Robust Beamforming in Cache-Enabled Cloud Radio Access Networks

Transcription

1 1 Roust Beaforing in Cache-Enaled Cloud Radio Access Networks Oussaa Dhifallah, Student Meer, IEEE, ayssa Dahrouj, Senior Meer, IEEE, Tareq Y. Al-Naffouri, Meer, IEEE, Mohaed-Sli Alouini, Fellow, IEEE arxiv: v2 [cs.it] 31 May 2016 Astract Popular content caching is expected to play a ajor role in efficiently reducing ackhaul congestion and achieving user satisfaction in next generation oile radio systes. Consider the downlink of a cache-enaled cloud radio access network CRAN), where each cache-enaled ase-station BS) is equipped with liited-size local cache storage. The central coputing unit cloud) is connected to the BSs via a liited capacity ackhaul link and serves a set of single-antenna oile users MUs). This paper assues that only iperfect channel state inforation CSI) is availale at the cloud. It focuses on the prole of iniizing the total network power and ackhaul cost so as to deterine the eaforing vector of each user across the network, the quantization noise covariance atrix, and the BS clustering suject to iperfect channel state inforation and fixed cache placeent assuptions. The paper suggests solving such a difficult, non-convex optiization prole using the sei-definite relaxation SDR). The paper then uses the l 0-nor approxiation to provide a feasile, su-optial solution using the ajorizationiniization MM) approach. Siulation results particularly show how the cache-enaled network significantly iproves the ackhaul cost especially at high signal-to-interference-plus-noise ratio SINR) values as copared to conventional cache-less CRANs. I. INTRODUCTION Cloud radio access network CRAN) is recognized as a proising network architecture to eet the treendous increase in data traffic for future wireless networks [1] [3]. In CRANs, a central coputing unit cloud) is connected to several BSs through ackhaul links which allow joint signal processing of user signals. This allows for effective interference anageent and significant energy consuption reduction. Further, to eet the increasing deands in oile data traffic, an increase in the network density is expected, which adds stringent constraints on the ackhaul load. This paper considers the strategy of proactively caching the popular users content in the local eory of the BSs during the off-peak tie. Such strategy is shown to significantly alleviate ackhaul congestion and satisfy users deand [4]. Consider the downlink of a cache-enaled CRAN, where each BS is equipped with a local eory with liited-size. Each cache-enaled BS is connected to the cloud via liitedcapacity ackhaul link. The central coputing unit serves a set of pre-known single-antenna MUs. The paper assues that only iperfect CSI is availale at the cloud. The paper further considers the quantization noise induced y the eployed copression schees due to the capacity-liited ackhaul links. The perforance of the syste ecoes, therefore, a function of the copression schee, as well as the eaforing vector of each user across the network. The prole considered in this paper is related to the recent state-of-art on CRANs. For instance, references [5] [9] consider a CRAN scenario and allow all BSs across the network to fetch the requested data fro the cloud. In dense data networks, however, such assuption ay no longer e feasile ecause of the high-cost of, and the difficulties in, deploying proliferated high-capacity ackhaul networks. As a result, the ackhaul congestion is expected to ecoe a liiting factor in futuristic dense networks perforance. To truly access the advantages harvested y CRAN, this paper considers the ore practical scenario of a cache-enaled CRAN where each BS is equipped with a local eory and connected to the cloud via capacityliited ackhaul links. Such a scenario is particularly related to reference [10], which addresses a ulticast cache-enaled CRAN and forulates the total network cost iniization prole in order to find the eaforing vectors and BS clustering. owever, reference [10] assues perfect CSI availale at the cloud and neglects the effect of the eployed copression schees. The optiization prole forulated in the current paper is also related to the works in [7] [9]. Reference [7] assues a conventional cache-less) CRAN and focuses on solving the total network power iniization prole for oth the data-sharing strategy and copression strategy. Reference [8] focuses on solving the utility axiization prole for oth the dynaic and static BS clustering. Reference [9] assues hyrid connections etween the cloud and the BSs and focuses on iniizing the total network power iniization to jointly design the eaforing vector of each user across the network, the quantization noise covariance atrix of the wireline ackhaul links, and the transit power of the wireless ackhaul links. The works in [7], [8] and [9], however assue a conventional CRAN where all the BSs fetch the requested data fro the cloud, and that perfect CSI is availale at the cloud, unlike the current paper. This paper considers a cache-enaled CRAN where the popular content is cached in the local storage of the BSs. If the content requested y a MU is availale in the local cache, it is directly transitted fro the cache to the user with no need for ackhauling and copression. owever, if the data is not availale in the local cache, the BS fetches the data fro the cloud which perfors joint precoding and copression. Such pre-processing of the data introduces quantization noise, which degrades the syste perforance. The paper forulates the prole of iniizing the total network power and ackhaul cost suject to per-bs power constraint, quality of service constraints, total ackhaul capacity constraint, and iperfect CSI constraints. The paper highlight is that it proposes solving such a non-convex optiization prole using the sei-definite relaxation SDR) [11], and the S-Procedure ethod [12]. The proposed heuristic algorith uses the l 0 -nor approxiation to provide a feasile solution using the ajorization-iniization MM) approach [13]. Siulation results particularly show how the cache-enaled network significantly iproves the ackhaul cost especially at high SINR values as copared to conventional cache-less CRANs. II. SYSTEM MODEL AND PROBLEM A. Syste Model Consider the downlink of a cache-enaled CRAN, where the cloud is connected to B BSs through capacity-liited ackhaul links. Each BS serves U single-antenna MUs. Further, we assue that each BS is equipped with a local cache with liited storage. For the siplicity of analysis, we assue that each BS

2 2 is equipped with single-antenna, all the files have the sae size, and that the knowledge of cache and request status are known for the cloud. Figure 1 illustrates an exaple of the considered network with six cache-enaled BSs and nine MUs. Figure 1. Cloud Base Station Moile User Liited Capacity Backhaul Link Cache The considered CRAN architecture. Let B = {1,..,B} e the set of BSs connected to the cloud, andu = {1,..,U} e the set of users across the network. Let the eaforing vector fro all the BSs to user u U e w u = [w 1u, w 2u,.., w Bu ] T C B and let the channel vector fro all the BSs to user u e h u = [h 1u, h 2u,.., h Bu ] T C B. Further, let the quantization noise vector ev= [v 1, v 2,, v B ] C B, where v is non-unifor white Gaussian with diagonal covariance atrixq C B B of diagonal entriesq q 0, B). The syste odel considers that each cache-enaled BS has a local storage with size F. The cache placeent is descried through the cache placeent atrix P R B F, where p f = 1 when the content f is cached at, BS and p f = 0 otherwise, i.e. F f=1 p f < F, B. The atrix P is fixed throughout the paper. Considering it in the overall optiization increases the coplexity of our original prole, and is left for future work. The received signal y u C at user u can e written as follows: y u = h u w us u + h u w u s u + α h u v +n u, 1) u U u B where U u = U\{u}, s u denotes the transitted data syol for user u, n u CN0,σu 2 ) denotes the additive white Gaussian noise which is independent fro the transitted data syols s u and the quantization noise v, and where α = 0 when the contents requested is cached at BS, and 1 otherwise, since if the content f u requested y user u fro BS is availale in the local cache of BS, the BS transits the data directly without ackhauling and copression. owever, when the content f u is not in the cache of BS, it needs to e retrieved fro the cloud. B. Backhaul Cost and Power Model This paper considers the prole of iniizing the total network cost, which consists of the ackhaul cost and network power consuption. Firstly, the ackhaul cost is proportional to the transission rate of user u y eans of ackhauling. The ackhaul cost associated with BS serving user u can therefore e expressed as follows: { log2 1+δ u ) if p fu = 0 and w u 2 > 0 B u = 0 if p fu = 1 and w u 2 = 0. 2) where δ u denotes the target SINR to e achieved y user u. Secondly, the power consuption of BS can e odeled y: { 1 ν P = P,t +P,a +p P,c +U p )P,cl if P,t > 0 P,s if P,t = 0, 3) where P,t = w ) u 2 +α q denotes the total transit power consuption, ν denotes the power aplifier efficiency, and p = p f u is the total nuer of files requested, yet availale at BS s local cache. Further, P,a denotes the power consued y BS in the active ode, P,s denotes the power consued y BS in the sleep ode,p,c denotes the axiu power needed y BS to retrieve the requested data fro its local cache and P,cl denotes the axiu power needed y BS to retrieve the requested data fro the cloud. Therefore, the total power consuption can e rewritten as follows: P t = { } 1 w u 2 0 +α q )+ P,t P r +P,s, ν B 4) where P r = P,a +p P,c +U p )P,cl P,s denotes the relative power consuption. C. Prole Forulation The paper focuses on iniizing the total network cost suject to per-bs power consuption and total ackhaul capacity constraint, quality of service constraints, and CSI error constraints. This paper assues that the channel errors are ounded y an elliptical region. The true channel vector h u of user u can therefore e written as follows: h u = h u +e u, u U, 5) where e u denotes the CSI error vector of user u and it satisfies the following elliptical constraints e u E u e u < 1, 6) where h u denotes the estiated channel vector, and E u is a known positive definite atrix specifying the accuracy of the CSI. We assue that the user syols have unit power, i.e. E s u 2 ) = 1, u U, and independent fro each other, fro the quantization noise, and fro the additive noise. The SINR of user u can then e written as follows: h Γ u = u w u 2 h u w u 2 +. α h u 2 q +σ 2 7) u u U u B One of the constraints studied in this paper is the per-bs power constraint which is given y: w u 2 +α q P, B. 8) If the requested data fro a MU is not availale in the local cache of the BS, the data is fetched fro the cloud through the liited-capacity ackhaul link y eans of copression, which is assued to e independent aong users for the sake of siplicity. If the requested data is availale at the BS local cache, no copression is needed. Therefore, the eaforing vector associated with user u, the quantization noise level q, and the total ackhaul capacity C are related as follows: w u 2 +α q R C. 9) B 0 where R is given y: 1 p fu ) w u R = log ). 10) q

3 3 This paper iniizes the total network cost which consists of the total power consuption 4) and the ackhaul cost 2), i.e. C N = { 1 w u 2 +α q )+ w u 2 +α q r ν B 0P + } w u 2 0 R u 1 p fu ) 11) where R u = log 2 1+δ u ) denotes the service requireent rate target rate). The optiization prole studied in this paper can therefore e forulated as follows: in w,q C N 12) s.t. Constraints 5),6), 8), and 9) h Γ u = u w u 2 h u w u 2 +h u Qh δ u, u U. u +σu 2 u U u where the optiization is over the eaforing vectors w and the quantization noise vector q. The optiization prole 12) is difficult to handle due to the non-convexity of the cost function, the non-convexity of the total ackhaul capacity constraint 9), and the infinite set of possile CSI errors. This paper proposes a heuristic algorith instead, so as to solve 12) ased on the l 0 -nor approxiation using the MM approach. Siulation results suggest that such approach provides an appreciale perforance iproveent as copared to cacheless networks. Reark 1. If the contents requested fro BS y all the users across the network are availale in the local cache of BS, the corresponding quantization noise q is only in the SINR expression of the optiization prole 12). Thus, it is set to zero to increase the SINR value and satisfy the SINR constraints. The SINR expressions Γ u in 12) are, therefore, equivalent to the expressions in 7). III. A. Prole Relaxation PROPOSED SOLUTION To solve the difficult optiization prole 12), this section first relaxes the total ackhaul capacity constraint 9) to reduce the coplexity, y eans of a well-chosen l 0 -nor approxiation: 1+ǫ 1 x ) x 0 = Ix > 0) = li log1+ǫ 1, x 0. 13) ) The ackhaul capacity constraint 9) can then e approxiated as: { }) λ ǫ log 1+ǫ 1 w u 2 +α q R C, 14) B 1 where λ ǫ = log1+ǫ 1 ), and ǫ > 0 is a sall constant. As the approxiated constraint 14) is still non-convex, the paper next replaces the left hand side of 14) y an upper ound that helps reforulating 12) using the SDR afterwards. ǫ 0 log Theore 1. An upper-ound of the function on the left-hand side of the constraint 14) is given y: CM = λ w u 2 +α q ǫ R B wu 2 +α q +ǫ +a, 15) where a = log1 + ǫ 1 { w u 2 + α q }) { w u 2 +α q }/{ w u 2 +α q +ǫ}, and where wu and q denote the eaforing vector fro BS to user u and the quantization noise of BS of the previous iteration, respectively. Proof: Please Refer to Appendix A for details. After replacing the left-hand side of the constraint 14) y 15), the constraint 14) reains non-convex due to the presence of R. This paper, however, uses the trick of fixing R at each iteration +1 to R using the results fro the previous iteration, so as to convexify 14) at each iteration. Specifically, at iteration + 1, we solve the following optiization prole: in w,q C N 16) s.t. Constraints 5),6) and 8) ) λ ǫ γ R w u 2 +α q + a γ C B h Γ u = u w u 2 h u w u 2 +h u Qh δ u, u U u +σu 2 u U u γ 1 = B. wu 2 +α q +ǫ, Prole 16) reains a difficult, non-convex optiization prole despite convexifying constraint 9). In the next susection, the paper proposes solving the prole y intelligently approxiating it, and then solving it using the MM algorith to guarantee a stationary solution to the prole. B. SDP Reforulation First, define the rank-one atrix W u as W u = w u wu, u U. We now use the S-Procedure [12] and the rankone SDR approach [11] to facilitate the steps of our proposed algorith. Based on the introduced quadratic variales W u and after dropping the non-convex rank-one constraints, the relaxed iniization prole 16) can e reforulated as follows: in Ĉ N 17) s.t. B TrA W u )+α TrA Q) P ˆR ) TrA W u )+α TrA Q)+â C u 0, λ u 0, u U W u 0,Q 0,Q is diagonal, u U, where ĈN is given in 18), ˆR = γ R, â = a γ, A denotes the diagonal atrix with 1 at the ain diagonal entry and zeros otherwise, and where the optiization is over the eaforing atrices W u, the quantization noise covariance atrixq, and the introduced varialesλ u, and where the atrix u is denoted [ y: ] Gu +λ u = u E u G u hu h u G u h u G u hu σu 2, 19) λ u and where G u = 1 W u W u Q. 20) δ u u U u While the feasiility set of the optiization prole 17) is convex, the cost function 18) is still not convex. The paper, therefore, proposes deterining an approxiate solution to the relaxed optiization prole 17) y approxiating the l 0 - nor and, then, y using the MM algorith.

4 Ĉ N = { ) 1 TrA W u )+TrA Q) + TrA W u )+TrA Q) P r ν B 0 + } TrA W u ) R u 1 p fu ). 18) 0 4 C. Majorization-Miniization Approach To approxiate the l 0 -nor, use approxiation 13) for a sufficiently sall fixed ǫ. The cost function in the optiization prole 17), i.e., 18), can then e approxiated as follows: C N = { ) 1 TrA W u )+TrA Q) ν B }) +λ ǫ log 1+ǫ 1 { TrA W u )+TrA Q) P r +λ ǫ log 1+ǫ 1 TrA W u ) ) } R u 1 p fu ), 21) The aove cost function is not convex, and so the MM algorith is now used to find a stationary point to the otained optiization prole. The MM approach consists of first finding a surrogate function that ajorizes the cost function. Then, it iteratively iniizes the otained function until a local optial solution of the optiization prole with cost function 21) is reached. Theore 2. The surrogate function that ajorizes the function 21) at iteration +1 is given y: CN W u,q) = ) Bη TrA W u )+TrA Q) + βutra W u )+cwu,q ). 23) B where η, β u are given in 22) and cw u,q ) is a constant depending only on the eaforing atrices Wu and the quantization noise covariance atrix Q of the previous iteration. Proof: Please Refer to Appendix B for details. Using the aove theore, the MM approach solves the following optiization prole at the iteration + 1: in CNW u,q) 24) s.t. B TrA W u )+α TrA Q) P, B ˆR ) TrA W u )+α TrA Q)+â C u 0, λ u 0, u U W u 0,Q 0,Q is diagonal, u U, The aove optiization prole is a sei-definite prograing SDP). Therefore, it can e solved using efficient nuerical algoriths [12]. D. Proposed Iterative Algorith To solve the optiization prole, one can use a two loop algorith where the outer loop is responsile of updating ˆR and â, and the inner loop is responsile of solving the SDP prole 24) and then updating η and β. Such approach is nevertheless coputationally coplex in nature. The paper instead proposes an iterative algorith that coines the two loops and iterates etween two levels. At the first level, it solves +1 the optiization prole 24). Then, it updates R, γ +1, η +1 and β +1. We suarize the proposed relaxed MM algorith to find an approxiate solution to 12) in Tale 1. Algorith 1 The Iterative Relaxed MM Algorith Initialization : Initialize R 0, â0, η0 and β0 and the iteration index to = 0. Repeat : 1: Solve the optiization prole 24). If it is infeasile, go to End; +1 2: Update ˆR, â +1, η +1 and β +1 ; 3: Update the iteration index = +1; Until : Desired level of convergence. 4: Find the optial eaforing vectors wu, u U and the optial quantization noise covariance atrix Q. End Such algorith is further guaranteed to converge as shown in the following theore. Theore 3. The proposed iterative algorith is guaranteed to converge to a feasile, yet su-optial, solution of the original optiization prole 12) as ǫ tends to 0. Proof: Please Refer to Appendix C for details. The proposed algorith does not always lead to a rank-one solution. owever, a rank-one solution can always e guaranteed using the randoization techniques [12]. Siulation results in the next section show that the proposed iterative algorith leads to a rank-one solution in ost of the cases. E. Coputational Coplexity Analysis The ipleentation of the proposed iterative algorith requires to solve the SDP prole 24) with c = B +3U +3) SDP constraints and v = 2U + 1) SDP variales at each iteration. Further, the second step of the proposed algorith +1 consists of updating ˆR, â +1, η +1 and β +1. Therefore, the coputational coplexity of the proposed algorith coes ainly fro solving the SDP prole. If the otained solution does not satisfy the relaxed rank-one constraints, Gaussian randoization techniques [12] can e applied to estiate a rank-one solution. The coputational coplexity of the overall algorith then increases since, to guarantee a good solution, the randoization techniques need to e ran for a large nuer of saples. To solve the SDP proles generated at each iteration of the proposed algorith, one ay use the interior point ethod which is ipleented in ost advanced solvers, e.g. SDPT3 and SeDuMi. In large-scale network i.e., B and U are large), the coputational coplexity of the proposed algorith using the interior point ethod can e huge due to solving a largescale SDP prole at each iteration. To reduce coplexity, the splitting conic solver SCS) [14], which is ased on the ADMM algorith [15], can e applied to solve the large-scale SDP prole 24). The SCS solves the large-scale SDP prole

5 5 η = 1 λ ǫ P r + ν ǫ+ TrA Wu )+TrA Q ), β u = R u1 p fu )λ ǫ ǫ+tra W 22) u ). y perforing parallel cone projection and suspace projection which significantly reduces the coputational coplexity as copared to the interior point ased solvers. IV. SIMULATION RESULTS This section provides siulation exaples to illustrate the perforance of the proposed iterative algorith. It considers a cache-enaled CRAN scenario fored y B = 15 singleantenna cache-enaled BSs, where each BS is connected to U = 6 single-antenna MUs. The BSs and MUs are uniforily and independently distriuted in the square region[0 1000] [0 1000] eters. Further, the estiated channel vectors are generated using Rayleigh fading coponent and a distance-dependent path loss, odeled as Ld u ) = log 10 d u ), where d u denotes the distance etween BS and user u in kiloeters. Each user across the network randoly requests one content fro the BSs according to the content popularity, odeled as Zipf distriution with skewness paraeter 1. The noise power spectral density is set to σu 2 = 98 db u. We set the axiu transit power of BS to P = 1 Watt, the total ackhaul capacity liit to C = 560 Mps, the relative power consuption to P r = 38 Watts, ν = 2.5 and the accuracy atrix E u = 1 a I B where a > 0. Additionally, R0, â 0, η0 and β 0 B are initially all set to 1. The Cost Function 21) Figure Nuer of Iteration Convergence ehavior of the Iterative Relaxed MM Algorith. First, the SINR target is set to δ u = 10dB u U, the positive constant a to 0.01 and ǫ = Figure 2 shows the convergence ehavior of the proposed iterative algorith for different channel realizations. It can e noticed that the proposed algorith converges for all the considered channel realizations. Figure 2 further shows that the proposed algorith has a reasonale convergence speed around 20 iterations) for the considered channel realizations. It is especially rearkale that the cost function, i.e., function 21), is always driven downhill. This results validates the stateent in theore 3. Then, we assue that users across the network request different contents and the network is fored y B = 16 BSs and U = 8 MUs, where half of the scheduled users request a coon content, and the other half randoly request one content. Figure 3 shows the total ackhaul cost versus the target SINR. It can e noticed that the cache-enaled network significantly reduces the ackhaul cost especially at high SINR values as copared to the network without cache. It is especially Total Backhaul Cost Figure Popularity-Aware Cache, Size=6 Popularity-Aware Cache, Size=3 No-Cache SINR db) The total ackhaul cost as function of the target SINR. rearkale that increasing the cache size significantly reduces the total ackhaul cost. Tale I. PROBABILITY TAT TE OPTIMAL SOLUTION IS RANK-ONE. a = 0.8 a = 0.5 a = 0.1 a = 0.01 a = δ u = 5 db δ u = 10 db Finally, we assue that the network is fored y B = 10 BSs and U = 4 MUs. Tale I presents the proaility that the optial solution found using the proposed algorith is rankone for 500 channel realizations and various SINR targets and scalars a. It can e noticed that the solution found using the proposed algorith is rank-one with high proaility. Therefore, the Gaussian randoization technique is not used in ost of the cases, which reduces the coputational coplexity of the proposed iterative algorith. V. CONCLUSION Proactively caching popular content is expected to play a ajor role in alleviating ackhaul congestion proles. This paper considers the downlink of a cache-enaled CRAN, where each cache-enaled single-antenna BS serves a pre-known set of single-antenna MU. The paper assues that only iperfect CSI is availale at the cloud, and each BS is connected to the cloud through liited-capacity ackhaul link. The paper provides an iterative algorith to solve the total network power and ackhaul cost iniization prole. Siulation results show that the cache-enaled network significantly reduces the ackhaul cost especially at high SINR values as copared to networks without cache. APPENDIX A PROOF OF TEOREM 1 In this appendix, we will present a detailed proof of Theore 1. For ǫ > 0, the function x log1 + ǫ 1 x) is a concave function on the interval [0 + [. Based on the first order propriety of concave functions, the following inequality holds for any x 0 and x 0 log1+ǫ 1 x x) ǫ+x + log1+ǫ 1 x ) x ǫ+x. 25) Let wu and q denote the eaforing vector fro BS to user u and quantization noise of BS of the previous iteration

6 6 of the MM algorith and let x = w u 2 + α q and x = w u 2 +α q. Sustituting x and x in 25), the right hand side of 25) can e rewritten as follows w u 2 +α q T = ǫ+ wu 2 +α q wu 2 +α q ǫ+ wu 2 +α q + log1+ǫ 1 { wu 2 +α q }) w u 2 +α q = ǫ+ +a wu 2 +α q. 26) The following inequality{ holds for any w u }) and q 0 λ ǫ log 1+ǫ 1 w u 2 +α q R B w u 2 +α q 27) λ ǫ B ǫ+ +a wu 2 +α q R. Further, we can readily see that the aove inequality is achieved with equality when w u = wu and q = q, B, u U. An upper-ound of the function in the left hand side of the constraint 9) is the right hand side of the inequality 27). This copletes the proof of theore 1. APPENDIX B PROOF OF TEOREM 2 In this appendix, we will present a detailed proof of Theore 2. To this end, let Wu and Q denote the eaforing atrix associated with user u and the quantization noise covariance atrix of the previous iteration of the MM algorith, respectively. Based on the fact that the function x log1 + ǫ 1 x) is a concave function on the interval [0 + [, for ǫ > 0, we have }) and log 1+ǫ 1 { TrA W u )+TrA Q) TrA W u )+TrA Q) ǫ+ TrA Wu )+TrA Q ) +c 1Wu,Q ), 28) log 1+ǫ 1 TrA W u ) ) TrA W u ) ǫ+tra Wu ) +c u2wu ), 29) where c 1 Wu,Q ) and c u2 Wu ) are constants depending only on the eaforing atrices Wu and the quantization noise covariance atrix Q of the previous iteration. Based on 28) and 29), we have C N CNW u,q), 30) where cwu,q ) = λ ǫ c 1 Wu,Q )P r B +λ ǫ c u2 Wu )R u 1 p fu ). 31) B This copletes the proof of theore 2. APPENDIX C PROOF OF TEOREM 3 In this appendix, we will present a detailed proof of Theore 3. We start y proving that the Iterative Relaxed MM Algorith is guaranteed to converge. Specifically, we will first prove that the ojective function 21) is driven downhill, i.e. C N Wu +1,Q +1 ) C N Wu,Q ), 32) where denotes the iteration index. To this end, let Wu, Q, and Q +1 denote the optial solution of optiization prole at iteration and +1, respectively. We consider two particular cases in ters of whether Wu and Q are feasile for the optiization prole 24) at the iteration +1. Case one: when Wu and Q are feasile for the optiization prole 24) at the iteration +1. In this case, we have Wu +1 C N W +1 u,q +1 ) CN W+1 u,q +1 ) 33) = in CN W u,q) 34) s.t. Constraints in 24). CNW u,q ) 35) = C N Wu,Q ), 36) where 31) follows ecause Wu and Q are feasile for the optiization prole 24) at the iteration + 1. Case two: when Wu and Q are not feasile for the optiization prole 24) at the iteration +1. In this case, Wu and Q only violate the following constraint ) TrA W u )+α TrA Q)+â C. 37) B ˆR Given the optiization prole considered in this paper, the per-bs power constraints and the aove constraint 37) are only feasiility constraints and the SINR constraints are oth feasiility and optiality constraints. This eans that the SINR constraints are the only constraints that deterine the optial solution of the optiization prole 24). We know that Wu and Q verifies the SINR constraints and the per-bs power constraints of the optiization prole 24) ecause Wu and Q are the optial solution of 24) at the iteration. Further, we know that Wu +1 and Q +1 are the optial solution of 24) at the iteration +1. Since violating the constraints 37) does not influence the optiality of 24), we have CNW u +1,Q +1 ) = in C NW u,q) s.t. Constraints in 24) C NW u,q ). 38) This proves that the ojective function 21) is driven downhill. Thus, the Iterative Relaxed MM Algorith is guaranteed to converge. The relaxed constraint 37) is an upper ound of the approxiated total ackhaul capacity constraint 14). Thus, the solution of the Iterative Relaxed MM Algorith is a feasile solution of the original optiization prole as ǫ tends to 0. This copletes the proof of theore 3. REFERENCES [1] J. Andrews, S. Buzzi, W. Choi, S. anly, A. Lozano, A. Soong, and J. Zhang, What will 5G e? IEEE Journal on Selected Areas in Counications, vol. 32, no. 6, pp , June [2] Y. Shi, J. Zhang, and K. Letaief, Group sparse eaforing for green cloud-ran, IEEE Transactions on Wireless Counications, vol. 13, no. 5, pp , May [3]. Dahrouj, A. Douik, O. Dhifallah, T. Y. Al-Naffouri, and M.-S. Alouini, Resource allocation in heterogeneous cloud radio access networks: advances and challenges, IEEE, Wireless Counications, vol. 22, no. 3, pp , June 2015.

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