Joint Resource Optimization for Multicell Networks with Wireless Energy Harvesting Relays
Ali A. Nasir, Duy T. Ngo, Xiangyun Zhou, Rodney A. Kennedy, Salman Durrani
JJoint Resource Optimization for Multicell Networkswith Wireless Energy Harvesting Relays
Ali A. Nasir, Duy T. Ngo, Xiangyun Zhou, Rodney A. Kennedy, and Salman Durrani
Abstract
This paper first considers a multicell network deployment where the base station (BS) of each cell communicateswith its cell-edge user with the assistance of an amplify-and-forward (AF) relay node. Equipped with a powersplitter and a wireless energy harvester, the self-sustaining relay scavenges radio frequency (RF) energy from thereceived signals to process and forward the information. Our aim is to develop a resource allocation scheme thatjointly optimizes (i) BS transmit powers, (ii) received power splitting factors for energy harvesting and informationprocessing at the relays, and (iii) relay transmit powers. In the face of strong intercell interference and limited radioresources, we formulate three highly-nonconvex problems with the objectives of sum-rate maximization, max-minthroughput fairness and sum-power minimization. To solve such challenging problems, we propose to apply thesuccessive convex approximation (SCA) approach and devise iterative algorithms based on geometric programmingand difference-of-convex-functions programming. The proposed algorithms transform the nonconvex problems into asequence of convex problems, each of which is solved very efficiently by the interior-point method. We prove that ouralgorithms converge to the locally optimal solutions that satisfy the Karush-Kuhn-Tucker conditions of the originalnonconvex problems. We then extend our results to the case of decode-and-forward (DF) relaying with variabletimeslot durations. We show that our resource allocation solutions in this case offer better throughput than that ofthe AF counterpart with equal timeslot durations, albeit at a higher computational complexity. Numerical resultsconfirm that the proposed joint optimization solutions substantially improve the network performance, comparedwith cases where the radio resource parameters are individually optimized.
Ali A. Nasir, Xiangyun Zhou, Rodney A. Kennedy and Salman Durrani are with the Research School of Engineering, the AustralianNational University, Canberra, ACT 2601, Australia (Email: { ali.nasir, xiangyun.zhou, rodney.kennedy, salman.durrani } @anu.edu.au). DuyT. Ngo is with the School of Electrical Engineering and Computer Science, the University of Newcastle, Callaghan, NSW 2308, Australia(Email: [email protected]). a r X i v : . [ c s . I T ] M a r ndex Terms Convex optimization, multicell interference, resource allocation, successive convex approximation, wirelessenergy harvesting
I. I
NTRODUCTION
Multicell networks with universal frequency reuse play an important role in meeting the ever increasingdemand of ubiquitous wireless coverage and high data throughput in the near future [1]–[3]. One of thechallenges in such networks is to maintain the quality of service requirements for cell-edge users dueto the interference from the neighboring cells [1], [2]. The deployment of relays is regarded as a viablesolution in eliminating coverage holes in areas that are otherwise difficult for BSs’ signals to penetrate [4],[5]. In addition, the performance of multicell networks can be further enhanced by utilizing coordinatedmultipoint transmission and reception (CoMP) techniques [6], [7], in which BSs and relays cooperatewith one another to best serve the cell-edge users.Due to random positions and mobility of users, relays need to be opportunistically deployed wheremost needed. This can be achieved if relays do not require a wired power connection and are poweredusing alternative ‘green’ energy resources. Recently, radio frequency (RF) or wireless energy harvestinghas emerged as an attractive solution to power wireless nodes [8]. While energy harvesting from ambientsources may not be sufficient to power relay nodes, carefully designed wireless power transfer links can beused to power relay nodes [8]–[10]. In this regard, it is crucial to ensure that the very different informationdecoding and power transfer power sensitivity requirements are met at the receiver (e.g., − dBm forinformation receivers and − dBm to − dBm for energy receivers [8]).A multicell network with energy harvesting relays poses interesting design challenges, such as: (i)How to effectively manage intercell interference, (ii) How to allocated limited power at the base stations(BSs), (iii) How to design wireless power transfer links for amplify-and-forward (AF) and decode-and-forward (DF) relays, and (iv) How the harvested RF energy is utilized at the relays. Existing research in theliterature has partially addressed these important issues. The design of wireless energy harvesting relays inoint-to-point single-cell systems is considered in [11]–[17]. Assuming simultaneous wireless informationand power transfer in a single-cell network, the power control problem for multiuser broadband wirelesssystems without relays is studied in [18]. In [19], a similar problem is examined, albeit in the context ofmultiuser multi-input-multi-output (MIMO) systems. Considering relays in a single-cell network, resourceallocation schemes for the remote radio heads are specifically developed in [20]. In the downlink ofa multicell multiuser interference network, coordinated scheduling and power control algorithms for themacrocell BSs only are proposed in [21], [22]. Recently, in [23], an optimal power splitting rule is devisedfor energy harvesting and information processing at the self-sustaining relays of multiuser interferencenetworks. However, [23] does not consider the important issue of allocating the transmit powers at theBSs and the relays.In this paper, we consider a multicell network in which the BS of each cell communicates with itscell-edge user via a wireless energy harvesting relay node. The relay is equipped with an energy harvestingreceiver and information transceiver. We assume that the energy harvesting receiver implements a power-splitting (PS) based receiver architecture [24], i.e., the relay uses a portion of the received signal powerfor energy harvesting and the remaining signal energy as input to the information transceiver. Using theharvested energy, the information transceiver employs either AF or DF relaying to forward the receivedsignal to its corresponding user. The BSs in the multicell network adopt CoMP, i.e., they share the channelquality measurements and schedule the transmissions, allowing for more efficient radio resource utilization.First, we formulate three new resource optimization problems for multicell networks with EH-enabledAF relays, namely, sum-rate maximization, minimum-throughput maximization, and sum-power minimiza-tion . The objective is to jointly optimize the transmit powers at the BSs and the relays and also find theoptimal power splitting rule at the relays. Our formulations directly target the critical issue of multicellinterference, at the same time as meeting the stringent constraints on the available transmit powers at A preliminary version of this work, which considers the sum-rate maximization problem for AF relaying only, has been accepted forpresentation at the 2015 IEEE International Conference on Communications (ICC), London, U.K. [25]. he BSs and the relays. Since the optimization variables are strongly coupled with many nonlinear cross-multiplying terms, the formulated problems are highly nonconvex. To the best of our knowledge, thereexists no practical method that guarantees to offer the true global optimality to these challenging problems.Then, we exploit the problem structure and adopt the successive convex approximation (SCA) methodto transform the highly nonconvex problems into a series of convex subproblems. Here, we specificallytailor the generic SCA framework via the applications of geometric programming (GP) and difference-of-convex-functions (DC) programming. At each step of our proposed iterative algorithms, we efficientlysolve the resulting convex problem by the interior-point method. We analytically prove that our developedalgorithms generate a sequence of improved feasible solutions, which eventually converge to a locallyoptimal solution satisfying the Karush-Kuhn-Tucker (KKT) conditions of the original problems. Note thatthe general convergence analysis of SCA method is established in [26] and SCA-based solutions have beenempirically shown to often achieve the global optimality in many practical applications, e.g., in wirelineDSL networks [27], wireless interference networks [28], [29], and small-cell heterogeneous networks [30].Finally, we show that the proposed SCA-based approach can be extended to the more general caseof variable timeslot durations with DF relaying. Numerical examples with realistic network parametersconfirm that our joint optimization solutions significantly outperform those where the radio resourceparameters are individually optimized.The rest of this paper is organized as follows: Sec. II presents the system model and states the keyassumptions used throughout this work. Sec. III presents the signal model for AF relaying and equaltimeslot durations. Sec. IV formulates the nonconvex resource allocation problems and introduces thegeneric SCA framework. Secs. V and VI propose the GP-based and DC-based SCA solutions for AFrelaying, respectively. Sec. VII extends our results to the case of variable timeslot durations with DFrelaying. Sec. VIII presents numerical results to confirm the advantages of our proposed algorithms. AndSec. IX concludes the paper. II. S
YSTEM M ODEL AND A SSUMPTIONS
P Cell 1 Cell 2Cell N
Base Station Relay User
Other Cells h , g , h , g , h N , N g N , N h , h N , g , g N , intended signalinterfering signal Fig. 1. A multicell network consists of N cells and a central processing (CP) unit. Each cell has a base station, a relay and a cell-edgeuser. For clarity, we only show the interfering scenarios in Cell 1, i.e., at relay and user . In general, interference happens at all N relaysand N users. Consider the downlink transmissions in an N -cell network with universal frequency reuse, i.e., thesame radio frequencies are used in all cells. Adopting CoMP, we assume that the base stations (BSs)are connected to a central processing (CP) unit which coordinates the multicellular transmissions andradio resource management. The network under consideration is illustrated in Fig. 1. Note that althoughsquare-cells are shown in Fig. 1, the analysis and proposed solutions in this paper are valid for any cellularnetwork geometry.Let N = { , . . . , N } denote the set of all cells. In each cell i ∈ N , the BS attempts to establishcommunication with its cell-edge users. We assume that these users are located in the ‘signal dead zones’,here no direct signal from their serving BS can reach. A relay node is deployed in each cell to assist inforwarding the signal from the BS, extending the network coverage to the distant users. We assume thatorthogonal channels are assigned to users in each cell (e.g., by means of TDMA, FDMA or OFDMA);hence, the intracell interference is eliminated. Therefore, we only focus on the resource allocation in onechannel, which corresponds to only one user in a cell. By BS i , relay i and user i , we mean the BS, therelay and the single user of cell i ∈ N , respectively.We assume that the relays are energy-constrained nodes and they harvests energy from the RF signalsof all BSs, using the power-splitting based receiver architecture. While each BS has a maximum powerlimit P max available for transmission, it must transmit with a minimum transmit power P min to ensure thatthe energy harvesting circuit at the relay is activated. The harvested energy is used by a relay transceiverto process and forward the BS signal to its intended user. We further assume that the relays are mountedon the building rooftops to have a line-of-sight link from the serving BSs.Let h i,j be the channel coefficient from the BS i to relay j and g j,k be the channel coefficient from therelay j to user k . We assume that all the BSs send the available channel state information (CSI) to theCP unit via a dedicated control channel. In this paper, we assume perfect knowledge of CSI at the BSs,allowing for a benchmark performance to be determined.III. S IGNAL M ODEL WITH AF RELAYING
We first consider the case of AF relaying where we divide the total transmission block time T into twoequal timeslots. The first timeslot includes BS-to-relay transmissions and energy harvesting at the relays.During the first timeslot, the relays do not transmit. The second timeslot includes signal processing at therelays and relay-to-user transmissions. In this second timeslot, the BSs do not transmit. The operationsin each timeslot are illustrated in Fig. 2, which will be further discussed in the following. ireless EnergyHarvesterPowerSplitter AF InformationTransceiver + + y R i y R Ii n ai n ri √ α i y R i √ − α i y R i E i P i p i BS i User ih i,i g i,i interference interference Relay i T Time slot 1: (cid:2) , T (cid:3) T Time slot 2: (cid:2) T , T (cid:3) Fig. 2. BS-to-user communication assisted by a wireless energy harvesting AF relay.
A. BS-to-Relay Transmissions and Wireless Energy Harvesting at Relay Receivers
In the first timeslot [0 , T / , let x i be the normalized information signal to be sent by BS i , i.e., E {| x i | } = 1 , where E {·} denotes the expectation operator and | · | the absolute value operator. Let P min ≤ P i ≤ P max denote the transmit power of BS i , d hi,j the distance between BS i and relay j , and β the path-loss exponent. Assuming that n ai is the zero-mean additive white Gaussian noise (AWGN) withvariance σ ai at the receiving antenna of relay i , the received signal at relay i can be expressed as: y R i = h i,i (cid:113)(cid:0) d hi,i (cid:1) β (cid:112) P i x i + N (cid:88) j =1 ,j (cid:54) = i h j,i (cid:113)(cid:0) d hj,i (cid:1) β (cid:112) P j x j + n ai . (1)We assume that each relay is equipped with a power splitter that determines how much received signalenergy should be dedicated to the energy harvester and the signal processing receiver [11], [12], [23],[24]. As shown in Fig. 2, the power splitter at relay i ∈ N divides the power of y R i into two parts in theproportion of α i : (1 − α i ) . Here, α i ∈ (0 , is termed as the power splitting factor. The first part √ α i y R i is processed by the energy harvester and stored as energy (e.g., by charging a battery at relay i ) for theuse in the second timeslot. The amount of energy harvested at relay i is given by: E i = ηα i T N (cid:88) j =1 P j ¯ h j,i , (2)here η ∈ (0 , is the efficiency of energy conversion and ¯ h j,i (cid:44) | h j,i | (cid:0) d hj,i (cid:1) − β , ∀ i, j ∈ N , is the effectivechannel gain from BS j to relay i (including the effects of both small-scale fading and large-scale pathloss).The second part √ − α i y R i of the received signal is passed to an information transceiver. In Fig. 2, n ri denotes the AWGN with zero mean and variance σ ri introduced by the baseband processing circuitry.Since antenna noise power σ ai is very small compared to the circuit noise power σ ri in practice [31], n ai has a negligible impact on both the energy harvester and the information transceiver of relay i . Thus, forsimplicity, we will ignore the effect of n ai in the following analysis by setting σ ai = 0 . The signal at theinput of the information transceiver of relay i can be written as: y IR i = √ − α i y R i + n ri = √ − α i h i,i (cid:113)(cid:0) d hi,i (cid:1) β (cid:112) P i x i + √ − α i N (cid:88) j =1 ,j (cid:54) = i h j,i (cid:113)(cid:0) d hj,i (cid:1) β (cid:112) P j x j + n ri , (3)where the first term in (3) is the desired signal from BS i , and the second term is the total interferencefrom all other BSs. B. Signal Processing at Relays and Relay-to-User Transmissions
In the second timeslot [ T / , T ] , the information transceiver amplifies the signal y IR i prior to forwardingit to user i . Denote the transmit power of relay transceiver i as p i . With the harvested energy E i in (55),the maximum power available for transmission at relay i is given by E i T/ = E i T , which means that: p i ≤ E i T = ηα i N (cid:88) j =1 P j ¯ h j,i . (4)The transmitted signal from relay i to user i can then be written as: x R i = √ p i y IR i (cid:118)(cid:117)(cid:117)(cid:116) (1 − α i ) N (cid:88) j =1 P j ¯ h j,i + σ ri , (5)where the denominator of (5) represents an amplifying factor that ensures power constraint (4) be met.Now, the received signal at user i is: y U i = g i,i (cid:113)(cid:0) d gi,i (cid:1) β x R i + N (cid:88) j =1 ,j (cid:54) = i g j,i (cid:113)(cid:0) d gj,i (cid:1) β x R j + n ui , (6)here d gi,j denotes the distance between relay i and user j , and n ui the AWGN with zero mean and variance σ ui at the receiver of user i . Substituting x R i in (5) into (6) yields: y U i = g i,i √ p i y IR i (cid:118)(cid:117)(cid:117)(cid:116)(cid:0) d gi,i (cid:1) β (cid:34) (1 − α i ) N (cid:88) k =1 P k ¯ h k,i + σ ri (cid:35) + N (cid:88) j =1 ,j (cid:54) = i g j,i √ p j y IR j (cid:118)(cid:117)(cid:117)(cid:116)(cid:0) d gi,i (cid:1) β (cid:34) (1 − α j ) N (cid:88) k =1 P k ¯ h k,j + σ rj (cid:35) + n ui . (7)With y IR i defined in (3), we can then write (7) explicitly as: y U i = g i,i h i,i (cid:112) p i P i (1 − α i ) x i (cid:118)(cid:117)(cid:117)(cid:116)(cid:0) d gi,i d hi,i (cid:1) β (cid:34) (1 − α i ) N (cid:88) k =1 P k ¯ h k,i + σ ri (cid:35) + g i,i (cid:112) p i (1 − α i ) N (cid:88) j =1 ,j (cid:54) = i h j,i (cid:113)(cid:0) d hj,i (cid:1) β (cid:112) P j x j (cid:118)(cid:117)(cid:117)(cid:116)(cid:0) d gi,i (cid:1) β (cid:34) (1 − α i ) N (cid:88) k =1 P k ¯ h k,i + σ ri (cid:35) + g i,i √ p i n ri (cid:118)(cid:117)(cid:117)(cid:116)(cid:0) d gi,i (cid:1) β (cid:34) (1 − α i ) N (cid:88) k =1 P k ¯ h k,i + σ ri (cid:35) + N (cid:88) j =1 ,j (cid:54) = i g j,i √ p j y IR j (cid:118)(cid:117)(cid:117)(cid:116)(cid:0) d gj,i (cid:1) β (cid:34) (1 − α j ) N (cid:88) k =1 P k ¯ h k,j + σ rj (cid:35) + n ui . (8)The first term in (8) represents the desired signal from BS i to its serviced user i , whereas other termsrepresent the intercell interference and the noise.Without loss of generality, let us assume σ ri = σ ui = σ, ∀ i ∈ N . The signal-to-interference-plus-noiseratio (SINR) at the receiver of user i can be derived from (8) as: γ i = φ i,i P i p i (1 − α i ) N (cid:88) j =1 ,j (cid:54) = i φ i,j P j p i (1 − α i ) + N (cid:88) j =1 (cid:0) φ i,j P j (1 − α i ) + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j (1 − α i ) + 1 , (9)where we define φ i,j (cid:44) ¯ g i,i ¯ h j,i σ ; φ i,j (cid:44) ¯ h j,i σ ; φ i,j (cid:44) ¯ g j,i σ ; φ i,j,k (cid:44) ¯ g j,i ¯ h k,i σ . (10)where ¯ g j,i (cid:44) | g j,i | (cid:0) d gj,i (cid:1) − β , ∀ i, j ∈ N . For notational convenience, let us also define P (cid:44) [ P , . . . , P N ] T , p (cid:44) [ p , . . . , p N ] T , and α (cid:44) [ α , . . . , α N ] T . From (9), the achieved throughput in bps/Hz (bits per second perHz) of cell i is given by τ i ( P , p , α ) = 12 log (1 + γ i ) . (11)n important observation from (9) and (11) is that by dedicating more received power at relay i for energyharvesting (i.e. increasing α i ), one might actually decrease the end-to-end throughput in cell i . This canbe verified upon dividing both the numerator and the denominator of γ i in (9) by (1 − α i ) . However if oneopts to decrease α i , the transmit power available at the information transceiver of relay i will be furtherlimited [see (4)], thus potentially reducing the corresponding data rate τ i . Similarly, increasing the BStransmit power P i or the relay transmit power p i does not necessarily increase the throughput τ i of cell i .The reason is that P i and p i appear in the positive terms in both the numerator and the denominator of γ i .This suggests the importance of the resource allocation problem in this context, which will be addressedin the next section.IV. J OINT R ESOURCE O PTIMIZATION P ROBLEMS FOR
AF R
ELAYING
In this paper, we aim to devise an optimal tradeoff of all three parameters, transmit power at BSs, P ,transmit power at relays, p , and power splitting factor at relays, α , to maximize the performance of themulticell network under consideration. Specifically, we will study the following problems which jointlyoptimize ( P , p , α ) for three different design objectives. A. Problem (P1): Sum-Rate Maximization
We assume that P max is the maximum power available for transmission at each BS. Also, P min is theminimum transmit power required at each BS to ensure the activation of energy harvesting circuitry atthe relay. The problem of sum throughput maximization is formulated as follows. max P , p , α N (cid:88) i =1 τ i (12a)s.t. ≤ α i ≤ , ∀ i ∈ N (12b) P min ≤ P i ≤ P max , ∀ i ∈ N (12c) ≤ p i ≤ ηα i N (cid:88) j =1 P j ¯ h j,i , ∀ i ∈ N . (12d)n this formulation, (12a) is the total network throughput whereas (12b) are the constraints for the powersplitting factors for all relays. Also, (12c) and (12d) ensure that the transmit powers at the BSs and relaysdo not exceed the maximum allowable. B. Problem (P2): Max-Min Throughput Fairness
In Problem (P1), the network sum-rate is maximized without any consideration given to the throughputactually achieved by the individual users. It might happen that users with more favorable links conditionsare allocated with most of the radio resources, leaving nothing for others to fulfill their bare minimum QoSrequirements. The latter includes cell-edge users who are the victims of strong intercell interference. Inthe following, we formulate a max-min fairness problem where the throughput of the most disadvantageduser is maximized. max P , p , α min i ∈N τ i (13a)s.t. (12b) − (12d) . From the network design perspective, (13) can be regarded as the problem of maximizing a commonthroughput: max P , p , α ,τ τ (14a)s.t. τ i ≥ τ ≥ , ∀ i ∈ N (14b)(12b) − (12d) , where τ is an auxiliary variable that denotes the common throughput. . Problem (P3): Sum-Power Minimization Different from Problems (P1) and (P2), our objective here is to minimize the total transmit powerconsumption subject to guaranteeing some minimum data throughput τ min for each user: min P , p , α N (cid:88) i =1 P i (15a)s.t. τ i ≥ τ min , ∀ i ∈ N (15b)(12b) − (12d) , This problem is of particular interest for “green” communications, where one wishes to reduce theenvironmental impacts of the large-scale deployment of wireless communication networks. At the sametime, the performance of all cell-edge users is protected with constraint (15b).All three problems (P1), (P2) and (P3) are highly nonconvex in ( P , p , α ) because the throughput τ i in(11) is highly nonconvex in those variables. Even if we fix p and α and try to optimize the BS transmitpower P alone, τ i would still be highly nonconvex in the remaining variable P due to the cross-cellinterference terms. Simultaneously optimizing P , p and α will be much more challenging due to the nonlinearity introduced by the cross-multiplying terms, e.g., P k p j α i in (9) and α i P j in (12d).To efficiently solve Problems (P1), (P2) and (P3), we propose to adopt the successive convex approxi-mation (SCA) approach [26]–[30], [32] to transform the original nonconvex problems into a sequence ofrelaxed convex subproblems. The key steps of the generic SCA approach are summarized in Algorithm 1for our formulated optimization problems. However, in applying the SCA approach, there remain two keyquestions: (i) How to perform the approximation in Step 2 in generic Algorithm 1? (ii) Given that theapproximation is known, how to prove that the iterative algorithm is convergent to an optimal solution?We will provide the answers for those questions in the following sections. Specifically, we will exploitthe structure of the formulated problems to propose two types of approximations, one based on GPprogramming and the other DC programming. We will demonstrate that with the given objective functionsand constraints, it is possible to apply both approximations to solve the formulated nonconvex problems lgorithm 1 Generic Successive Convex Approximation Algorithm Initialize with a feasible solution ( P [0] , p [0] , α [0] ) . At the m -th iteration, form a convex subproblem by approximating the nonconcave objective functionand constraints of (P1), (P2) and (P3) with some concave function around the previous point ( P [ m − , p [ m − , α [ m − ) . Solve the resulting convex subproblem to obtain an optimal solution ( P [ m ] , p [ m ] , α [ m ] ) at the m -thiteration. Update the approximation parameters in Step 2 for the next iteration. Go back to Step 2 and repeat until ( P , p , α ) converges.under the same SCA framework.V. S OLUTIONS FOR
AF R
ELAYING : SCA M
ETHOD U SING
GPTo implement Step 2 in Algorithm 1, in this section we will make use of the single condensationapproximation method [28] to form a relaxed geometric program (GP), instead of directly solving thenonconvex Problems (P1), (P2) and (P3). A GP is expressed in the standard form as [33, p. 161]: min y f ( y ) (16a)s.t. f i ( y ) ≤ , i = 1 , . . . , m (16b) h (cid:96) ( y ) = 1 , (cid:96) = 1 , . . . , M (16c)where f i ( y ) , i = 0 , . . . , m are posynomials and h (cid:96) ( y ) , (cid:96) = 1 , . . . , M are monomials . A GP in standardform is a nonlinear and nonconvex optimization problem because posynomials are not convex functions.However, with a logarithmic change of the variables and multiplicative constants, one can easily turn itinto an equivalent nonlinear and convex optimization problem (using the property that the log-sum-expfunction is convex) [28], [33]. A monomial ˆ q ( y ) is defined as ˆ q ( y ) (cid:44) cy ˆ a y ˆ a . . . y ˆ a n n , where c > , y = [ y , y , . . . , y n ] T ∈ R n ++ , and ˆ a = [ˆ a , ˆ a , . . . , ˆ a n ] T ∈ R n .A posynomial is a nonnegative sum of monomials. [33] . GP-based Approximated Solution for Problem (P1) First, we express the objective function in (12a) as: max P , p , α N (cid:88) i =1
12 log (1 + γ i ) ≡ max P , p , α log N (cid:89) i =1 (1 + γ i ) (17a) ≡ min P , p , α N (cid:89) i =1
11 + γ i , (17b)where (17b) follows from (17a) since log ( · ) is monotonically increasing function. Upon substituting γ i in (9) to (17b) and replacing − α i by an auxiliary variable t i , it is shown that Problem (P1) in (12) isequivalent to: min P , p , α , t N (cid:89) i =1 N (cid:88) j =1 ,j (cid:54) = i φ i,j P j p i t i + N (cid:88) j =1 (cid:0) φ i,j P j t i + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j t i + 1 N (cid:88) j =1 (cid:0) φ i,j P j p i t i + φ i,j P j t i + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j t i + 1 (18a)s.t. t i + α i ≤ , ∀ i ∈ N (18b) t i ≥ , ∀ i ∈ N (18c) ≤ p i ηα i (cid:80) Nj =1 P j ¯ h j,i ≤ , ∀ i ∈ N . (18d)(12b) , (12c) , where t (cid:44) [ t , · · · , t N ] T .It can be seen that (18) is not yet in the form of (16) because (18a) and (18d) are not posynomials.For notational convenience, let us define: u i ( x ) (cid:44) N (cid:88) j =1 ,j (cid:54) = i φ i,j P j p i t i + N (cid:88) j =1 (cid:0) φ i,j P j t i + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j t i + 1 , (19) v i ( x ) (cid:44) N (cid:88) j =1 (cid:0) φ i,j P j p i t i + φ i,j P j t i + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j t i + 1 , (20)where x = [ P T , p T , t T ] T ∈ R N + . The objective function in (18a) can then be expressed as: N (cid:89) i =1 u i ( x ) v i ( x ) . (21)ince u i ( x ) and v i ( x ) are both posynomials, u i ( x ) /v i ( x ) is not necessarily a posynomial, confirming that(18a) is not a posynomial.To transform Problem (P1) into a GP of the form in (16), we would like the objective function (21) tobe a posynomial. To this end, we propose to apply the single condensation method [28] and approximate v i ( x ) with a monomial ˜ v i ( x ) as follows. Given the value of x [ m − at the ( m − -th iteration, we applythe arithmetic-geometric mean inequality to lower bound v i ( x ) at the m -th iteration by a monomial ˜ v i ( x ) as [28, Lem. 1]: v i ( x ) ≥ ˜ v i ( x ) = N (cid:89) j =1 (cid:40) (cid:32) v i ( x [ m − ) P j p i t i P [ m − j p [ m − i t [ m − i (cid:33) φi,j P [ m − j p [ m − i t [ m − ivi ( x [ m − × (cid:32) v i ( x [ m − ) P j t i P [ m − j t [ m − i (cid:33) φi,j P [ m − j t [ m − ivi ( x [ m − × (cid:32) v i ( x [ m − ) p j p [ m − j (cid:33) φi,j p [ m − jvi ( x [ m − (cid:41) × v i ( x [ m − ) vi ( x [ m − × N (cid:89) j =1 ,j (cid:54) = i N (cid:89) k =1 (cid:32) v i ( x [ m − ) P k p j t i P [ m − k p [ m − j t [ m − i (cid:33) φi,j,k P [ m − k p [ m − j t [ m − ivi ( x [ m − . (22)It is straightforward to verify that v i ( x [ m − ) = ˜ v i ( x [ m − ) . In fact, ˜ v i ( x ) is the best local monomialapproximation to v i ( x ) near x [ m − in the sense of the first-order Taylor approximation. With (22), theobjective function u i ( x ) /v i ( x ) in (18a) is approximated by u i ( x ) / ˜ v i ( x ) . The latter is a posynomial because ˜ v i ( x ) is a monomial and the ratio of a posynomial to a monomial is a posynomial. The upper bound (cid:81) Ni =1 ( u i ( x ) / ˜ v i ( x )) of (21) is also a posynomial because the product of posynomials is a posynomial.Next, we will approximate constraint (12d) by a posynomial to fit into the GP framework (16). Again,we lower bound posynomial ηα i (cid:80) Nj =1 P j ¯ h j,i by a monomial as [28, Lem. 1]: ηα i N (cid:88) j =1 P j ¯ h j,i ≥ w i ( α i , P ) (cid:44) ηα i N (cid:89) j =1 (cid:32) P j (cid:80) Nk =1 P [ m − k ¯ h k,i P [ m − j (cid:33) P [ m − j ¯ hj,i (cid:80) Nk =1 P [ m − k ¯ hk,i . (23)It is clear that the ratio p i /w i ( α i , P ) is now a posynomial. Upon substituting (22) and (23) into (18), wean formulate an approximated subproblem at the m -th iteration for Problem (P1) as follows: min x , α N (cid:89) i =1 u i ( x )˜ v i ( x ) (24a)s.t. ≤ p i w i ( α i , P ) ≤ , ∀ i ∈ N (24b)(12b) , (12c) , (18b) , (18c) . Comparing with (16), we see that (24) belongs to the class of a geometric program, i.e., a convexoptimization problem. In (24a), since v i ( x ) ≥ ˜ v i ( x ) [see (22)], we are actually minimizing the upperbound of the original objective function in (18a). With (23), constraint (24b) is stricter than (12d) as: p i ηα i (cid:80) Nj =1 P j ¯ h j,i ≤ p i w i ( α i , P ) ≤ . (25) B. GP-based Approximated Solution for Problem (P2)
By substituting τ i in (11) and carrying out simple algebraic manipulations, constraint (14b) of Problem(P2) can be rewritten as: e τ ln 2 γ i ≤ , ∀ i ∈ N ; and τ ≥ , (26)where ln( · ) denotes the natural logarithm. By introducing the auxiliary variable t and with u i ( x ) and v i ( x ) defined in (19)-(20), it is shown that Problem (P2) is equivalent to: max x , α ,τ τ (27a)s.t. u i ( x ) e τ ln 2 v i ( x ) ≤ , ∀ i ∈ N (27b) τ ≥ , (27c)(12b) − (12d) , (18b) , (18c) . As seen, (27) is not yet in the form of the standard GP (16) because constraints (27b) and (12d) are notposynomials. Using the similar approach in Sec. V-A, we can transform (27b) and (12d) into posynomialsy the approximations in (22) and (23). The resulting subproblem at the m -th iteration of Problem (P2)can be expressed in the standard GP form as: max x , α ,τ τ (28a)s.t. u i ( x ) e τ ln 2 ˜ v i ( x ) ≤ , ∀ i ∈ N (28b) τ ≥ , (28c)(12b) , (12c) , (18b) , (18c) , (24b) , where (28b) follows directly from (27b) by replacing v i ( x ) with ˜ v i ( x ) [see in (22)], and (24b) is used inlieu of (12d). C. GP-based Approximated Solution for Problem (P3)
By introducing an auxiliary variable t and applying monomial approximation ˜ v i ( x ) [in (22)] for v i ( x ) [in (20)], we can transform the nonconvex constraint (15b) in Problem (P3) into a posynomial form as: u i ( x ) e τ min ln 2 ˜ v i ( x ) ≤ . (29)Again, we use (24b) instead of (12d) and arrive at the following GP, which is an approximated problemfor Problem (P3) at the m -th iteration: min x , α N (cid:88) i =1 P i (30a)s.t. u i ( x ) e τ min ln 2 ˜ v i ( x ) ≤ , ∀ i ∈ N (30b)(12b) , (12c) , (18b) , (18c) , (24b) . D. Proposed GP-based SCA Algorithm for Joint Resource Allocation
It should be noted that GP problems (24), (28) and (30) are the convex approximations of the originalProblems (P1), (P2) and (P3), respectively. In Algorithm 2, we propose an SCA algorithm in which a(convex) GP is optimally solved at each iteration. lgorithm 2
Proposed GP-based SCA Algorithm Initialize m := 1 . Choose a feasible point (cid:16) x [0] (cid:44) (cid:0) P [0] , p [0] , t [0] (cid:1) ; α [0] (cid:17) . Compute the value of v i ( x [0] ) , ∀ i ∈ N according to (20). repeat Using v i ( x [ m − ) , form the approximate monomial ˜ v i ( x ) according to (22). Using the interior-point method, solve one GP, i.e., (24) or (28) or (30) to find the m -th iterationapproximated solution (cid:16) x [ m ] (cid:44) (cid:0) P [ m ] , p [ m ] , t [ m ] (cid:1) ; α [ m ] (cid:17) for Problem (P1) or (P2) or (P3), respectively. Compute the value of v i ( x [ m ] ) , ∀ i ∈ N according to (20). Set m := m + 1 . until Convergence of ( x , α ) or no further improvement in the objective value (24a) or (28a) or (30a) Proposition 1:
Algorithm 2 generates a sequence of improved feasible solutions that converge to apoint ( x (cid:63) , α (cid:63) ) satisfying the KKT conditions of the original problems (i.e., Problems (P1), (P2) and (P3)). Proof:
We will prove that Proposition 1 holds for the case of GP (24) and its corresponding Problem(P1). The proofs for GP (28) (hence Problem (P2)) and GP (30) (hence Problem (P3)) are similar and willbe omitted. From (23), we have that p i (cid:46) (cid:16) ηα i (cid:80) Nj =1 P j ¯ h j,i (cid:17) ≤ p i /w i ( α i , P ) . This means that the optimalsolution of the approximated problem (24) always belongs to the feasible set of the original Problem (P1).Next, since v i ( x ) ≥ ˜ v i ( x ) , ∀ x ∈ R N + , it follows that: N (cid:89) i =1 u i ( x [ m ] ) v i ( x [ m ] ) ≤ N (cid:89) i =1 u i ( x [ m ] )˜ v i ( x [ m ] ) = min x N (cid:89) i =1 u i ( x )˜ v i ( x ) ≤ N (cid:89) i =1 u i ( x [ m − )˜ v i ( x [ m − ) = N (cid:89) i =1 u i ( x [ m − ) v i ( x [ m − ) , (31)where the last equality holds because ˜ v i ( x [ m − ) = v i ( x [ m − ) . As the actual objective value of Problem(P1) is non-increasing after every iteration, Algorithm 2 will eventually converge to a point ( x (cid:63) , α (cid:63) ) .inally, it can be verified that ∇ (cid:18) u i ( x ) v i ( x ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x = x [ m − = ∇ (cid:18) u i ( x )˜ v i ( x ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x [ m − , (32) ∇ (cid:32) p i ηα i (cid:80) Nj =1 P j ¯ h j,i (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) α i = α [ m − i ; P = P [ m − = ∇ (cid:18) p i w i ( α i , P ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i = α [ m − i ; P = P [ m − , (33)where ∇ denotes the gradient operator. The results in (32)-(33) imply that the KKT conditions of theoriginal Problem (P1) will be satisfied after the series of approximations involving GP (24) converges tothe point ( x (cid:63) , α (cid:63) ) . This completes the proof.VI. S OLUTIONS FOR
AF R
ELAYING : SCA M
ETHOD U SING
DC P
ROGRAMMING
A. DC-based Approximated Solution for Problem (P1)
In the GP-based approach proposed in Sec. V, we have eliminated the logarithm function in the objectivefunction to form a posynomial [see (17)] and solve the resulting (convex) GP. In the current approach,we propose to keep the logarithm function and rewrite the throughput expression as: log (1 + γ i ) = log (cid:32) N (cid:88) j =1 (cid:0) φ i,j P j p i (1 − α i ) + φ i,j P j (1 − α i ) + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j (1 − α i ) + 1 (cid:33) − log (cid:32) N (cid:88) j =1 ,j (cid:54) = i φ i,j P j p i (1 − α i ) + N (cid:88) j =1 (cid:0) φ i,j P j (1 − α i ) + φ i,j p j (cid:1) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 φ i,j,k P k p j × (1 − α i ) + 1 (cid:33) =¯ v i ( x ) − ¯ u i ( x ) , (34)where we define ¯ u i ( x ) (cid:44) log ( u i ( x )) and ¯ v i ( x ) (cid:44) log ( v i ( x )) with u i ( x ) and v i ( x ) given in (19) and(20), respectively. We also recall that x = [ P T , p T , t T ] T ∈ R N + , and t = − α ∈ R N + .Using the following logarithmic change of variables: ¯ P i (cid:44) ln P i ; ¯ p i (cid:44) ln p i ; ¯ t i (cid:44) ln t i ; ¯ φ i,j (cid:44) ln φ i,j ; ¯ φ i,j (cid:44) ln φ i,j ; ¯ φ i,j (cid:44) ln φ i,j ; ¯ φ i,j,k (cid:44) ln φ i,j,k , (35)or all i, j, k ∈ N , we can further write ¯ u i ( · ) and ¯ v i ( · ) in terms of the sums of exponentials in ¯ x : ¯ u i (¯ x ) = log (cid:32) N (cid:88) j =1 ,j (cid:54) = i e ¯ P j +¯ p i +¯ t i + ¯ φ i,j + N (cid:88) j =1 (cid:16) e ¯ P j +¯ t i + ¯ φ i,j + e ¯ p j + ¯ φ i,j (cid:17) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 e ¯ P k +¯ p j +¯ t i + ¯ φ i,j,k + 1 (cid:33) (36) ¯ v i (¯ x ) = log (cid:32) N (cid:88) j =1 (cid:16) e ¯ P j +¯ p i +¯ t i + ¯ φ i,j + e ¯ P j +¯ t i + ¯ φ i,j + e ¯ p j + ¯ φ i,j (cid:17) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 e ¯ P k +¯ p j +¯ t i + ¯ φ i,j,k + 1 (cid:33) , (37)where ¯ x (cid:44) [ ¯ P T , ¯ p T , ¯ t T ] T , ¯ P (cid:44) [ ¯ P , . . . , ¯ P N ] T , ¯ p (cid:44) [¯ p , . . . , ¯ p N ] T , and ¯ t (cid:44) [¯ t , . . . , ¯ t N ] T . Since thelog-sum-exp function is convex [33], both ¯ u i (¯ x ) and ¯ v i (¯ x ) are convex in ¯ x . However, their difference ¯ v i (¯ x ) − ¯ u i (¯ x ) = log (1 + γ i ) in (34) is not necessarily concave.Using the first-order Taylor series expansion around a given point ¯ x [ m − , we propose to approximate ¯ v i (¯ x ) by an affine function as follows [29]: ¯ v i (¯ x ) ≈ ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) , (38)where the (cid:96) -th element of gradient ∇ ¯ v i (¯ x ) is given by: ∇ ( (cid:96) ) ¯ v i (¯ x ) = 1 v i (¯ x ) ln 2 × e ¯ P (cid:96) +¯ p i +¯ t i + ¯ φ i,(cid:96) + e ¯ P (cid:96) +¯ t i + ¯ φ i,(cid:96) + N (cid:88) j =1 ,j (cid:54) = i e ¯ P (cid:96) +¯ p j +¯ t i + ¯ φ i,j,(cid:96) , if (cid:96) ∈ { , . . . , N } e ¯ p i + ¯ φ i,i + N (cid:88) j =1 e ¯ P j +¯ p i +¯ t i + ¯ φ i,j , if (cid:96) = N + ie ¯ p (cid:96) − N + ¯ φ i,(cid:96) − N + N (cid:88) k =1 e ¯ P k +¯ p (cid:96) − N +¯ t i + ¯ φ i,(cid:96) − N,k , if (cid:96) ∈ { N + 1 , . . . , N } \ { N + i } N (cid:88) j =1 (cid:16) e ¯ P j +¯ p i +¯ t i + ¯ φ i,j + e ¯ P j +¯ t i + ¯ φ i,j (cid:17) + N (cid:88) j =1 ,j (cid:54) = i N (cid:88) k =1 e ¯ P k +¯ p j +¯ t i + ¯ φ i,j,k , if (cid:96) = 2 N + i , otherwise . (39)With the affine approximation (38) and the convex function ¯ u i (¯ x ) , it is clear that the throughput can nowbe approximated by a concave function as: log (1 + γ i ) ≈ ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) − ¯ u i (¯ x ) . (40)By the variable change ¯ α i (cid:44) ln α i , ∀ i ∈ N (41)nd upon denoting ¯ α (cid:44) [ ¯ α , . . . , ¯ α N ] T , the nonconvex constraint (12d) of Problem (P1) can be rewrittenas: e ¯ p i ≤ ηe ¯ α i N (cid:88) j =1 e ¯ P j ¯ h j,i . (42)Applying the arithmetic-geometric inequality, we have that: N (cid:88) j =1 e ¯ P j ¯ h j,i ≥ N (cid:89) j =1 (cid:32) e ¯ P j ¯ h j,i λ [ m − j,i (cid:33) λ [ m − j,i , (43)where P [ m − is a fixed point and λ [ m − j,i (cid:44) e ¯ P [ m − j ¯ h j,iN (cid:88) k =1 e ¯ P [ m − j ¯ h k,i . (44)As such, (42) can be replaced by a stricter constraint: e ¯ p i ≤ ˜ w i ( ¯ α i , ¯ P ) (cid:44) ηe ¯ α i N (cid:89) j =1 (cid:32) e ¯ P j ¯ h j,i λ [ m − j,i (cid:33) λ [ m − j,i , (45)which is equivalent to the following affine constraint: ¯ p i − ¯ α i − N (cid:88) j =1 λ [ m − j,i ¯ P j − c i ≤ , (46)where c i (cid:44) ln η + (cid:80) Nj =1 λ [ m − j,i (cid:16) ln ¯ h j,i − ln λ [ m − j,i (cid:17) is a constant.From (40) and (46), we now have the following convex optimization problem which gives an approxi-mated solution to Problem (P1) at the m -th iteration: max ¯ x , ¯ α N (cid:88) i =1 ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) − ¯ u i (¯ x ) (47a)s.t. e ¯ t i + e ¯ α i ≤ , ∀ i ∈ N (47b) e ¯ α i ≤ , ∀ i ∈ N (47c) e ¯ t i ≤ , ∀ i ∈ N (47d) P min ≤ e ¯ P i ≤ P max , ∀ i ∈ N (47e) ¯ p i − ¯ α i − N (cid:88) j =1 λ [ m − j,i ¯ P j − c i ≤ , (47f)where ¯ x [ m − is known from the ( m − -th iteration. . DC-based Approximated Solution for Problems (P2) and (P3) In this case, we apply the same logarithmic change of variables in (35) and (41). We also make use ofthe results in (40) and (46) to show that Problem (P2) in (14) is approximated by: max ¯ x , ¯ α ,τ τ (48a)s.t. ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) − ¯ u i (¯ x ) ≥ τ ≥ , ∀ i ∈ N (48b)(47b) − (47f) . It is clear that (48) is a convex optimization problem for any given point ¯ x [ m − .By a similar approach, Problem (P3) in (15) can be approximated by following convex problem: min ¯ x , ¯ α N (cid:88) i =1 P i (49a)s.t. ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) − ¯ u i (¯ x ) ≥ τ min , ∀ i ∈ N (49b)(47b) − (47f) , where ¯ x [ m − is known from the ( m − -th iteration. C. Proposed DC-based SCA Algorithm for Joint Resource Allocation
In Algorithm 3, we propose an SCA algorithm in which a convex problem based on the DC approxi-mation is optimally solved at each iteration.
Proposition 2:
Algorithm 3 generates a sequence of improved feasible solutions that converge to apoint ( x (cid:63) ; α (cid:63) ) satisfying the KKT conditions of the original problems (i.e., Problems (P1), (P2) and (P3)). Proof:
We will prove that Proposition 2 holds for the case of (47) and its corresponding Problem(P1). The proofs for (48) (hence Problem (P2)) and (49) (hence Problem (P3)) are similar and will beomitted. From (43), we have that e ¯ p i (cid:46) (cid:16) ηe ¯ α i (cid:80) Nj =1 e ¯ P j ¯ h j,i (cid:17) ≤ e ¯ p i / ˜ w i ( ¯ α i , ¯ P ) . Imposing a stricter constraintmeans that the optimal solution of the approximated problem (47) always belongs to the feasible set ofthe original Problem (P1). lgorithm 3 Proposed DC-based SCA Algorithm Initialize m := 1 . Choose a feasible point (cid:16) x [0] (cid:44) (cid:0) P [0] , p [0] , t [0] (cid:1) ; α [0] (cid:17) and evaluate (cid:16) ¯ x [0] (cid:44) (cid:0) ¯ P [0] , ¯ p [0] , ¯ t [0] (cid:1) ; ¯ α [0] (cid:17) using (35) and (41). Compute ¯ v i (¯ x [0] ) , ∇ log ¯ v i (cid:0) ¯ x [0] (cid:1) and λ [0] j,i , ∀ i, j ∈ N using (36), (39) and (44), respectively. repeat Given ¯ v i (¯ x [ m − ) , ∇ log ¯ v i (cid:0) ¯ x [ m − (cid:1) and λ [ m − j,i , form one convex problem, i.e., (47) or (48) or(49). Using the interior-point method to solve (47) or (48) or (49) for an approximated solution (cid:16) ¯ x [ m ] (cid:44) (cid:0) ¯ P [ m ] , ¯ p [ m ] , ¯ t [ m ] (cid:1) ; ¯ α [ m ] (cid:17) of Problem (P1) or (P2) or (P3) at the m -th iteration, respectively. Update ¯ v i (¯ x [ m ] ) , ∇ log ¯ v i (cid:0) ¯ x [ m ] (cid:1) and λ [ m ] j,i , ∀ i, j ∈ N using (36), (39) and (44), respectively. Set m := m + 1 . until Convergence of (¯ x , ¯ α ) or no further improvement in the objective value (47a) or (48a) or (49a) Recover the optimal solution ( x (cid:63) ; α (cid:63) ) from (¯ x (cid:63) ; ¯ α (cid:63) ) via (35) and (41).Because the gradient of the convex function ¯ v i (¯ x ) is its subgradient [33], it follows that: ¯ v i (¯ x ) ≥ ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) , ∀ x ∈ R N + . (50)We now have the following relations for the approximated objective value (47a) at the m -th iteration: N (cid:88) i =1 ¯ v i (¯ x [ m ] ) − ¯ u i (¯ x [ m ] ) ≥ N (cid:88) i =1 ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v Ti (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x [ m ] − ¯ x [ m − (cid:1) − ¯ u i (¯ x [ m ] )= max ¯ x N (cid:88) i =1 ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v Ti (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) − ¯ u i (¯ x ) ≥ N (cid:88) i =1 ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v Ti (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x [ m − − ¯ x [ m − (cid:1) − ¯ u i (¯ x [ m − )= N (cid:88) i =1 ¯ v i (¯ x [ m − ) − ¯ u i (¯ x [ m − ) (51)It is clear that the actual objective value of Problem (P1) is non-decreasing after every iteration. Therefore,Algorithm 3 will eventually converge to a point ( x (cid:63) ; α (cid:63) ) = (cid:0) e ¯ x (cid:63) ; e ¯ α (cid:63) (cid:1) .inally, it can be verified that ∇ (¯ v i (¯ x ) − ¯ u i (¯ x )) (cid:12)(cid:12)(cid:12)(cid:12) ¯ x =¯ x [ m − = ∇ (cid:16) ¯ v i (cid:0) ¯ x [ m − (cid:1) + (cid:0) ∇ ¯ v i (cid:0) ¯ x [ m − (cid:1)(cid:1) T (cid:0) ¯ x − ¯ x [ m − (cid:1) − ¯ u i (¯ x ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ¯ x =¯ x [ m − , (52) ∇ (cid:32) e ¯ p i ηe ¯ α i (cid:80) Nj =1 e ¯ P j ¯ h j,i (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ α i =¯ α [ m − i ; ¯ P = ¯ P [ m − = ∇ (cid:18) e ¯ p i ˜ w i ( ¯ α i , ¯ P ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ α i =¯ α [ m − i ; ¯ P = ¯ P [ m − . (53)The results in (52)-(53) imply that the KKT conditions of the original Problem (P1) will be satisfied afterthe series of approximations involving convex problem (47) converges to (¯ x (cid:63) ; ¯ α (cid:63) ) . This completes theproof. Remark 1:
As discussed in Secs. V and VI, we use the SCA framework to propose two differentmethods, i.e., GP and DC programming, to solve the three problems (P1), (P2), and (P3). In this remark,we present the computational complexity of the two solutions. We first use the big- O notation to findthe computational complexity of the convex subproblems in an iteration [34]. To solve problem (P1),the complexity of solving both convex subproblems (24) (in Algorithm 2) and (47) (in Algorithm 3) is O ((4 N ) N ) because they both have N optimizing variables and N constraints. Multiplying this factorby the number of iterations required for convergence, we can obtain the overall computational complexityof Algorithms 2 and 3. This implies that the order of complexity for both proposed algorithms is the same.Second, in order to compare the exact computational time for the proposed algorithms, we evaluate theCPU execution time [35]. For a fair comparison, the MATLAB codes of the two algorithms are optimizedto run on the same computer equipped with Intel Core i7-2670QM, 2.20 GHz processor and 8 GB ofRAM. We have observed that the GP-based algorithm is slightly more efficient than DC programmingbased algorithm, e.g., in solving Problem (P1), Algorithms 2 and 3 on average require sec and . sec,respectively.VII. S YSTEM M ODEL AND P ROPOSED S OLUTION FOR
DF R
ELAYING WITH V ARIABLE T IMESLOT D URATIONS
In this section, we extend our work to decode-and-forward (DF) relaying. With DF relaying, we have theflexibility to vary the time duration of BS-to-relay and relay-to-user transmissions. In what follows, we williscuss the signal model, sum-rate maximization problem with GP-based solution and the correspondingcomplexity analysis for DF relaying.
A. Signal Model
Let (cid:15)T define the fraction of the block time used for relay-to-user transmissions. The remaining blocktime (1 − (cid:15) ) T is used for BS-to-relay energy harvesting and information transmissions. With the signalat the input of information transceiver at relay i in (3), the SINR at the receiver of relay i is given by γ DF-R i = (1 − α i )¯ h i,i P i (1 − α i ) (cid:80) Nj =1 ,j (cid:54) = i ¯ h j,i P j + σ (54)The amount of energy harvested at DF relay i is then: E i = ηα i (1 − (cid:15) ) T N (cid:88) j =1 P j ¯ h j,i , (55)The maximum power available for transmission at DF relay i is E i (cid:15)T , which means that p i ≤ E i (cid:15)T = ηα i − (cid:15)(cid:15) N (cid:88) j =1 P j ¯ h j,i . (56)DF relay i will decode the signal from the BS i and forward it to user i . Let ¯ x i be the decoded versionof the signal x i sent by the BS i . The received signal at user i in DF relaying is y U i = g i,i (cid:113)(cid:0) d gi,i (cid:1) β √ p i ¯ x i + N (cid:88) j =1 ,j (cid:54) = i g j,i (cid:113)(cid:0) d gj,i (cid:1) β √ p j ¯ x j + n ai . (57)The SINR at the receiver of user i is thus γ DF-U i = ¯ g i,i p i (cid:80) Nj =1 ,j (cid:54) = i ¯ g j,i p j + σ (58)The achievable throughput in bps/Hz of cell i is then given by τ DF i ( P , p , α , (cid:15) ) = (cid:15) log (1 + γ DF i ) , (59)where γ DF i (cid:44) min { γ DF-R i , γ DF-U i } . . Sum-Rate Maximization Problem and GP-based Solution The problem of sum throughput maximization for DF relaying is formulated as follows. max P , p , α ,(cid:15) (cid:15) N (cid:88) i =1 log (1 + min { γ DF-R i , γ DF-U i } ) (60a)s.t. ≤ α i ≤ , ∀ i ∈ N (60b) P min ≤ P i ≤ P max , ∀ i ∈ N , (60c) ≤ p i ≤ ηα i − (cid:15)(cid:15) N (cid:88) j =1 P j ¯ h j,i , ∀ i ∈ N . (60d) ≤ (cid:15) ≤ . (60e)We will now demonstrate that GP-based SCA approach can be used to solve the nonconvex problem(60) . To transform problem (60) into a GP of the form in (16), we first fix (cid:15) to find the optimal solutionof other parameters and then optimize (cid:15) later. By introducing a new auxiliary variable z i , problem (60)is equivalently expressed as max P , p , α , z ¯ (cid:15) N (cid:88) i =1 log (1 + z i ) (61a)s.t. γ DF-R i ≥ z i , ∀ i ∈ N (61b) γ DF-U i ≥ z i , ∀ i ∈ N (61c) ≤ p i ≤ ηα i − ¯ (cid:15) ¯ (cid:15) N (cid:88) j =1 P j ¯ h j,i , ∀ i ∈ N . (61d)(60b) , (60c) , where z (cid:44) [ z , . . . , z N ] T . The objective function in (61a) is rewritten as max P , p , α , z ¯ (cid:15) N (cid:88) i =1 log (1 + z i ) ≡ min P , p , α , z N (cid:89) i =1
11 + z i (62) Note that the other problems, i.e., max-min throughput and sum-power minimization, can be similarly formulated and solved for DFrelaying. For brevity, they are not presented here. ext, we approximate the expression z i in (62) by a posynomial to fit into the GP framework (16). Tothis end, we lower bound z i by a monomial as [28, Lem. 1]: z i ≥ (1 + z [ m − i ) z [ m − i (cid:32) (1 + z [ m − i ) z i z [ m − i (cid:33) z [ m − i z [ m − i . (63)By using (62) and (63) and ignoring the constant terms, we further reduce (62) to ≡ min P , p , α , z N (cid:89) i =1 z − z [ m − i z [ m − i i (64)Upon substituting γ DF-R i and γ DF-U i from (54) and (58) into (61), replacing − α i by an auxiliary variable t i , applying arithmetic-geometric mean inequality to lower bound z i and (cid:80) Nj =1 P j ¯ h j,i in (62) and (60d)by monomials, we can formulate an approximated subproblem at the m -th iteration for problem (60) asfollows: min P , p , α , t , z N (cid:89) i =1 z − z [ m − i z [ m − i i (65a)s.t. z i (cid:16) t i (cid:80) Nj =1 ,j (cid:54) = i ¯ h j,i P j + σ (cid:17) t i ¯ h i,i P i ≤ , ∀ i ∈ N (65b) z i (cid:16)(cid:80) Nj =1 ,j (cid:54) = i ¯ g j,i p j + σ (cid:17) ¯ g i,i p i ≤ , ∀ i ∈ N (65c) ≤ ¯ (cid:15)p i (1 − ¯ (cid:15) ) w i ( α i , P ) ≤ , ∀ i ∈ N (65d) ≤ t i ≤ , ∀ i ∈ N (65e) α i + t i ≤ , ∀ i ∈ N (65f)(60b) , (60c) , where w i ( α i , P ) (cid:44) ηα i (cid:81) Nj =1 (cid:18) P j (cid:80) Nk =1 P [ m − k ¯ h k,i P [ m − j (cid:19) P [ m − j ¯ hj,i (cid:80) Nk =1 P [ m − k ¯ hk,i is defined in (23). Compared with (16),problem (65) belongs to the class of geometric programs, i.e., a convex optimization problem. Theconvergence of the iterative algorithms that solves convex subproblem (65) for DF relaying can be provedusing similar steps as stated in Proposition 1.Using the optimized values of P , p , and α , we have to optimize the time fraction (cid:15) in the originalproblem (60). Although (60) is linear in (cid:15) , constraint (60d) is met with equality at convergence. No further
50 100 150 200 250 300050100150200250300 x−coordinate (m) y − c oo r d i na t e ( m ) Base Station (BS)Relay (R)User (U)
Cell 1 Cell 2Cell 3 Cell 4
Fig. 3. Topology of the multicell network used in the numerical examples. improvement of (cid:15) can be achieved by solving (60) with the optimized values of P , p , and α . Moreover,constraint (60d) is not monotonic in (cid:15) . Hence, the only available option is to apply exhaustive search tofind the optimal value of (cid:15) in (60) for given optimized values of P , p , and α . Remark 2:
In the numerical results in Sec. VIII, we will show that DF relaying with an optimizedtimeslot fraction results in more than twice the throughput that is otherwise achieved by AF relaying withequal timeslot durations. However, this performance improvement is at the expense of a much highercomputational complexity due to the required exhaustive search.VIII. N
UMERICAL R ESULTS
Fig. 3 shows an example multicell network consisting of four m-by- m cells. In each cell, thegeographical distance between the servicing BS and its corresponding relay and that between the relayand the cell-edge user is both √ ≈ . m, i.e., the relay in each cell is located midway between theBS and the cell-edge user. At the relays, we set the energy harvesting efficiency to η = 0 . . To model thewireless channels we assume independently and identically distributed block fading. Channel coefficients h i,j and g ¯ j,k , ∀ i, j, ¯ j, k and i (cid:54) = j , are circularly symmetric complex Gaussian random variables with zero The value of η is typically in the range of . − . for practical energy harvesting circuits [8]. T o t a l t h r o u g hpu t , P N i = τ i ( bp s / H z ) GPDC P max = 40 dBm P max = 46 dBm (a) Fixed ς = 0 . T o t a l t h r o u g hpu t , P N i = τ i ( bp s / H z ) GPDC ς = 0 . ς = 0 . (b) Fixed P max = 46 dBmFig. 4. Convergence of Algorithms 2 and 3 in Problem (P1) for AF relaying. mean and unit variance. The channel coefficients between the servicing BS and its corresponding relay,i.e., h i,i ∀ i , are modeled by Rician fading with the Rician factor of dB. We assume that the randomly-generated values of h i,j and g j,k remain unchanged during each time block where the radio resourceallocation process takes place. To model large scale fading, we assume that the path loss exponent is β = 3 . This results in a maximum path loss of dB between the BS and the associated relay in eachcell. In order to activate RF energy harvesting with η = 0 . and assuming that the input power at theenergy harvesting relay has to be greater than − dBm [8], [36] , we set P min = −
25 + 51 = 26 dBm.Using a channel bandwidth of kHz and assuming a noise power density of − dBm/Hz, the totalnoise power is calculated as σ = − dBm [37]. We initialize the proposed Algorithms 2 and 3 with P [0] i = ςP max ; α [0] i = ς ; t [0] i = 1 − α [0] i ; p [0] i = ςηα [0] i (cid:80) Nj =1 P [0] j ¯ h j,i , ∀ i ∈ N , where ς is a real numbertaken between and . To solve each convex problem in Algorithms 2 and 3, we use CVX, a packagefor specifying and solving convex programs [38], [39]. Energy conversion efficiency of around has been reported in the ISM band (900 MHz, 2.4 GHz) with an RF input power of − dBm and using 13 nm CMOS technology [8], [36]. M i n i m u m t h r o u g hpu t , τ ( bp s / H z ) GPDC
Iterations P max = 46 dBm P max = 40 dBmZooming only GPcurves for clarity (a) Fixed ς = 0 . M i n i m u m t h r o u g hpu t , τ ( bp s / H z ) GPDC ς = 0 . ς = 0 . (b) Fixed P max = 46 dBmFig. 5. Convergence of Algorithms 2 and 3 in Problem (P2) for AF relaying. A. Convergence of the Proposed Algorithms for AF Relaying
In this subsection, we present numerical results to demonstrate the convergence behavior of the proposedalgorithms under different parameter settings. Regarding Problem (P1), Fig. 4 plots the convergence ofthe sum throughput (cid:80) Ni =1 τ i by the proposed solutions. In our simulations, each iteration corresponds tosolving of a GP (24) in Algorithm 2 or a DC program (47) in Algorithm 3 by CVX. It is clear from Fig.4 that both algorithms exhibit similar convergence behaviors. In our example, they converge within iterations and achieve the same optimal throughput. As observed from Fig. 4(a), the sum rate is increasedby if we allow a higher BS transmit power budget of dBm instead of dBm. In an interference-limited multicell multiuser network setting, increasing the transmit powers may trigger the ‘power racing’phenomenon among the users, which in turn adversely affect the total achieved throughput. Our numericalresults, on the other hand, confirm that the proposed algorithms effectively manage the strong intercellinterference and maximize the network performance. For a fixed power budget P max = 46 dBm, Fig.4(b) demonstrates that the final performance of our algorithms is insensitive to the initial points, furthersuggesting that the solution corresponds to the actual global optimum in our example [27]–[29].We demonstrate the performance of our developed algorithms in Figs. 5 and 6 for Problems (P2) and T o t a l B S t r a n s m i t p o w e r , P N i = P i ( d B m ) GPDC τ min = 0 . τ min = 0 . (a) Fixed ς = 0 . T o t a l B S t r a n s m i t p o w e r , P N i = P i ( d B m ) GPDC ς = 0 . ς = 0 . (b) Fixed τ min = 0 . Fig. 6. Convergence of Algorithms 2 and 3 in Problem (P3) for AF relaying. (P3), respectively, which plot the convergence of the minimum throughput τ and total BS transmit power (cid:80) Ni =1 P i , respectively. Again, the proposed algorithms converge quickly to the corresponding optimalvalues. Different from the results for Problem (P1), increasing P max from dBm to dBm in Fig. 5(a)marginally improves the achieved minimum throughput. This signifies the challenge of enhancing theperformance of the most disadvantaged user, who is typically located in the cell-edge areas and suffersfrom the strong intercell interference. In this situation, simply increasing the total allowable transmit powerat the BSs would not be helpful. On the other hand, Fig. 6(a) verifies that the total required transmit powerdrops to the minimum value possible, i.e., N × P min = 32 dBm for different values of minimum throughput.Similar to Fig. 4(b), Figs. 5(b) and 6(b) show that initializing the algorithms with different values of ς ,again, does not affect the final solutions.As seen from Figs. 4, 5 and 6, both Algorithms 2 and 3 achieve the same optimal values. However,it is impractical to compare their performance with a globally optimal solution. There is no globaloptimization approach available in the literature to solve our highly nonconvex optimization problems.A direct exhaustive search would incur a prohibitive computational complexity. It is noteworthy that theworks of [27]–[29] have shown that the SCA approach often empirically achieves the global optimality a) Total throughput in Problem (P1) (b) Minimum throughput in Problem (P2) (c) Total transmit power in Problem (P3)Fig. 7. Performance comparison of the proposed joint optimization algorithms and the separate optimization approaches. in most practical network applications. Also since we assume perfect knowledge of CSI at the BSs, theachieved performance corresponds to the theoretical bound that can be obtained. The actual performancewith channel estimation errors is out of the scope of this work—a potential future research direction. B. Importance of the Proposed Joint Optimization Algorithms for AF Relaying
Fig. 7 demonstrates the advantages of jointly optimizing ( P , p , α ) as in Algorithms 2 and 3 overoptimizing those three parameters individually. In the latter approach, we only optimize one parameter(i.e., P or p or α ) while fixing the remaining two parameters where applicable as: P i = P max ; p i = ηα i (cid:80) Nj =1 P j ; α i = 0 . , ∀ i ∈ N . Note that for the total power minimization problem (P3), P is optimizedwhile p and α must be fixed. Also in the individual optimization approach, we only present the resultsof GP-based solutions because both GP and DC approaches achieve similar outcomes.The results presented in Fig. 7 have been averaged over , independent simulation runs and we set ς = 0 . and P max = 46 dBm. As expected, the proposed joint optimization algorithms outperform the soleoptimization approach in all cases. The significant gain is observed in Fig. 7(a), where the total throughputis increased by . Regarding Problem (P2), Fig. 7(b) shows that the minimum throughput in Problem(P2) is increased by with the proposed Algorithms 2 and 3. The performance improvement is lesspronounced here. This is because since max-min fairness problem (P2) deals with the most disadvantagedcell-edge user, it is more difficult to support the QoS requirements of such a user compared to only P max (dBm) T o t a l t h r o u g hpu t , P N i = τ i ( bp s / H z ) DF, optimized ǫ DF, ǫ = ¯ ǫ = 0.5AF Fig. 8. Average sum throughput vs. P min = { , , , , , } dBm of the proposed joint optimization algorithm (solving Problem(P1)) for AF and DF relaying. The results for DF relaying include both the fixed timeslot case, i.e., (cid:15) = ¯ (cid:15) = 0 . and the optimized (cid:15) case. maximizing the overall network performance. Finally, with the minimum throughput τ min = 0 . requiredby the most disadvantaged user in Problem (P3), Fig. 7(c) shows that the proposed algorithms reduce thetotal BS transmit power by dB i.e., almost times over optimizing P alone. C. Comparison of AF and DF Relaying
Fig. 8 plots the average sum throughput against different values of P max = { , , , , , } dBmobtained by the proposed joint optimization algorithm, while solving Problem (P1) for AF and DF relaying.The results for DF relaying include both the equal timeslot case, i.e., (cid:15) = ¯ (cid:15) = 0 . and the optimized (cid:15) case.With the equal timeslot assumption for BS-to-relay and relay-to-user transmissions, i.e., (cid:15) = ¯ (cid:15) = 0 . , DFrelaying increase the throughput by at P max = 35 dBm. With an optimized value of (cid:15) , the throughputenhancement can be as high as at P max = 35 dBm.IX. C ONCLUSIONS AND F UTURE R ESEARCH D IRECTIONS
In this paper, we have considered the challenging problems for jointly optimizing the BS transmit pow-ers, the relay power splitting factors and the relay transmit powers in a multicell network. It is assumed herehat the relay (operating in either AF mode or DF mode) is equipped with a PS receiver architecture thatcan split the received power in order to scavenge RF energy and to process the information signal from itsrespective BS. To resolve the highly nonconvex problem formulations, we have proposed SCA algorithmsbased on geometric programming and DC programming that offer sum-throughput maximization, max-min throughput optimization and sum-power minimization. We have proven that the devised algorithmsconverge to the solutions that satisfy the KKT conditions of the original nonconvex problems. Illustrativeexamples have demonstrated the clear advantages of our developed solutions.In case of multiple relays in a cell, two additional problems can be considered for future research (i)in the first timeslot, beamforming design at the BS toward multiple relays, (ii) in the second time slot,relay selection to choose which relay to forward the BS message to which users and over which channel.While these problems are outside the scope of this paper, our proposed solution for the case of one relayand one user per cell can serve as a first building block toward a joint design in more general cases.R
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