IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 2, FEBRUARY 2015
195
A First-Order Optimal Zero-Forcing Beamformer Design for Multiuser MIMO Systems via a Regularized Dual Accelerated Gradient Method Bin Li, Hai Huyen Dam, Antonio Cantoni, Life Fellow, IEEE, and Kok Lay Teo, Senior Member, IEEE
Abstract—A first-order zero-forcing beamformer design is proposed in this letter for MU-MIMO systems under per-antenna power constraints (PAPC). By forming the regularized dual problem, first-order methods can be applied. To achieve the limit convergence rate of first-order methods, an accelerated method is introduced to solve the regularized dual problem. Simulation results are provided to show the effectiveness of the proposed method. Index Terms—Zero-forcing beamforming (ZFBF), per-antenna power constraint (PAPC), MIMO systems, first-order method, accelerated gradient method.
I. I NTRODUCTION
II. P ROBLEM F ORMULATION Consider a MISO multiuser broadcast channel ym = hH m x + nm ,
Manuscript received August 5, 2014; accepted December 7, 2014. Date of publication December 18, 2014; date of current version February 6, 2015. This work was supported by a Discovery Grant from the Australia Research Council (No. DP120103859). The associate editor coordinating the review of this paper and approving it for publication was M. C. Gursoy. B. Li, H. H. Dam, and K. L. Teo are with the Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia (e-mail: bin.li@curtin. edu.au;
[email protected];
[email protected]). A. Cantoni is with the School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, WA 6009, Australia (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LCOMM.2014.2381245
(1)
where ym is the received signal of the mth user, hm is the channel vector of length N of the mth user, x is the transmitted vector of length N, and nm is the complex Gaussian noise with mean 0 and variance σ2 , and (·)H denotes the conjugate transpose. Equation (1) can be written in a compact form given below:
I
N this letter, we consider a zero-forcing beamformer design problem under per antenna power constraints [1]–[4]. This type of constraint is more realistic for real world applications, because each antenna of the transmitter has its own amplifier and the linear operation is important for modern efficient modulation, e.g. Orthogonal Frequency Division Multiplexing (OFDM) [1]. The minimum user information rate is taken as the performance measure. This problem can be solved by applying the interior point method [5]. However, it is expensive to realize this algorithms with hardware due to the fact that a lot of complex computations are involved, such as the calculations of a Hessian matrix and the inverse of a high dimensional matrix in each iteration [5]. Thus, we shift our focus to first-order methods, for which the iteration complexity is low. Furthermore, first-order methods are more robust in practice, especially when the problem size becomes larger [6]. Here, a regularized dual method together with an accelerated method is proposed. By introducing a regularized Lagrangian, an approximated dual problem of the original problem is formed. The newly formed problem can be solved by applying first-order methods. Instead of using the gradient method, which yields a convergence rate of O(1/k), we introduce an accelerated method [7]. By applying this method, the limit convergence rate of first-order methods, i.e., O(1/k2 ), can be achieved [8].
m = 1, 2, . . . , M
y = Hx + n
(2)
where y = [y1 , y2 , . . . , yM ] , H = [h1 , h2 , . . . , hM ]H , n = [n1 , n2 , . . . , nM ] , and (·) denotes the transpose. Zero-forcing pre-coding transmitter is applied, i.e., x = Ws √ HW = Λ
(3) (4)
where s is the information vector of length M such that E{ssH } = I, I denotes the identity matrix of appropriate dimension, W is an M × N complex matrix, and Λ denotes a real and positive diagonal matrix. The per-antenna power constraints are imposed as follows: M
2
∑ eTn wm
≤ P,
n = 1, 2, . . . , N
(5)
m=1
where wm is the mth column vector of W, en is a vector of length N with an 1 in the nth element while 0 in the other elements, and P is the maximum allowable power on each antenna. The information rate for each user is denoted by r(m) and is given by r(m) = log2 (1 + SINR(m)) ,
j = 1, 2, . . . , M
(6)
where SINR(m) is the signal-to-interference-plus-noise ratio (SINR) for each user, which is given by SINR(m) =
|(HW)m,m |2 2 ∑ j, j=m (HW) j,m + σ2m
(7)
m, j = 1, 2, . . . , M. From (4), we have ∑ j, j=m |(HW) j,m |2 = 0.
1089-7798 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
196
IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 2, FEBRUARY 2015
In this letter, our objective is to maximize the minimum user information rate subject to (4) and (5), which can be formally stated below: Problem 2.1: max
s.t.
log2 M
H 2 h wm 1+ m 2 ≥ r0 σ 2
∑ en wm
max
d(λ, v, µ)
s.t.
λ 0
λ,v,µ
m = 1, . . . , M
r0
wm ,r0
Then, the dual problem of Problem 2.2 can be written as Problem 3.1:
≤ P,
µ 0 where a 0 means each element of a is greater than or equal to 0. Problem 2.2 can be solved through solving its dual problem, i.e., Problem 3.1, since the Problem 2.2 is convex and the Slater’s condition holds [9]. As (10) is non-differentiable, we introduce ρt 2 and ρ x 2 , where ρ is a smoothing parameter and · denotes Euclidian norm. By appending ρt 2 and ρ x 2 into (9), we obtain
n = 1, . . . , N
m=1
∀ j = m
hHj wm = 0,
According to [2], [5] and by letting x = w1Re w1Im . . . wMRe wMIm = x1 x2 . . . x2M−1 x2M
Lρ (t, x, λ, v, µ) = L(t, x, λ, v, µ) + ρt 2 + ρ x 2
(11)
(8)
The optimal solution of min Lρ (t, x, λ, v, µ) can be written in a
where x ∈ R2NM , Problem 2.1 can then be simplified as the following convex optimization problem. Problem 2.2:
closed form as M 1 1 1− ∑ λm , x∗ = − cρ (µ) H (12) t∗ = 1 λ + H2 v 2ρ 2 m=1
min x,t
−t
x,t
∈ R2NM , g = where cρ (µ) = [g g · · · g ] 1 1 1 N [ ρ+µ1 ρ+µ2 · · · ρ+µN ] ∈ R and denotes the pointwise product. By considering (10), (11), and (12), we derive the regularized dual function as follows
s.t. H1 x −t1 x Ai x ≤ P,
i = 1, 2, . . . , N
H2 x = 0 where 1 is a vector of ones with appropriate dimension, a 0 means each element of a is less than or equal to 0, H1 ∈ RM×2NM and H2 ∈ R(2M(M−1)+M)×2NM . For the structures of H1 and H2 , the readers are referred to [5].
dρ (λ, v, µ) = min Lρ (t, x, λ, v, µ) x,t
= L(t ∗ , x∗ , λ, v, µ) + ρ(t ∗ )2 + ρ x∗ 2 =
III. A R EGULARIZED D UAL ACCELERATED G RADIENT M ETHOD
1 − 4ρ
A. Regularization of the Lagrangian In this section, we develop a regularized dual accelerated gradient method to solve Problem 2.2. To begin with, we introduce the following Lagrangian of Problem 2.2 as L (t, x, λ, v, µ) = −t + λ (H1 x + t1) N
+ v H2 x + ∑ µn
x An x − P
(9)
The corresponding dual function is denoted as d (λ, v, µ) = min L(t, x, λ, v, µ) = min x,t
+ ∑ µn (x An x − P) . n=1
2
∑ λm − 1
m=1
N
− P ∑ µn n=1
2 1 = − cρ (µ) H λ + H v 1 2 4 1 − 4ρ
M
∑
m=1
λ,v,µ
M
max
− t + λ (H1 x + t1) + v H2 x N
2 λm − 1
N
− P ∑ µn
(13)
n=1
cρ (µ) denotes taking the square root of cρ (µ) where componentwise. Then, the regularized dual problem which is referred to as Problem (3.1(ρ)) can be stated as follows.
n=1
x,t
1 cρ (µ) H H1 λ+H 2v 1 λ+H2 v 4
s.t. (10)
dρ (λ, v, µ) λ 0 µ 0.
LI et al.: A FIRST-ORDER OPTIMAL ZERO-FORCING BEAMFORMER DESIGN FOR MULTIUSER MIMO SYSTEMS
Remark 3.1: From x Ai x ≤ P, we have
197
TABLE I C OMPUTATIONAL C OMPLEXITY C OMPARISON
N
∑ x Ai x ≤ NP.
(14)
i=1 N
Note that ∑ Ai = IN , where IN is an N × N identity matrix.
Now we state our algorithm as follows:
i=1
Then, it follows that x 2 ≤ NP.
(15)
In fact, (15) provides a bound to x . By considering (11), the error introduced by the regularization is related to N and P. B. An Accelerated Gradient Algorithm In this section, we shall present an algorithm based on the accelerated gradient method [7] to solve Problem (3.1(ρ)). More specifically, for the kth iteration, we introduce the following variables ζkλ , ζkv , ζkµ defined as below: k−1 k (λ − λk−1 ) k+2 k−1 k (v − vk−1 ) ζkv = vk + k+2 k−1 k µ − µk−1 . ζkµ = µk + k+2
ζkλ = λk +
Then, we update λ, v, and µ as follows:
∂d 1 ρ λk+1 = max ζkλ + ,0 ∂λ ζk ,ζk ,ζk L v v µ vk+1 = ζkv +
1 ∂dρ ∂v ζk ,ζk ,ζk L v
v
dρ∗ = max dρ (y) = dρ (y∗ ). y
Proof: The proof follows from Proposition 2 of [7]. C. Computational Complexity Analysis (16)
(17)
v
v
µ
where M ∂dρ 1 k 1 − ∑ ζλ m 1 = ∂λ ζkv ,ζkv ,ζkµ 2ρ m=1 1 H1 cρ ζkµ pk 2 1 = − H2 cρ ζkµ pk 2 −
∂dρ ∂v ζkv ,ζkv ,ζkµ
∂dρ 1 k k = 2 (p ) An p − P ∂µn ζk ,ζkv ,ζkµ k 4 ζµ n + ρ λ k k pk = H 1 ζλ + H2 ζv
is the Lipschitz constant. and L
Theorem 3.1: Let yk = {λk , vk , µk } be the vector generated by Algorithm 1 in the kth iteration. Then 0 − y∗ 2 1 2L y = O dρ (yk ) − dρ∗ ≤ (k + 1)2 k2 where
µ
∂d 1 ρ µk+1 = max ζkµ + ,0 ∂µ ζk ,ζk ,ζk L
Algorithm 1 Initialize λ0 = λ−1 , v0 = v−1 , µ0 = µ−1 , x0 , and t 0 . Set ε for the tolerance and the smoothing parameter ρ. For the kth iteration Step 1 Compute xk and t k according to (12). Step 2 Compute f k = −t k + ρ(t k )2 + ρ xk 2 and dρk . If the duality gap, i.e., | f k − dρk | meet the tolerances ε, then stop. Otherwise, goto Step 3. Step 3 Compute λk+1 , vk+1 , µk+1 according to (16)–(18).
(18)
For the interior point method, the main computation is the calculation of the Newton step [5] and its iteration complexity is O(M 3 N 3 ). Algorithm 1’s main computation is (16)–(18) and the computational complexity is O(M 2 N 2 ). For the gradient method, although the iteration complexity remains the same as Algorithm 1, the convergence rate drops to O(1/ε). The details of the comparison are shown in Table I, where “IPM” stands for the interior point method. IV. N UMERICAL R ESULTS In this section we consider a base station array with a uniform planar circular array of N elements. M users are spread uniformly in a circle around the BS. The noise and the transmit levels are set as σ2 = 0.001 and P = 0.01. The simulation is implemented in Matlab environment. The channel adopted is from [10]. Unless otherwise specified, ε is set as 10−6 and ρ = 0.4 in the optimization algorithm. Fig. 1 plots the residual error of the duality gap in different scenarios. We compare the convergence rate of Algorithm 1 and the gradient method with different number of transmit antennas N and different number of users M in Fig. 1. As is shown in Fig. 1, the proposed method converges much faster than the gradient method in all scenarios. As it is observed, the curves generated by the accelerated algorithm fluctuate when they approach the optimal solutions. This is because the gradient we use is not a “real” gradient and hence it does not always give a descent direction.
198
IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 2, FEBRUARY 2015
TABLE II I TERATION C OMPUTATIONAL C OMPLEXITY C OMPARISON
V. C ONCLUSION
Fig. 1. Convergence rate comparison.
A first-order zero-forcing beamformer design, which is based on a regularized dual accelerated method, is proposed. As it √ was shown in this letter, an O(1/ ε) convergence rate can be achieved by applying this method. This convergence rate was proved to be the limit of the first-order methods, and it is much faster than the gradient method, i.e., O(1/ε). The iteration complexity of the proposed method is O(M 2 N 2 ), which is the same as that of the gradient method while much lower than the interior point method with O(M 3 N 3 ). ACKNOWLEDGMENT The first author would like to thank Dr. Changzhi Wu for the helpful discussions. This work was supported by a grant from the Australia Research Council (No. DP120103859). R EFERENCES
Fig. 2. Performance with different ρ.
Fig. 2 plots the minimum user information rate of the proposed beamformer as function of N with different ρ. Here, M = 3 and we use the performance of the optimal beamformer as a benchmark and compare the performance with different ρ. In Fig. 2, it can be seen that the performance converges to the optimal beamformer as ρ decreases. This is consistent with (11) which indicates that the effect of the regularization will decrease as we decrease ρ. One interesting phenomenon shown in Fig. 2 is that the performance loss becomes larger as N is increasing. This implies that the regularization error is related to the number of antennas N which is consistent with Remark 3.1 and (11). As expected, we also observe that the performance increases as N is increasing. Table II shows the iteration computational complexity for different scenarios. It can be seen that the iteration complexity of Algorithm 1 is much lower than that of the interior point method.
[1] A. Wiesel, Y. C. Eldar, and S. Shamai (Shitz), “Zero-forcing precoding and generalized inverses,” IEEE Trans. Signal Process., vol. 56, no. 9, pp. 4409–4418, Sep. 2008. [2] K. Karakayali, R. Yates, G. Foschini, and R. Valenzuela, “Optimal zeroforcing beamforming with per-antenna power constraints,” in Proc. IEEE Int. Symp. Circuits Syst. Inf. Theory, 2007, pp. 101–105. [3] H. H. Dam and A. Cantoni, “Interior point method for optimum zeroforcing beamforming with per-antenna power constraints and optimal step size,” Signal Process., vol. 106, pp. 10–14, Jan. 2015. [4] S. R. Lee, J. S. Kim, S. H. Moon, H. B. Kong, and I. Lee, “Zero-forcing beamforming in multiuser MISO downlink systems under per-antenna power constraint and equal-rate metric,” IEEE Trans. Wireless Commun., vol. 12, no. 1, pp. 20–24, Jan. 2013. [5] B. Li, H. H. Dam, A. Cantoni, and K. L. Teo, “A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints,” Optim. Lett., vol. 8, no. 6, pp. 1829–1843, Aug. 2014. [6] J. Koshal, A. Nedic, and U. V. Shanbhag, “Multiuser optimization: Distributed alogrithms and error analysis,” SIAM J. Optim., vol. 21, no. 3, pp. 1046–1081, 2011. [7] P. Tseng, “On accelerated proximal gradient methods for convex concave optimization,” SIAM J. Optim., May 2008, submitted for publication. [8] L. Vandenberghe, “Optimization methods for large-scale systems.” Lecture Notes, UCLA, Spring 2013–2014. [9] S. Boyd and L. Vandenberghe, Covex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [10] H. Suzuki, D. Robertson, N. L. Ratnayake, and K. Ziri-Castro, “Prediction and measurement of multiuser MIMO-OFDM channel in rural Australia,” in Proc. IEEE 75th Veh. Technol. Conf., 2012, pp. 1–5.
