. Moreover, since linearfractional functions are nonconvex, it needs special mathe-matical processing to obtain the global optimum for prob-lems with such functions. BB algorithms in [1]–[3] thusare incapable of globally optimizing (P1) because they donot handle this issue. To improve these defects, we proposea novel upper bounding scheme using PLR and PCH tobound the bilinear product and the log function, respectively.We then establish a MILFP model serving as a tight UBproblem of (P1). However, MILFPs are NP-hard in general.Therefore, we transform the MILFP model into its equivalentMILP problem using the method suggested in [17]. In thisway, we can find the UB of (P1) by exploiting cutting-edgeMILP solvers that have become highly effective over the pastdecades [26]–[28].

To remedy the shortcoming of RLT, the authors in [24]proposed fifteen PLRs to relax bilinear products. To use thePLRs in [24], we first divide the original domain of one ofthe variables involved in the bilinear product into smallersubdomains. Then we bound the bilinear term using moreaccurate linear subenvelopes in each subdomain, and employa set of newly defined binary variables to choose the opti-mal subdomain. The advantage of the PLRs is that they candramatically accelerate convergence of the BB algorithm byproviding much tighter relaxations. Results in [24] show thatthese PLRs can reduce the optimality gap (i.e., ε0) at theroot node of the BB procedure from 20%–40% (i.e., twosubdomains) to 75%–95% (i.e., fifteen subdomains) as com-pared with the RLT. Further, [25] have demonstrated that thePLRs can significantly curtail the number of nodes requiredfor investigation during the BB process, thus decreasing thecomputational time from several-fold to hundreds-fold com-pared to RLT. To improve the quality of UB, we propose

to use one of the fifteen PLRs to approximate the bilinearterm.

In (P1), as both SINRlt and I lt participate in the bilinearproduct, one can select either of them to partition. Neverthe-less, we observe that SINRlt also appears in the log functionwhile I lt does not. If we choose to divide on the dimension ofSINRlt , we can construct the PLR and the PCH for the bilinearproduct and the log function, respectively, by separating onlyone unique domain. This not only strengthens the linearrelaxations for both the bilinear term and the log function butalso avoids increasing the number of variables and constraintsin the relaxed formulation. Therefore, we opt to divide on thedomain of SINRlt .

In [24], the authors recommend four representatives thatare the most efficient among the fifteen PLRs, namely, nf4l,nf4r, nf6t, and nf7r. However, by carefully examining thefour most effective PLRs, we find only nf4l can accomplishthe above goal. Therefore, we apply nf4l to build a series ofsubenevelopes to bound the bilinear term over each subrangeof SINRlt . We then use the same partitioning to establish thePCH for the log function.

2) THE CONSTRUCTION OF MILFP PROBLEMFirst, we divide the domain of each SINRlt variable into Ksubdomains by defining a grid of points, SINRlt (k), such that

SINRlt (k) = SINRlt + kd ∀l ∈ L, t ∈ 0, k ∈ 0, 1, . . . ,K ),

(14)

where d = (SINRlt − SINRlt )/K , and SINRlt and SINR

lt are

lower and upper bounds of SINRlt , respectively. We call thenumber of subdomains K the partition level.We then introduce a set of binary variables λlt (k), that are

equal to one if and only if SINRlt lies within the k th segmentas shown in (15),

λlt (k) ={1; if SINRlt (k − 1) ≤ SINRlt ≤ SINR

lt (k)

0; otherwise(∀l ∈ L, t ∈ 0, k ∈ 9), (15)

where 9 = {1, . . . ,K }. Since SINRlt only lies in one of theK subdomains, we have the following constraint∑K

k=1λlt (k) = 1 (∀l ∈ L, t ∈ 0). (16)

Further, we define a set of continuous variables αlt (k)

(∀l ∈ L, t ∈ 0, k ∈ 9), where 0 ≤ αlt (k) ≤ dλlt (k).These variables denote the deviation of SINRlt from grid pointSINRlt (k − 1), if SINRlt (k − 1) ≤ SINR

lt ≤ SINR

lt (k), or zero

otherwise. We thus model SINRlt as (17).

SINRlt =∑K

k=1[SINRlt (k − 1)λlt (k)+ α

lt (k)]

(∀l ∈ L, t ∈ 0) (17)

We also reformulate I lt by introducing another set of con-tinuous variables β lt (k) (∀l ∈ L, t ∈ 0, k ∈ 9), where0 ≤ β lt (k) ≤ (I lt − I lt )λ

lt (k), and I

lt and I lt are lower and

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upper bounds of I lt , respectively. Here, βlt (k) signifies the

deviation of I lt from I lt if SINRlt (k − 1) ≤ SINR

lt ≤ SINR

lt (k);

otherwise, it is zero. We hence denote I lt as follows

I lt = I lt +∑K

k=1β lt (k) (∀l ∈ L, t ∈ 0). (18)

Finally, we define variable wlt = I lt SINRlt and rewrite (13)

as follows

σWSINRlt + wlt = hlplt (∀l ∈ L, t ∈ 0). (19)

We then bound wlt (∀l ∈ L, t ∈ 0) using the nf4l givenin (20)-(23).

wlt ≥ I lt SINRlt +

∑K

k=1{SINRlt (k − 1)β lt (k)} (20)

wlt ≥ I lt SINRlt +

∑K

k=1{SINRlt (k − 1)β lt (k)}

+

∑K

k=1

{dβ lt (k)

}+(I lt − I

lt )∑K

k=1{αlt (k)− dλ

lt (k)}

(21)

wlt ≤ I lt SINRlt +

∑K


+(I lt − Ilt )∑K

k=1αlt (k), (22)

wlt ≤ I lt SINRlt +

∑K


+

∑K

k=1dβ lt (k), (23)

To eliminate the nonlinearity of the log function, we firstdefine variable clt = ln(1+ SINRlt ) and rewrite (7) as follows∑

s∈Sf ls,t ≤

Wln2

clt (∀l ∈ L, t ∈ 0). (24)

We then bound clt by the PCH using the same partitioningof SINRlt (i.e., (14)-(17)), as depicted in Fig. 2. For everysubdomain k ∈ 9, the PCH consists of three tangents that aretangential at points SINRlt (k − 1), γ = 1

3 [SINRlt (k − 1) +

SINRlt (k)], and SINRlt (k), as well as one secant betweenSINRlt (k − 1) and SINRlt (k). To derive formulas of the PCH,we take the tangent at SINRlt (k − 1) as an example. If SINRltlies within subdomain k , as elaborated in Fig. 2, we have

clt ≤SINRlt − SINR

lt (k − 1)

1+ SINRlt (k − 1)+ ln

[1+ SINRlt (k − 1)

].

(25)

This is because clt is smaller than the tangent at SINRlt (k − 1)denoted by the right hand side of (25). By using (17), we canrewrite this tangent as αlt (k)

1+SINRlt (k−1)+ ln[1+ SINRlt (k − 1)].

We then equivalently express (25) as (26) using the methoddeveloped in [29].

clt ≤∑K

k=1

{αlt (k)

1+ SINRlt (k − 1)

+ λlt (k)ln[1+ SINRlt (k − 1)

]}(∀l∈L, t ∈0)

(26)

In (26), since only one of the K subdomains is active, onlythe tangent in the active subdomain exists, while tangentsin all other subdomains are discarded. Therefore, no matterwhich subdomain is active, we can always represent the tan-gent at SINRlt (k − 1) (∀k ∈ 9) using (26). Similarly, one canfind formulas of tangents at γ and SINRlt (k), and the secantbetween SINRlt (k − 1) and SINRlt (k) as (27)-(29). Note thatIn Fig.2, we draw the PCH with 3 subdomains and theSCH used in [1]–[3]. As illustrated in the figure, the proposedPCH is much tighter than the SCH and thus can improve thequality of the UB.

clt ≤K∑k=1

{αlt (k)+ λ

lt (k)

[SINRlt (k − 1)− γ

]1+ γ

+ λlt (k) ln(1+ γ )} (∀l ∈ L, t ∈ 0) (27)

clt ≤K∑k=1

{αlt (k)+ λ

lt (k)

[SINRlt (k − 1)− SINRlt (k)

]1+ SINRlt (k)

+ λlt (k) ln[1+ SINRlt (k)

]}(∀l ∈ L, t ∈ 0),

(28)

clt ≥M∑m=1

{λlt (k) ln[1+ SINRlt (k − 1)]

+ln[1+ SINRlt (k)]− ln[1+ SINRlt (k−1)

]SINRlt (k)− SINR

lt (k−1)

αlt (k)}

(∀l ∈ L, t ∈ 0) (29)

Using the above relaxation scheme, we form UB problem(P2) from (P1), as follows

Max

∑s∈S rs∑L

l=1∑T

t=1 plt

(P2)

subject to constraints (3), (5), (6), (8), (10), (12), (16)-(24),(26)-(29),

x lt ∈ {0, 1} , 0 ≤ plt ≤ pmax , 0 ≤ SINR

lt ≤ hlpmax/σW ,

0 ≤ clt ≤ ln(1+ hlpmax/σW ),wlt ≥ 0,

0 ≤ I lt ≤ pmax∑

l′∈L,l′ 6=lh(TX l′ ,RX l) (∀l ∈ L, t ∈ 0) ;

f ls,t ≥ 0 (∀l ∈ L, s ∈ S, t ∈ 0); rs ≥ 0 (∀s ∈ S) ;

λlt (k) ∈ {0, 1} , 0 ≤ αlt (k) ≤ dλ

lt (k),

0 ≤ β lt (k) ≤ (I lt − Ilt )λ

lt (k) (∀l ∈ L, t ∈ 0, k ∈ 9).

3) THE MILFP TO MILP TRANSFORMATIONBy examining (P2), we discover that the objective functionis a linear fractional function and all constraints are linearexcept for some variables that are restricted to binary values.Therefore, it is a MILFP problem. Owing to the nonconvexityof the objective function and combinatorial feature, MILFPsare nonconvex MINLPs and NP-hard in general. To solveMILFPs for global optimality, the authors in [17] proposea novel approach to transform MILFPs into their equivalent

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MILPs which are then optimized using efficient MILP tech-niques. The results of extensive computational experimentsin [17] demonstrate that this approach requires significantlyless computational efforts than several commercial MINLPoptimization packages. Hence, we apply this method to refor-mulate (P2) into its equivalent MILP model.To perform the reformulation, we first utilize a proce-

dure similar to the Charnes–Cooper transformation [17], [18]except that the variable transformations are only appliedto continuous variables. We define variable u and a setof variables rs such that u = 1/

∑Ll=1

∑Tt=1 p

lt and

rs = rs/∑L

l=1∑T

t=1 plt = rsu, where u and rs are positive

because∑L

l=1∑T

t=1 plt > 0 and rs > 0 for all feasible solu-

tions of (P1). According to this definition, we can transformthe fractional objective function to the linear function givenby∑

s∈S rs/∑L

l=1∑T

t=1 plt =

∑s∈S rs.

We then multiply both sides of all constraints of (P2) by u.The resulting constraints are given in (30)-(54)∑

l∈IL(i)∪OL(i)ux lt ≤ u (∀i ∈ N , t ∈ 0), (30)∑

l∈OL(i),t∈{1,2,...,T }f ls,t = rsT (∀i ∈ N , s ∈ IS (i)),

(31)∑l∈IL(i),t∈{1,2,...,T }

f ls,t =∑

l∈OL(i),t∈{1,2,...,T }f ls,t

(∀i ∈ N , s ∈ S − (IS (i) ∪ OS (i))), (32)rmins u ≤ rs ≤ rmaxs u (∀s ∈ S) (33)0 ≤ plt ≤ pmaxux

lt (∀l ∈ L, t ∈ 0), (34)

˙I lt =∑


l′t (∀l ∈ L, t ∈ 0), (35)∑K

k=1uλlt (k) = u (∀l ∈ L, t ∈ 0), (36)

˙SINRlt =∑K

k=1[SINRlt (k − 1)uλlt (k)+ α

lt (k)]

(∀l ∈ L, t ∈ 0) (37)

˙I lt = I lt u+∑K

k=1˙β lt (k) (∀l ∈ L, t ∈ 0), (38)

σW ˙SINRlt + wlt = hl plt (∀l ∈ L, t ∈ 0), (39)

wlt ≥ Ilt˙SINRlt +

∑K

k=1{SINRlt (k − 1) ˙β lt (k)}

(∀l ∈ L, t ∈ 0) (40)


∑K


+

∑K

k=1

{d ˙β lt (k)

}+ (I lt − I

lt )∑K

k=1

× {αlt (k)− duλlt (k)} (∀l ∈ L, t ∈ 0), (41)

wlt ≤ Ilt˙SINRlt +

∑K


+ (I lt − Ilt )∑K

k=1αlt (k) (∀l ∈ L, t ∈ 0) (42)

wlt ≤ Ilt˙SINRlt +

∑K


+

∑K

k=1d ˙β lt (k) (∀l ∈ L, t ∈ 0). (43)∑

s∈Sf ls,t ≤

Wln2

clt (∀l ∈ L, t ∈ 0), (44)

clt ≤∑K

k=1{

αlt (k)

1+ SINRlt (k − 1)+ uλlt (k)

× ln[1+ SINRlt (k − 1)]} (∀l ∈ L, t ∈ 0)

(45)

clt ≤∑K

k=1{αlt (k)+ uλ

lt (k) [SINR

lt (k − 1)− γ ]

1+ γ+ uλlt (k) ln(1+ γ )} (∀l ∈ L, t ∈ 0)

(46)

clt ≤∑K

k=1{αlt (k)+ uλ

lt (k)


]1+ SINRlt (k)

+ uλlt (k) ln[1+ SINRlt (k)

]}

(∀l ∈ L, t ∈ 0) (47)

clt ≥∑M

m=1{uλlt (k) ln[1+ SINR

lt (k − 1)]

+ln[1+SINRlt (k)]− ln[1+SINR

lt (k−1)

]SINRlt (k)− SINR

lt (k − 1)

}

× αlt (k) (∀l ∈ L, t ∈ 0), (48)

0 ≤ ˙SINRlt ≤ hlpmaxu/σW (∀l ∈ L, t ∈ 0), (49)

0 ≤ f ls,t (∀l ∈ L, s ∈ S, t ∈ 0), (50)

0 ≤ ˙I lt ≤ [pmax∑

l′∈L,l′ 6=lh(TX l′ ,RX l)]u (∀l ∈ L, t ∈ 0),

(51)

0 ≤ clt ≤ ln(1+ hlpmax

/σW

)u (∀l ∈ L, t ∈ 0), (52)

0 ≤ αlt (k) ≤ duλlt (k) (∀l ∈ L, t ∈ 0, k ∈ 9), (53)

0 ≤ β lt (k) ≤(I lt − I

lt

)uλlt (k) (∀l ∈ L, t ∈ 0, k ∈ 9),

(54)

where plt = uplt , fls,t = uf ls,t ,

˙I lt = uI lt ,˙SINRlt = uSINRlt ,

wlt = uwlt , clt = uclt ,∀l ∈ L, t ∈ 0, and αlt (k) = uαlt (k),

β lt (k) = uβ lt (k) ,∀l ∈ L, t ∈ 0, k ∈ 9.Additionally, as u = 1/

∑Ll=1

∑Tt=1 p

lt , we have the

following constraint∑L

l=1

∑T

t=1plt = 1. (55)

Note that nonlinear terms in (30)–(55) are ux lt and uλlt (k),

which are products of a binary variable and a continu-ous variable. We can exactly linearize such nonlinear termsusing Glover’s linearization scheme [17]. To achieve thisend, we first introduce auxiliary variables qlt = ux lt andvlt (k) = uλlt (k), and rewrite (30), (34), (36), (37), (41),(45)–(48), (53) and (54) as (56)-(66).∑

l∈IL(i)∪OL(i)qlt ≤ u (∀i ∈ N , t ∈ 0), (56)

0 ≤ plt ≤ pmax qlt (∀l ∈ L, t ∈ 0), (57)∑K

k=1vlt (k) = u (∀l ∈ L, t ∈ 0), (58)

˙SINRlt =∑K

k=1[SINRlt (k − 1)vlt (k)+ α

lt (k)]

(∀l ∈ L, t ∈ 0), (59)

VOLUME 4, 2016 935





∑K


+

∑K

k=1

{d ˙β lt (k)

}+(I lt −I

lt )∑K

k=1{αlt (k)− dv

lt (k)}

(∀l ∈ L, t ∈ 0) (60)

clt ≤∑K

k=1{

αlt (k)

1+ SINRlt (k − 1)+ vlt (k)

× ln[1+ SINRlt (k − 1)]} (∀l ∈ L, t ∈ 0),

(61)

clt ≤∑K

k=1{αlt (k)+ v

lt (k)

[SINRlt (k − 1)− γ

]1+ γ

+ vlt (k)ln(1+γ )} (∀l∈ L, t ∈ 0) (62)

clt ≤∑K

k=1

{αlt (k)+ v

lt (k)


]1+ SINRlt (k)

+ vlt (k) ln[1+SINRlt(k)

]}(∀l∈ L, t ∈ 0),

(63)

clt ≥∑M

m=1{vlt (k)ln[1+ SINR

lt (k − 1)]

+ln[1+SINRlt (k)]− ln[1+SINRlt (k−1)

]SINRlt (k)−SINR

lt (k − 1)

}

× αlt (k) (∀l ∈ L, t ∈ 0), (64)

0 ≤ αlt (k) ≤ dvlt (k) (∀l ∈ L, t ∈ 0, k ∈ 9), (65)

0 ≤ ˙β lt (k) ≤ (I lt − Ilt )v

lt (k) (∀l ∈ L, t ∈ 0, k ∈ 9).

(66)

Next, we append the new constraints given below

0 ≤ qlt ≤ H · xlt (∀l ∈ L, t ∈ 0), (67)

u− H · (1− x lt ) ≤ qlt ≤ u (∀l ∈ L, t ∈ 0), (68)

0 ≤ vlt (k) ≤ H · λlt (k) (∀l ∈ L, t ∈ 0, k ∈ 9), (69)

u− H ·(1− λlt (k)

)≤ vlt (k) ≤ u (∀l∈ L, t ∈ 0, k ∈ 9),

(70)

where H is a sufficiently large number. Constraints(67) and (68) imply that if x lt is zero, qlt should be zero.Otherwise, if x lt is one, qlt should be equal to u. Hence,(67)–(68) are linearization constraints for qlt = ux lt . Similarly,(69)–(70) are formulations to linearize vlt (k) = uλlt (k).After performing the above transformation, we can recon-

struct an equivalent MILP model (P3) for (P2) as follows

Max∑

s∈Srs (P3)

subject to constraints (31)-(33), (35), (38)-(40), (42)-(44),(49)-(52), (55)-(70),

u ≥ 0; x lt ∈ {0, 1} , plt ≥ 0, ˙I lt ≥ 0, ˙SINRlt ≥ 0, wlt ≥ 0,

f ls,t ≥ 0 (∀l ∈ L, s ∈ S, t ∈ 0) ; λlt (k) ∈ {0, 1} ,

clt ≥ 0, qlt ≥ 0 (∀l ∈ L, t ∈ 0); rs ≥ 0 (∀s ∈ S);

αlt (k) ≥ 0, β lt (k) ≥ 0, vlt (k) ≥ 0 (∀l ∈ L, t ∈ 0, k ∈ 9).

In [17], the authors have proven that under this transfor-mation, both MILFP and MILP models are mathematicallyequivalent. They also have proven that this transformationpreserves the one-to-one mapping relationship between solu-tions of MILFP and MILP problems. We refer readers to [17]for details of the proofs. Manifestly, based on these prop-erties, (P3) is equivalent to (P2) and can serve as the UBproblem of (P1). Hence, one can easily find the UB of (P1)using state-of-the-art MILP solvers to optimize (P3).

C. LOWER BOUNDING STRATEGYBecause any feasible solution of (P1) yields a valid LB, wepropose a scheme using the optimal solution of (P2) as astarting point to search for a good feasible solution of (P1).We first optimize (P3) globally and transform its solutionto that of (P2). The resulting values of x lt and plt (∀l ∈ L,t ∈ 0) always provide a set of feasible solutions to thesame variables in (P1). We thus fix x lt and plt variables attheir optimal values from (P2). Given feasible x lt and plt ,we can compute values of SINRlt (∀l ∈ L, t ∈ 0) using(12) and (13). Constraints (3) and (10) are also satisfied andthus can be omitted. Furthermore, the denominator of (9)becomes a constant. Thus, optimizing (9) is equivalent tomaximizing its numerator. Consequently, we can equivalentlytransform (P1) to LP problem (P4) as follows

Max∑

s∈Srs (P4)

subject to constraints (5)-(8),

rs ≥ 0 (∀s ∈ S); f ls,t ≥ 0 (∀s ∈ S, l ∈ L, t ∈ 0).

If (P4) is feasible, its solution is also feasible to f ls,t and rsvariables in (P1). Therefore, we can find a feasible solutionto (P1) by combining values of x lt , p

lt , f

ls,t with rs (∀s ∈ S,

l ∈ L, t ∈ 0) using the above scheme. Otherwise, it exhibitsthere does not exist a session rate satisfying the minimumsustained and peak rate requirements for all sessions. In thiscase, we do not update the LB.

D. BRANCHING RULETo partition the current subproblem, we have to choosea branching variable and corresponding branching point.Because both bilinear terms and log functions are relaxedin (P2), we select variables involved with these relaxationsfor branching. Therefore, we have I lt and SINR

lt as branching

candidates. We define the relaxation error of the bilinear term

and log function as REwl,t =∣∣wlt − I lt SINR

lt

∣∣ and REcl,t =∣∣clt − ln(1 + SINRlt)∣∣, respectively, where wlt , I lt , SINRlt , andclt are values of w

lt , I

lt , SINR

lt , and c

lt variables, respectively,

at the solution of (P2).We also define the largest relaxation error (LRE) as

max{REwl,t ,RE

cl,t ; ∀l ∈ L, t ∈ 0

}. To decide the exact

branching variable, we select the one associated with theLRE, because dividing on such a variable most decreasesrelaxation errors of (P2). In case the LRE coincides witha log function, we let the branching variable be the

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TABLE 1. Locations of nodes.

corresponding SINRlt and the branching point be SINRlt .

However, if the LRE corresponds to a bilinear product, weneed to decide a single variable for branching. Specifically,

we choose branching variable vb as (71), where I lt and Ilt are

the current upper and lower bounds of I lt , respectively, andSINR

lt and SINR

lt are the current upper and lower bounds of

SINRlt , respectively. After choosing a specific vb for branch-ing, we partition at vb. Note that in (71), we choose the vbwhose relaxation solution value is closest to the midpoint ofits current range. This is intended to cause a large disturbancein the created subproblem and decrease the UB as much aspossible.

After deciding the branching variable, we split the currentsearch space into two subregions along the branching variableaxis at the branching point and create two subproblems.Whensolving either of the two subproblems, we replace the originalrelaxations of both bilinear terms and log functions withstronger ones by exploiting the range reduction.

IV. NUMERICAL RESULTS AND DISICUSSIONSIn this section, we describe the evaluation of optimal EEof the MANET using our proposed BB algorithm. We firstcompare its computational efficiency versus a referenceBB algorithm. In the reference BB algorithm, we usedthe relaxation methods proposed in [1]–[3] to establish theUB problem. Specifically, we used the RLT to approximatethe bilinear term and the SCH to bound the log function.We created another MILFP model accordingly, which wasthen transformed into its equivalent MILP problem usingthe same technique described in Section 3.2.3. The otherelements of the reference algorithm were the same as thoseof our proposed BB algorithm. Subsequently, we experimenton optimal EE over different values of T and various routingstrategy to gain more insights on how power control, trafficscheduling and routing influence the optimal solution.

For the proposed and reference BB algorithms, weused CPLEX [30] to solve the UB and LB problems(i.e., (P3) and (P4)). We ran the programs in a machinewith two Intel Xeon 3.3 GHz processors. Each processor hadeight cores. Since CPLEX supports a parallel optimizationmode, we solved (P3) and (P4) using all available cores.We consider a MANET with twenty nodes randomly locatedin a 250 × 250 m2 region. Table 1 shows the locationsof these nodes. There are four sessions among the nodes.Table 2 presents the source node, destination node, rmins ,and rmaxs for the sessions. We set maximum transmis-sion power to pmax = 100 mW and channel bandwidth

W = 5 MHZ. We assume propagation gain hl = ϑd−4l ,where dl is the distance between transmitter and receiver oflink l, and ϑ = 0.002 is a constant characterizing the antennagain and average channel attenuation [31].

TABLE 2. Traffic profiles of sessions.

A. COMPUTATION EFFICIENCY EVALUATIONTable 3 illustrates computational outcomes for the T = 5case, including the reference BB algorithm and our proposedBB algorithm with different partitioning levels K . We ran allexperimental instances with a CPU time limit of 40 minutes.In Table 3, columns UBroot and CPUroot are the UB andcomputational time of the root node of the BB tree, respec-tively. Columns UB and LB show the best upper and lowerbounds found within the time limit, respectively. The remain-ing columns denote the optimality gap and the number ofinvestigated BB nodes when the time limit is reached.

From these results, we first observe that at the root node,as discussed in Section 3.2.1, our proposed scheme with dif-ferent values of K always provides a more rigorous UB thanthat of the reference BB algorithm, and the UB tightens as Kenlarges. Usually, larger K also leads to fewer nodes requiredto explore in the BB procedure as it provides tighter relax-ations. However, we also find that the computational time ofthe root node grows with an increased value ofK , because thenumber of variables and constraints in (P3) increases when Kaugments. Therefore, larger K increases the size and solutiontime of the relaxation problem. Apparently, an importanttradeoff exists between the computational time of each nodeand the number of nodes requiring investigation during theBB process. One thus has to carefully select the value of Kto lessen the solution time of the BB algorithm as much aspossible. We tested many cases to select a proper value for K ,and we observed that our algorithm minimizes the solutiontime when three partitions are used. Therefore, we usedK = 3 for all simulations in this section. Second,our proposed scheme with different values of K alwaysachieved a smaller optimality gap and better feasible solutionwithin the same time limit as compared with the referenceBB algorithm. The optimality gap of our proposedapproach with K = 3 and the reference algorithm

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TABLE 3. Computational outcomes for T = 5.

were 8.74% and 48.49%, respectively. The best feasiblesolution found by our proposed algorithm with K = 3and the reference algorithm were 4.771∗108 and 3.593∗108,respectively. This shows that our proposed method respec-tively decreased the optimality gap 81.98% and increasedthe best feasible solution 32.79%, which achieved significantcomputational improvements.

Note that the reason that our proposed scheme took tensof minutes to solve (P1) is due to its NP-hard nature. Forthe MANET considered in this simulation, the number of xltvariables is 1900 and thus there are 21900 (≈10572) possiblecombinations of link activations. In every possible combi-nation (i.e., fixed xlt value, ∀l ∈ L, t ∈ 0), (P1) becomesa nonconvex nonlinear programming (NLP) problem that isNP-hard. This means if a brute force method is used, one hasto solve 10572 NP-hard problems. It is very difficult to solvean NP-hard problem because we cannot find a polynomialtime algorithm approaching such a problem, let alone tackle10572 NP-hard problems, which is far toomany to realisticallyaddress. However, our proposed scheme (with K = 3) canobtain a solution which is at least 91.26% optimal within40 minutes. Obviously, compared with the reference algo-rithm and a brute force method, our proposed scheme canprovide good solutions while preserving high computationalefficiency.

Table 4 presents details of the optimal solution obtainedusing our proposed algorithm for the above T = 5 case.As shown in Table 4.1, at time slot 1, there were two activelinks, namely, links 20 → 13 and 2 → 9 correspondingto sessions 1 and 3, respectively. The transmission powersof the two links were 1.16 mw and 4.17 mw, respectively.Since both links transmitted concurrently, they interferedwith each other. Hence, the resulting capacities of thetwo links were 7.36 Mbps and 12.96 Mbps, respectively.The capacities also limited the flow rates of the two linksas 7.35 Mbps and 12.95 Mbps, respectively. Similarly,Tables 4.2 through 4.5 display the results of time slots 2 to 5,respectively. Note that we only show outcomes of active linksfor the five time slots in Table 4. Fig. 3 illustrates the routingtopology corresponding to Table 4. From Fig. 3, we can findthe routing path for each session. For instance, the path ofsession 1 consisted of 3 links, namely, 20 → 13, 13 → 6,and 6 → 3, which were scheduled to transmit at

TABLE 4. Table 4. Optimization results of different time slots.Table 4.1. Optimal EE plan for time slot 1. Table 4.2. Optimal EE plan fortime slot 2. Table 4.3. Optimal EE plan for time slot 3. Table 4.4. OptimalEE plan for time slot 4. Table 4.5. Optimal EE plan for time slot 5.

time slot 1, 2 and 3, respectively. The instantaneous flow ratesof the three links were all 7.35Mbps, which brought about theaverage session rate being equal to 1.47 Mbps. One can alsodiscover that the average session rates of sessions 2 through 4were 1.12 Mbps, 0.97 Mbps, and 2.59 Mbps, respectively.The total depleted power was 12.89mw. Hence, the optimalEE was 4.771∗108 (bits/Joule).

B. THE IMPACT OF TRAFFIC SCHEDULINGFig.4 shows the curve of optimal EE versus different numbersof T for the same example MANET considered in Fig. 3.We observe that when T increased, optimal EE enlargedup to a maximum point that occurred at T = 5. Thiswas because each node had more choices of traffic schedul-ing and routing to mitigate interference from other nodes.

938 VOLUME 4, 2016




FIGURE 3. Routing topology for T = 5.

FIGURE 4. Optimal EE versus different numbers of T .

Therefore, EE increased as T augmented. However, beyondthe maximum point, optimal EE decreased as T increased.This was because as T enlarged, each session had to sendmore traffic to satisfy the minimum sustained and peakrate requirements. Hence, every active node consumed morepower to send the increased amount of traffic. However, whenEE reached its maximal point, interference of each activenode had been diminished to be very small. Therefore, whenT increased, it was hard to further reduce interference ofeach active node. Nevertheless, each active node must spendmore power to transmit the increased data, thus magnifyingdissipated power of the entire network. Because the totalsession rate was limited to a fixed range and the total amountof power increased as T augmented, optimal EE lessened asT enlarged. This result indicates that it is essential to selecta suitable value of T to enable the network operating aroundthe optimal EE. To the best of our knowledge, this is still anopen issue. Thus, it is also a direction for future research onoptimizing EE of MANETs.

TABLE 5. Optimization results of the minimum hop-count routing.

C. THE INFLUENCE OF ROUTING STRATEGYDue to the simplicity of implementation in practice and thelessened wastage of MANET resources such as network linksand buffers, the minimum hop count has become the mostwidespread metric used by MANET routing protocols. It hasbeen chosen as the default metric for many routing protocols,for instance, [19]–[23], etc. To investigate the influence ofrouting methodologies on the optimal EE, we fixed the valueof T as 5 and selected the minimum hop count as the perfor-mance comparison index. In Fig. 3, if theminimumhop-countrouting was used, the destination node of each session couldbe reached from the source node with 1 hop. However,we had 5 time slots. We thus let the session 1 having thelongest distance among the 4 sessions use 2 hops that occu-pied 2 time slots respectively. For the other sessions, eachsession used one time slot to send data between source anddestination nodes (i.e. one-hop routing).1 Table 5 presentsthe optimal results for the minimum hop-count routing. Thetotal session rate and power consumption were 6.02 Mbpsand 90.82 mw, respectively. The optimal EE thus was6.629∗107 (bits/Joule). Compared to the results shown inTable 4, our proposed cross-layer optimization approachimproved the EE by 7.87 times over the minimum hop-count routing, which revealed a substantial enhancement.This was because the minimum hop-count routing usuallyprefers longer distance for each hop, which causes higherpower consumption for transmitting the same amount of data.

V. CONCLUSIONIn this paper, we addressed the EE optimization problem forMANETs, jointly considering routing, traffic scheduling, andpower control. According to the cross-layer design principle,we formulated this problem as a nonconvex MINLP modelthat is intrinsically NP-hard. We proposed a customizedBB algorithm for the global optimization of the problemby exploiting the specific structure of this type of model.In our algorithm, we approximated bilinear terms withpowerful PLRs and log functions with PCHs to build aMILFP model serving as a tight UB problem. We theneffectively transformed the MILFP model to its equivalentMILP formulation, allowing for the global optimization ofthe MILFP using efficient MILP methods. We obtainedtight UBs on the global optimum by solving the resulting

1Identifying the allocation of time slot in advance is the same as knowingvalues of xlt variables. Furthermore, constraints (5) and (6) can also be sim-plified if the routing protocol is already known. Hence, one can equivalentlytransform (P1) to a nonconvex NLP problem that can be globally optimizedby modifying our proposed BB algorithm.

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MILP-based relaxations. Our proposed BB algorithm alsoincorporated a novel lower bounding strategy and branchingrule designed to accelerate convergence.

We presented our numerical results of applying our pro-posed BB algorithm to EE optimization of a MANET.We also compared the efficiency of our proposed BB algo-rithm with another scheme. Our algorithm performed quitewell with respect to computational complexity. In the future,we aim to concentrate on developing distributed protocolsand algorithms that can be realistically implemented to opti-mize EE of MANETs. Moreover, we plan to compare theperformance of distributed protocols and mechanisms withthat found using the global optimization technique of thispaper, thus encouraging the development of novel distributedprotocols and algorithms to improve EE of MANETs.

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WEN-KUANG KUO received the B.S. degreein electrical engineering from National ChengKung University, Tainan, Taiwan, in 1994,the M.S. degree in electrical engineering fromNational Taiwan University, Taipei, Taiwan,in 1996, and the Ph.D. degree in electricalengineering from the University of SouthernCalifornia, Los Angeles, in 2003. Since 2003,he has been with the Institute of Computer andCommunication Engineering, Department of Elec-

trical Engineering, National Cheng Kung University, where he is currentlyan Assistant Professor. His current research interests include multimediatransmission over wireless networks, cross-layer optimization of wirelesscommunication systems, protocol design and performance evaluation, andheterogeneous wireless/wireline network integration.

SHU-HSIEN CHU received the B.S. degree inmathematics and the M.S. degree in applied math-ematics from National Cheng Kung University,Tainan, Taiwan, in 2007 and 2009, respectively.He is currently pursuing the Ph.D. degree in elec-trical and computer engineering with the Univer-sity of Minnesota, Minneapolis, USA. His currentresearch interests include mathematical modeling,optimization, magnetic resonance imaging, imageprocessing, machine learning, signal processing,

network optimization, and wireless communication.

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