$L_2$-Box Optimization for Green Cloud-RAN via Network Adaptation

In this paper, we propose a reformulation for the Mixed Integer Programming (MIP) problem into an exact and continuous model through using the $\ell_2$-box technique to recast the binary constraints into a box with an $\ell_2$ sphere constraint. The …

Authors: Fan Zhang, Qiong Wu, Hao Wang

$L_2$-Box Optimization for Green Cloud-RAN via Network Adaptation
1 L 2 -Box Optimization for Green Cloud-RAN via Network Adaptation Fan Zhang, Qiong W u, Hao W ang, and Y uanming Shi Abstract —In this paper , we propose a reformulation f or the Mixed Integer Programming (MIP) problem into an exact and continuous model through using the ` 2 -box technique to recast the binary constraints into a box with an ` 2 sphere constraint. The ref ormulated problem can be tackled by a dual ascent algorithm combined with a Majorization-Minimization (MM) method for the subproblems to solve the network power con- sumption problem of the Cloud Radio Access Network (Cloud- RAN), and which leads to solving a sequence of Difference of Con vex (DC) subpr oblems handled by an inexact MM algorithm. After obtaining the final solution, we use it as the initial result of the bi-section Group Sparse Beamf orming (GSBF) algorithm to promote the group-sparsity of beamformers, rather than using the weighted ` 1 /` 2 -norm. Simulation results indicate that the new method outperforms the bi-section GSBF algorithm by achieving smaller network power consumption, especially in sparser cases, i.e., Cloud-RANs with a lot of Remote Radio Heads (RRHs) but fewer users. Index T erms —Cloud-RAN, ` 2 -box, DC problems, MM algo- rithms, dual ascent methods, inexact algorithms, group-sparsity . I . I N T R O D UC T I O N I T Cloud Radio Access Network (Cloud-RAN) is a network architecture proposed to meet the explosi ve growth of mobile data traffic. An important problem of Cloud-RAN is the energy ef ficiency consideration, due to the increasing power consumption of a large number of Remote Radio Heads (RRHs) as well as the fronthaul links. W e focus on the power consumption problem of green Cloud-RAN by jointly in volving the power consumption of the transport network and RRHs. Sev eral methods have been proposed to solve the Cloud-RAN power minimization problem. The network power minimization problem can be formu- lated into a Mixed Integer Programming (MIP) problem, which is mainly solv ed by three strategies. First of all, a global opti- mal solution can achiev e by the branch-and-bound method [1], but it may suffer from an exponential worst-case complexity and work slowly in practice. In order to alleviate the computa- tional burden, Y ang et. al. [1] derives an approximation of the MIP problem by relaxing the binary constraint to a [0 , 1] box constraint. The most related method is a three-stage approach named Group Sparse Beamforming (GSBF) algorithm [2], [3], which balances between the computational complexity and the accuracy of solution. This algorithm exploits the group sparsity structure of beamformers with the priori knowledge. Fan Zhang, Qiong Wu, Hao W ang, and Y uanming Shi are with School of Information Science and T echnology , ShanghaiT ech Univ er- sity , Shanghai, P .R. China (email: { wuqiong, zhangfan4, wanghao1, shiym } @shanghaiT ech.edu.cn). Specifically , in the first stage, it solves a conv ex weighted ` 1 /` 2 norm relaxation of the MIP problem to induce the sparsity of the beamformers. The second stage generates an ordering rule to decide which RRH has a higher priority to be switched off. In the third stage, a selection procedure is performed to determine the best combination of the activ e and the sleep set of RRHs. Ho wever , the GSBF algorithm generally can not guarantee to provide a high accuracy solution. In this paper, we propose a new formulation of the Cloud- RAN po wer consumption problem along with a dual ascent method combined with an inexact Majorization-Minimization (MM) algorithm. The major idea of this recast is the ` 2 - box technique, introduced recently in [4] as a continuous equiv alent formulation of the binary constraints. By using this technique to replace the binary constraint with the intersection of a box and ` 2 sphere, we obtain a new formulation of the Cloud-RAN po wer consumption problem. As a result, a local optimal solution can be quickly found by continuous algo- rithms, while for the original mixed-binary problem, excessi ve computational effort may be needed to find a comparable solution. Therefore, the solution of the proposed reformulated problem can be employed to initialize the bi-section GSBF algorithm, which can be a more powerful sparsity-promoting tool than the weighted ` 1 /` 2 norm relaxation. It should be emphasized that our exact and continuous formulation of the network power consumption problem, in contrast to a relaxation model, can enable a better solution be reached. The reformulated problem can be addressed by our pro- posed MM dual ascent algorithm and test by the numerical experiments. The numerical results manifest that our proposed framew ork obviously improves the network energy ef ficiency , especially in the case of more RRHs but fe wer users. I I . P RO B L E M D E S C R I P T I O N A. System model W e consider a Cloud-RAN with L RRHs and K single- antenna Mobile Users (MUs), where the l -th RRH is equipped with N l antennas. In this architecture, all the Baseband Units (BBU) are mov ed in to a single BBU pool, creating a set of shared processing resources, and enabling ef ficient interfer - ence management and mobility management. All the RRHs are connected to the BB U pool through fronthaul links. In a beamforming frame work, let v lm ∈ C N l be the transmit beamforming vector from the l -th RRH to the k -th user, and 2 s k be the data symbol for user k with E [ | s k | 2 = 1] . The transmit signal at RRH l is given by x l = K X k =1 v lm s k , ∀ l ∈ L . (1) The channel propagation between user k and RRH l is denoted as h lm ∈ C N l , and n k ∈ C N (0 , σ 2 k ) is the additive Gaussian noise at user k . Therefore, the receiv ed signal at user k is then y k = X l ∈L h H kl v lk s k + X i 6 = k X l ∈L h H kl v li s i + n k . (2) W e assume that all the users treat the interference as noise [5]. The corresponding signal-to-interference-plus-noise ratio (SINR) for user k is Γ k =   P l ∈L h H kl v lk   2 P i 6 = k   P l ∈L h H kl v li   2 + σ 2 k . (3) Each RRH has its own transmit po wer constraint K X k =1 k v lk k 2 2 ≤ p P l , ∀ l ∈ L , (4) where P l is the maximum transmit power of the l -th RRH. B. Network power consumption minimization Due to the high density of RRHs and their joint transmis- sion, the energy used for signal transmission can be reduced significantly . Howe ver , the power consumption of the transport network becomes numerous and cannot be ignored. In order to reduce the network po wer consumption, it is essential to put some RRHs into sleep whenever possible. W e introduce a binary vector z = ( z 1 , ..., z L ) T to represent the activ e RRH, i.e., z l = 1 denotes the l -th RRH is activ e, and z l = 0 means the l -th RRH is sleeping. Denote the relativ e fronthaul link power consumption by P c l , and the inefficient of drain efficienc y of the radio frequenc y po wer amplifier by η l . Then the network power consumption p ( z , v ) is the sum of total transmit power consumption and the total relativ e fronthaul links power consumption: p ( z , v ) = X l ∈L P c l z l + X l ∈L 1 η l k ˜ v l k 2 2 . (5) where, for con venience, let ˜ v l = [ v T l 1 , ..., v T lK ] T ∈ C K N l × 1 . W ith target SINRs γ = ( γ 1 , ..., γ K ) T , the SINR constraint for user k as a second-order cone (SOC) constraint [6] must be satisfied. Therefore, the power minimization problem can be formulated as a MIP problem [7] min ( z , v ) p ( z , v ) s.t. s X i 6 = k k h H k v i k 2 2 + σ 2 k ≤ 1 γ k < ( h H k v k ) , k ∈ S , k ˜ v l k ≤ z l p P l , z l = { 0 , 1 } , l ∈ L , (6) where < ( · ) denotes the real part. Fig. 1: Geometric illustration of the ` 2 -box technique. The hollow circle is the intersection of the box and ` 2 -norm. I I I . ` 2 - B OX O P T I M I Z AT I O N R E F O R M U L A T I O N In this section, we propose a new formulation of the Cloud- RAN power consumption problem. An ` 2 -box technique is proposed to replace the binary by the intersection between a box and an ` 2 sphere as described in (7). A geometric illustration of the ` 2 -box technique is depicted in Fig. 1. x ∈ { 0 , 1 } n ⇔ x ∈ [0 , 1] n ∩  x : k x − 1 2 1 n k 2 2 = n 4  , (7) where 1 n is an n -dimension all-one vector . Therefore, the MIP (6) can be recast into min ( z , v ) X l ∈L P c l z l + X l ∈L 1 η l k ˜ v l k 2 2 s.t. s X i 6 = k k h H k v i k 2 2 + σ 2 k ≤ 1 γ k < ( h H k v k ) , k ∈ S , k ˜ v l k ≤ z l p P l , 0 ≤ z l ≤ 1 , l ∈ L , k z − 1 2 1 L k 2 2 = L 4 . (8) The main difficulty of this problem comes from the non- con vexity of the sphere constraint, and we are well aware that it may be effort-consuming to find the global optimal solution. Howe ver , rather than solve (8) directly for the global optimal solution, rather than use the global nonlinear method to solve (8) until global optimal, we only use local nonlinear algorithm to address (8) and use the (local) solution as the initial point in the first stage of the bi-section GSBF algorithm, which can further induce the group sparsity of the beamformers. I V . M M D UA L A S C E N T A L G O R I T H M In this section, we design a dual ascent algorithm [8] incorporated with an inexact MM algorithm to solve our proposed ` 2 -box Cloud-RAN po wer minimization problem. Notice that (8) is a con vex problem except for the noncon vex ` 2 sphere constraint. Therefore, we focus on dealing with the sphere constraint to construct our algorithm. For simplicity , let φ ( v , z ) = X l ∈L P c l z l + X l ∈L 1 η l k ˜ v l k 2 2 , (9) and Ω = { ( z , v ) | q P i 6 = k k h H k v i k 2 2 + σ 2 k ≤ 1 γ k < ( h H k v k ) , k ∈ K ; 3 k ˜ v l k 2 ≤ z l √ P l , 0 ≤ z l ≤ 1 , l ∈ L} . Notice that Ω is a conv ex set. Now (8) can be stated as min ( z , v ) ∈ Ω φ ( z , v ) s.t. k z − 1 2 1 L k 2 2 = L 4 . (10) A natural way to solve such a problem is to dualize the sphere constraint. Letting λ be the multiplier associated with the sphere constraint, the Lagrangian of (10) is defined as L ( z , v , λ ) = φ ( z , v ) + λ ( L 4 − k z − 1 2 1 L k 2 2 ) , (11) for ( z , v ) ∈ Ω . An alternativ e option is to use the augmented Lagrangian, but we do not suggest such approach since it will sev erely increase the nonlinearity of the resulted subproblems by introducing a fourth-order polynomial in the objective. The dual of objective is then giv en by (12) g ( λ ) = inf ( z , v ) ∈ Ω L ( z , v , λ ) , (12) and we have the dual problem (13) max λ g ( λ ) = max λ inf ( z , v ) ∈ Ω L ( z , v , λ ) . (13) Now we are ready to provide our dual ascent frame work. The dual ascent method consists of two stages: the first stage is to update the primal v ariables by minimizing the Lagrangian for a fixed dual variable λ , ( z t +1 , v t +1 ) = arg min ( z , v ) ∈ Ω L ( z , v , λ t ) , (14) and then update the dual variable based on the constraint residual λ t +1 = λ t + α t ( L 4 − k z t +1 − 1 2 1 L k 2 2 ) , (15) where α t > 0 is the step size of the dual update. Since z is restricted in Ω , it holds true z t +1 ∈ [0 , 1] L . It follows that ( L 4 − k z t +1 − 1 2 1 L k 2 2 ) ≥ 0 , where the equality holds true if and only if z ∈ { 0 , 1 } L . As a result, the dual variable λ 0 should be initialized to be positi ve to penalize the violation of the sphere constraint. Since the step size α t > 0 , λ is maintaining positi ve and increasing incrementally during the solution. Consequently , λ t k z − 1 2 1 L k 2 2 is kept con vex with respect to z . In other words, the subproblem (14) is a DC problem [9]. At the s -th iteration of MM algorithm [10], a con vex surrogate objective ˆ L ( z , v , λ t , z ( s ) ) is generated by linearizing the second con vex function while keeping the first function unchanged ˆ L ( z , v , λ t , z ( s ) ) = φ ( z , v ) + λ t ( L 4 − k z ( s ) − 1 2 1 L k 2 2 − 2( z − 1 2 1 L ) T ( z − z ( s ) )) , (16) where ( z , v ) ∈ Ω and { ( z ( s ) , v ( s ) ) } represents a se- quence of primal iterates for the subproblems. T o obtain ( z ( s +1) , v ( s +1) ) in each iteration of MM algorithm, we need to solve min ( z , v ) ∈ Ω ˆ L ( z , v , λ t , z ( s ) ) , (17) which can easily be solved by CVX solver [11]. It should be noticed that generally the subproblem does not need to be solved exactly if sufficient improv ement on the primal variable can be achie ved. Therefore, we also solve (14) inexactly , meaning we only solve a few subproblems (17). This inexact strategy has prov en to be able to reduce the computational cost substantially The description of the entire MM dual ascent algorithm is stated in Algorithm 1. The conv ergence analysis of dual ascent method is provided by [12]. Since MM algorithm is proposed [13] as a generaliza- tion of the EM algorithm, MM algorithm inherits the con ver- gence properties of the EM algorithm [14]. The con vergence results of the EM algorithm includes: the lik elihood sequence of the EM algorithm is nondecreasing and con vergent [15], and that the limit points of the EM algorithm are stationary points of the likelihood [16]. After obtaining the sparse beamformer v ∗ , we use its group sparsity to generate the ordering criterion, and then adopt the same binary search procedure as the bi-Section GSBF algorithm in [6] to obtain the final results. Algorithm 1 MM dual ascent algorithm 1: Gi ven the tolerances  1 > 0 ,  2 > 0 and  3 > 0 . 2: Initialize z , v and λ > 0 . 3: while | λ t +1 − λ t | ≥  2 or k z t +1 − z t k 2 + k v t +1 − v t k F ≥  3 do 4: while k z ( s +1) − z ( s ) k 2 + k v ( s +1) − v ( s ) k F ≥  1 do 5: Update ( z , v ) : ( z ( s +1) , v ( s +1) ) = arg min z , v ˆ L ( z , v , λ t , z ( s ) ) 6: s = s + 1 7: end while 8: z t = z ( s ) , v t = v ( s ) 9: Update λ : λ t +1 = λ t + α t ( L 4 − k z t − 1 2 1 L k 2 2 ) 10: Set t = t + 1 11: end while 12: r eturn ˜ v l , for l = 1 , ..., L . V . S I M U L AT I O N R E S U LT S In this section, we describe the experimental setting in- cluding the initial point and the algorithm parameters. In our e xperiment we check the con vergence of the proposed method, and exhibits the effect iv eness of our proposed method compared with contemporary methods. The initial point plays a significantly important role while solving the noncon ve x problems. Instead of randomly choos- ing initial point, we remov e the sphere constraint and solve the approximation problem ( z 0 , v 0 ) = arg min ( z , v ) ∈ Ω φ ( z , v ) (18) to deriv e the initial point. This generally renders better es- timate than random initial point of the solution. The DC subproblem is solved inexactly under the stopping criterion k z t +1 − z t k 2 ≤  1 with  1 = 10 − 5 . The main algorithm is terminated whene ver the primal iterates or the dual iterates con verge, i.e., we use termination criterion | λ t +1 − λ t | ≤  2 4 or k z s +1 − z s k 2 + k v s +1 − v s k F ≤  3 with  2 = 10 − 2 ,  3 = 10 − 3 . In our experiment, we consider a network with L = 10 , K = 6 , 2 -antenna RRHs and single-antenna MUs uni- formly and independently distributed in the square region [ − 1000 , 1000] × [ − 1000 , 1000] meters. W e set all the relativ e transport link po wer consumption to be P c l = 13 W, l = 1 , ..., L , and the inefficient of power amplifier [17] at each RRH is η l = 1 4 . In our first experiment, we show the efficienc y of Al- gorithm 1. The evolution of tol t 1 = log k z t +1 − z t k 2 and tol t 2 = log k v t +1 − v t k F , where k · k F is the Frobenius norm, is depicted in Fig. 2.(a) and (b) to represent the differences between the current and the previous iterates. As shown in Fig. 2, both of the primal variables z and v are conv erging efficienc y . In particular , tol 1 and tol 2 decrease dramatically about 10 − 5 in the first five iterations. (a) Evolution of tol 1 (b) Evolution of tol 2 Fig. 2: Conv ergence of the primal and dual variables. W e also compare our proposed method with the existing methods including: MIP which is the branch-and-bound al- gorithm for solving the MIP problem (6) for global optimal solution, RMIP which is the algorithm in [1] for solving the relaxed MIP problem, and GSBF which is the bi-section GSBF algorithm [2], [3]. The av erage network po wer consumption with dif ferent target SINR is shown in Fig. 3. The simulation results indicate that the ` 2 -box algorithm outperforms the GSBF and RMIP algorithm for different target SINR. This advantage becomes obvious in situations with smaller SINR. V I . C O N C L U S I O N A N D F U T U R E W O R K In this paper , we have proposed a new formulation of the Cloud-RAN power consumption problem by using the ` 2 -box technique, which replaces the binary constraint to two continuous constraints: a box constraint and a sphere constraint. W e design a dual ascent algorithm to solv e the new ` 2 -box optimization problem leading a sequence of DC subproblems. W e apply MM algorithm to inexactly solve the subproblem. The effecti veness of our proposed reformulation and algorithm is demonstrated in numerical experiment. Our method exhibited lower network consumptions of different target SINR than the GSBF algorithm. Our in vestigation leads to a variety of open questions. The final solution found by a nonlinear solver is often sensitive to the initial point. Therefore, it would be useful to explore better estimate of the global optimal solution to initialize our algorithm. Furthermore, the binary constraint is also equi valent to the intersection of a box and an ` p sphere with p ∈ (0 , + ∞ ) . Fig. 3: A verage network power consumption versus target SINR. It would be interesting to inv estigate the performance of other ` p -box techniques, e.g., ` 1 -box or ` 1 2 -box. 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