Multiuser Downlink Beamforming: Rank-Constrained SDP Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong
Outline of Lecture Problem formulation: downlink beamforming SOCP formulation, SDP relaxation, and equivalent reformulation General framework: rank-constrained SDP Numerical results Summary Daniel P. Palomar 1
Problem Formulation: Downlink Beamforming Daniel P. Palomar 2
Downlink Beamforming Consider several multi-antenna base stations serving single-antenna users: Daniel P. Palomar 3
Signal Model Signal transmitted by the kth base station: x k (t) = m I k w m s m (t) where I k are the users assigned to the kth base station. Signal received by mth user: r m (t) = h H m,κ m x κm + k κ m h H m,kx k + n m (t) where κ m is the base station assigned to user m, h m,k is the channel vector from base station k to user m with correlation matrix R m,k. Daniel P. Palomar 4
SINR of the mth user is SINR Constraints SINR m = wmr H mm w m l m wh l R. mlw l + σm 2 We can similarly consider an instantaneous version of the SINR, as well as the SINR for a dirty-paper coding type of transmission. Optimization variables: beamvectors w l for l = 1,..., L. The SINR constraints are SINR m ρ m for m = 1,..., L. Daniel P. Palomar 5
Formulation of Optimal Beamforming Problem Minimization of the overall transmitted power subject to SINR constraints: minimize {w l } subject to L l=1 w l 2 wm H R mmw m l m wh l R mlw l +σm 2 ρ m, m = 1,..., L. This beamforming optimization is nonconvex!! Daniel P. Palomar 6
SOCP Formulation Daniel P. Palomar 7
Assume rank-one direct channels: R mm = h m h H m. By adding, w.l.o.g., the constraint w H mh m 0, we can write the beamforming problem as minimize {w l } subject to L l=1 w l 2 ( w H m h m ) 2 ρm ( l m wh l R mlw l + σ 2 m) w H mh m 0., m Observe we can write wl H R ml w l + σm 2 = l m 1/2 R m w m 2 Daniel P. Palomar 8
where w m = w 1. w L 1 and Rm = diag R m1. 0. R ml σ 2 m. The beamforming problem can be finally written as the SOCP: minimize {w l } subject to L l=1 w l 2 w H mh m ρ m R1/2 m w m, m. Daniel P. Palomar 9
SDP Relaxation Daniel P. Palomar 10
Let s start by rewriting the SINR constraint wmr H mm w m l m wh l R mlw l + σm 2 ρ m as w H mr mm w m ρ m l m w H l R ml w l ρ m σ 2 m. Then, define the rank-one matrix: X l = w l w H l. Notation: observe that w H l Aw l = Tr (AX l ) A X l. Daniel P. Palomar 11
The beamforming optimization can be rewritten as minimize L X 1,...,X l=1 I X l L subject to R mm X m ρ m l m R ml X l ρ m σm, 2 m X l 0 rank (X l ) = 1 This is still a nonconvex problem, but if we remove the rank-one constraint, it becomes an SDP! Thus, a relaxation of the beamforming problem is an SDP which can be solved in polynomial time. Daniel P. Palomar 12
Equivalent Reformulation Daniel P. Palomar 13
An equivalent reformulation was obtained in the seminal work: M. Bengtsson and B. Ottersten, Optimal and suboptimal trasmit beamforming, in Handbook of Antennas in Wireless Communications, ch. 18, L.C. Godara, Ed., Boca Raton, FL: CRC Press, Aug. 2001. M. Bengtsson and B. Ottersten, Optimal transmit beamforming using semidefinite optimization, in Proc. of 37th Annual Allerton Conference on Communication, Control, and Computing, Sept. 1999. Theorem: If the SDP relaxation is feasible, then it has at least one rank-one solution. The proof is quite involved as it requires tools from Lagrange duality and Perron-Frobenius theory on nonnegative matrices. Daniel P. Palomar 14
Lemma: The following two problems have the same unique optimal solution λ: minimize λ,{u m } subject to L l=1 λ lρ l σl 2 (I λ m R mm + ) l m λ lρ l R ml u H m u i = 0, m u m = 1 λ m 0 and maximize L λ l=1 λ lρ l σl 2 subject to I λ m R mm + l m λ lρ l R ml 0, m λ m 0. Daniel P. Palomar 15
The next two steps of the proof as the following: 1. show that the dual problem of the beamforming problem is exactly the maximization in the previous lemma (this is quite simple) 2. show that the beamforming problem can be rewritten as the minimization in the previous lemma (this is quite challenging and requires Perron-Frobenius theory). At this point, one just needs to realize that what has been proved is that strong duality holds. This implies that an optimal solution of the original problem is also an optimal solution of the relaxed problem; hence, the relaxed problem must have a rank-one solution. Daniel P. Palomar 16
More General Problem Formulation: Downlink Beamforming Daniel P. Palomar 17
Soft-Shaping Interference Constraints Constraint to control the amount of interference generated along some particular direction: N [ h H E m,k x k (t) 2] τ m for m = L + 1,..., M. k=1 Defining S ml = h m,κl h H m,κ l, we can rewrite it as L l=1 w H l S ml w l τ m for m = L + 1,..., M. We can similarly consider long-term constraints, high rank matrices S ml, total power transmitted from one particular base station to one external user. Daniel P. Palomar 18
Individual Shaping Interference Constraints Some interference constraints only affect one beamvector and we call them individual constraints. We consider the following two types of constraints: w H l B l w l = 0 for l E w H l B l w l 0 for l / E. The following global null shaping constraint is equivalent to individual constraints (assuming S ml 0): L l=1 w H l S ml w l 0. Daniel P. Palomar 19
Formulation of Optimal Beamforming Problem Minimization of the overall transmitted power subject to all the previous SINR and interference constraints: minimize {w l } subject to L l=1 w l 2 wm H R mmw m l m wh l R mlw l +σm 2 ρ m, L l=1 wh l S mlw l τ m, m = 1,..., L m = L + 1,..., M wl HB lw l = ( ) 0, l (/ ) E 1 wl HD lw l = ( ) 0, l (/ ) E 2. This beamforming optimization is nonconvex!! Daniel P. Palomar 20
SDP Relaxation Let s start by rewriting the SINR constraint wmr H mm w m l m wh l R mlw l + σm 2 ρ m as w H mr mm w m ρ m l m w H l R ml w l ρ m σ 2 m. Then, define the rank-one matrix: X l = w l w H l. Notation: observe that w H l Aw l = Tr (AX l ) A X l. Daniel P. Palomar 21
The beamforming optimization can be rewritten as minimize L X 1,...,X l=1 I X l L subject to R mm X m ρ m l m R ml X l ρ m σm, 2 m = 1,..., L L l=1 S ml X l τ m, m = L + 1,..., M B l X l = ( ) 0, l (/ ) E 1 D l X l = ( ) 0, l (/ ) E 2 X l 0, rank (X l ) = 1 l = 1,..., L. This is still a nonconvex problem, but if we remove the rank-one constraint, it becomes an SDP! Thus, a relaxation of the beamforming problem is an SDP which can be solved in polynomial time. Daniel P. Palomar 22
Existing Literature M. Bengtsson and B. Ottersten, Optimal and suboptimal trasmit beamforming, in Handbook of Antennas in Wireless Communications, ch. 18, L.C. Godara, Ed., Boca Raton, FL: CRC Press, Aug. 2001. M. Bengtsson and B. Ottersten, Optimal transmit beamforming using semidefinite optimization, in Proc. of 37th Annual Allerton Conference on Communication, Control, and Computing, Sept. 1999. D. Hammarwall, M. Bengtsson, and B. Ottersten, On downlink beamforming with indefinite shaping constraints, IEEE Trans. on Signal Processing, vol. 54, no. 9, Sept. 2006. G. Scutari, D. P. Palomar, and S. Barbarossa, Cognitive MIMO Radio, IEEE Signal Processing Magazine, vol. 25, no. 6, Nov. 2008. Z.-Q. Luo and T.-H. Chang, SDP relaxation of homogeneous quadratic optimization: approximation bounds and applications, Convex Optimization in Signal Processing and Communications, D.P. Palomar and Y. Eldar, Eds., Cambridge University Press, to appear in 2009. Daniel P. Palomar 23
Curiosity: Multicast Downlink Beamforming This small variation was proven to be NP-hard: N. D. Sidiropoulos, T. N. Davidson, Z.-Q. (Tom) Luo, Transmit Beamforming for Physical Layer Multicasting, IEEE Trans. on Signal Processing, vol. 54, no. 6, June 2006. Daniel P. Palomar 24
Rank-Constrained SDP Daniel P. Palomar 25
Separable SDP Consider the following separable SDP: minimize X 1,...,X L subject to L l=1 C l X l L l=1 A ml X l = b m, m = 1,..., M X l 0, l = 1,..., L. If we solve this SDP, we will obtain an optimal solution X 1,..., X L with an arbitrary rank profile. Can achieve a low-rank optimal solution? (recall that for the beamforming problem we want a rank-one solution) Daniel P. Palomar 26
Rank-Constrained Solutions A general framework for rank-constrained SDP was developed in Yongwei Huang and Daniel P. Palomar, Rank-Constrained Separable Semidefinite Programming for Optimal Beamforming Design, in Proc. IEEE International Symposium on Information Theory (ISIT 09), Seoul, Korea, June 28 - July 3, 2009. Yongwei Huang and Daniel P. Palomar, Rank-Constrained Separable Semidefinite Programming with Applications to Optimal Beamforming, IEEE Trans. on Signal Processing, vol. 58, no. 2, pp. 664-678, Feb. 2010. Theorem: Suppose the SDP is solvable and strictly feasible. Then, it has always an optimal solution such that L l=1 rank 2 (X l ) M. Daniel P. Palomar 27
Purification-Type Algorithm Algorithm: Rank-constrained solution procedure for separable SDP 1. Solve the SDP finding X 1,..., X L with arbitrary rank. 2. while L l=1 rank2 (X l ) > M, then 3. decompose X l = V l Vl H and find a nonzero solution { l } to L Vl H A ml V l l = 0 m = 1,..., M. l=1 4. choose maximum eigenvalue λ max of 1,..., L 5. compute X l = V l (I 1 λ max l ) V H l for l = 1,..., L. 6. end while Daniel P. Palomar 28
A Simple Consequence Observe that if L = 1, then rank (X) M (real-valued case by [Pataki 98]). Furthermore, if M 3, then rank (X) 1: minimize X 0 C X subject to A m X = b m, m = 1,..., M. Corollary: The following QCQP has no duality gap: minimize x x H Cx subject to x H A m x = b m, m = 1, 2, 3. Daniel P. Palomar 29
Rank-One Solutions for Beamforming Our result allows us to find rank-constrained solutions satisfying L l=1 rank2 (X l ) M. But, what about rank-one solutions for beamforming? Recall that to write the beamforming problem as an SDP we used X l = w l w H l. Corollary: Suppose the SDP is solvable and strictly feasible, and that any optimal primal solution has no zero matrix component. Then, if M L + 2, the SDP has always a rank-one optimal solution. Daniel P. Palomar 30
The following conditions guarantee that any optimal solution has no zero component: L M A ml 0, l m, m, l = 1,..., L b m > 0, m = 1,..., L. These conditions are satisfied by the beamforming problem: minimize L X 1,...,X l=1 I X l L subject to R mm X m ρ m l m R ml X l ρ m σm, 2 m = 1,..., L L l=1 S ml X l τ m, m = L + 1,..., M X l 0, l = 1,..., L. Daniel P. Palomar 31
Original result: Contribution Thus Far original seminal 2001 paper by Bengtsson and Ottersten showed strong duality for the rank-one beamforming problem with M = L the proof relied on Perron-Frobenius nonnegative matrix theory and on the uplink-downlink beamforming duality it seemed that strong duality was specific to that scenario. Current result: applies to a much wide problem formulation of separable SDP in the context of beamforming, it allows for external interference constraints proof based purely on optimization concepts (no Perron-Frobenius, no uplink-downlink duality). Daniel P. Palomar 32
Rank-Constrained SDP with Additional Individual Constraints Daniel P. Palomar 33
Separable SDP with Individual Constraints Consider the following separable SDP with individual constrains: minimize L X 1,...,X l=1 C l X l L L subject to l=1 A ml X l = b m, m = 1,..., M B l X l = ( ) 0, l (/ ) E 1 D l X l = ( ) 0, l (/ ) E 2 X l 0, l = 1,..., L. By directly invoking our previous result we get: L rank 2 (X l ) M + 2L. Can we do better? l=1 Daniel P. Palomar 34
Low-Rank Solutions Theorem: Suppose the SDP is solvable and strictly feasible. Then, it has always an optimal solution such that L l=1 rank (X l ) M. Observe that the main difference is the loss of the square in the ranks. This implies that solutions will have higher ranks. Can we do better? Daniel P. Palomar 35
We can sharpen the previous result is we assume that some of the matrices (B l, D l ) are semidefinite. Theorem: Suppose the SDP is solvable and strictly feasible, and that (B l, D l ) for l = 1,..., L are semidefinite. Then, the SDP has always an optimal solution such that L l=1 rank 2 (X l ) + L l=l +1 rank (X l ) M. Observation: individual constraints defined by semidefinite matrices do not contribute to the loss of the square in the rank. Daniel P. Palomar 36
A Simple Consequence Corollary: The following QCQP has no duality gap (assuming B and D semidefinite): minimize x x H Cx subject to x H A m x = ( ) b m, m = 1, 2, 3 x H Bx = ( ) 0 x H Dx = ( ) 0. Daniel P. Palomar 37
Rank-One Solutions for Beamforming What about rank-one solutions for beamforming? Recall that to write the beamforming problem as an SDP we used X l = w l w H l. Corollary: Suppose the SDP is solvable and strictly feasible, and that any optimal primal solution has no zero matrix component. Then, the SDP has a rank-one solution if one of the following hold: M = L M L + 2 and (B l, D l ) are semidefinite for l = 1,..., L. Daniel P. Palomar 38
Numerical Simulations Daniel P. Palomar 39
Simulation Setup Scenario: single base station with K = 8 antennas serving 3 users at 5, 10, and 25 with an angular spread of σ θ = 2. SINR constraints set to: SINR l 1 for l = 1, 2, 3. External users from another coexisting wireless system at 30 and 50 with no angular spread. Soft-shaping interference constraints for external users set to: τ 1 = 0.001 and τ 2 = 0.0001. Daniel P. Palomar 40
Simulation #1 With no soft-shaping interference constraints (15.36 dbm): Radiation pattern at the transmitter 15 20 25 dbw 30 35 40 45 90 60 30 0 30 60 90 azimut in degrees Daniel P. Palomar 41
Simulation #2 With soft-shaping interference constraints (18.54 dbm): Radiation pattern at the transmitter 10 15 20 25 dbw 30 35 40 45 50 90 60 30 0 30 60 90 azimut in degrees Daniel P. Palomar 42
Simulation #3 With soft- and null-shaping interference constraints (at 20, 30 and 50, 70 ) (20.88 dbm): Radiation pattern at the transmitter 10 20 30 dbw 40 50 60 70 90 60 30 0 30 60 90 azimut in degrees Daniel P. Palomar 43
Simulation #4 With soft- and null-shaping interference constraints (at 20, 50, and 70 with τ 1 = 0.00001, τ 2 = 0, τ 3 = 0.000001) as well as a derivative null constraint at 70 (15.66 dbm): Radiation pattern at the transmitter 20 30 40 dbw 50 60 70 80 90 60 30 0 30 60 90 azimut in degrees Daniel P. Palomar 44
Summary We have considered the optimal downlink beamforming problem that minimizes the transmission power subject to: SINR constraints for users within the system soft-shaping interference constraints to protect users from coexisting systems individual null-shaping interference constraints derivative interference constraints. The problem belongs to the class of rank-constrained separable SDP. We have proposed conditions under which strong duality holds as well as rank reduction procedures. Daniel P. Palomar 45