Globally Optimal Beamforming for Rate Splitting Multiple Access
aa r X i v : . [ c s . I T ] F e b GLOBALLY OPTIMAL BEAMFORMING FOR RATE SPLITTING MULTIPLE ACCESS
Bho Matthiesen ⋆ , Yijie Mao † , Petar Popovski ‡ ⋆ , and Bruno Clerckx † ⋆ University of Bremen, Deptartment of Communications Engineering, Otto-Hahn-Allee 1, 28359 Bremen, Germany † Imperial College London, Deptartment of Electical and Electronic Engineering, London, United Kingdom ‡ Aalborg University, Department of Electronic Systems, 9220 Aalborg, Denmark
ABSTRACT
We consider globally optimal precoder design for rate splitting mul-tiple access in Gaussian multiple-input single-output downlink chan-nels with respect to weighted sum rate and energy efficiency max-imization. The proposed algorithm solves an instance of the jointmulticast and unicast beamforming problem and includes multicast-and unicast-only beamforming as special cases. Numerical resultsshow that it outperforms state-of-the-art algorithms in terms of nu-merical stability and converges almost twice as fast.
Index Terms — rate splitting, global optimization, resource al-location, energy efficiency, interference networks
1. INTRODUCTION
Rate splitting multiple access (RSMA) is a powerful non-orthogonaltransmission and robust interference management strategy for be-yond 5G communication networks [1–3]. The key idea is to spliteach message into common and private parts and transmit themby superposition coding [4]. The common message is decodedby multiple users, while the private message is only decoded bythe corresponding user employing successive interference cancel-lation (SIC). This approach allows arbitrary combinations of jointdecoding and treating interference as noise by flexibly adjustingthe message split. Recent results show that RSMA outperformsexisting multiple access schemes such as space division multipleaccess, power-domain non-orthogonal multiple access, orthogonalmultiple access, and multicasting in terms of weighted sum rate(WSR) [2, 5, 6] and energy efficiency (EE) [6, 7].This paper treats the important question of downlink multiple-input single-output (MISO) beamforming for RSMA with respectto WSR and EE maximization. The corresponding optimizationproblem is related to joint multicast and unicast precoding that isknown to be NP-hard [8, 9]. Existing works on RSMA focus onsuboptimal strategies to obtain computationally tractable algorithms[2, 6, 7, 10–13]. While several globally optimal algorithms for uni-cast beamforming [14, 15] and multicast beamforming [16] exist,joint solution methods are scarce. In particular, the procedure in[17] solves the power minimization problem and [18] maximizes theWSR for joint multicast and unicast beamforming. All these meth-ods are based on branch and bound (BB) in combination with thesecond-order cone (SOC) transformation in [19]. However, as thistransformation moves the complexity into the feasible set, pure BBmethods are prone to numerical problems, see Section 3. Instead,
This work is supported in part by the German Research Foundation(DFG) under grant EXC 2077 (University Allowance), by the U.K. En-gineering and Physical Sciences Research Council (EPSRC) under grantsEP/N015312/1 and EP/R511547/1, and by the North-German Supercomput-ing Alliance (HLRN). in this paper we design a successive incumbent transcending (SIT)BB algorithm to solve this beamforming problem with improved nu-merical stability and faster convergence. To the best of the authorsknowledge, this is the first globally optimal solution algorithm foran instance of the joint unicast and multicast problem with respectto EE maximization. It is also the first global optimization methodspecifically targeted at RSMA.
2. SYSTEM MODEL & PROBLEM STATEMENT
Consider the downlink in a wireless network where an M antennabase station (BS) serves K single-antenna users. The received sig-nal at user k , k ∈ K = { , . . . , K } , for each channel use is y k = h Hk x + n k , where the transmit signal x ∈ C M × is subject to anaverage power constraint P , h k is the complex-valued channel fromthe BS to user k , and n k is circularly symmetric complex whiteGaussian noise with unit power at user k .The transmitter employs 1-layer rate splitting [2,10], i.e., it splitsthe message W k intended for user k into a common part W c,k anda private part W p,k . Then, the common messages are combined intoa single message W c and these K + 1 messages are encoded withindependent Gaussian codebooks into s c , s , . . . , s K , each havingunit power. These symbols are combined with linear precoding intothe transmit signal x = p c s c + P k ∈K p k s k . The BS is subject toan average power constraint, i.e., k p c k + P k ∈K k p k k ≤ P .Each receiver uses SIC to first recover s c and then s k , treatingall other messages as noise. Asymptotic error free decoding of W c and W p,k is possible if the rates of these messages satisfy R c ≤ log(1 + γ c,k ) and R p,k ≤ log(1 + γ p,k ) , with signal to interferenceplus noise ratios (SINRs) γ c,k = | h Hk p c | P j ∈K | h Hk p j | + 1 , γ p,k = | h Hk p k | P j ∈K\ k | h Hk p j | + 1 . (1)The rate R c is shared across the users, where user k is allocated aportion C k corresponding to the rate of W c,k such that P k ∈K C k = R c . Then, the total rate of user k is R k = C k + R p,k .Observe that this system model includes multi-user linear pre-coding and multicast beamforming as special cases. We consider the following resource allocation problem under mini-mum rate R thk quality of service constraints max p ,..., p K , p c , c , γ c , γ p P k ∈K u k ( C k + log(1 + γ p,k )) µ (cid:0) k p c k + P k ∈K k p k k (cid:1) + P c (2a)s.t. γ c,k and γ p,k as in (1) (2b) X k ′ ∈K C k ′ ≤ log(1 + γ c,k ) , ∀ k ∈ K (2c) k ≥ max n , R thk − log(1 + γ p,k ) o , ∀ k ∈ K (2d) k p c k + X k ∈K k p k k ≤ P (2e)with nonnegative weight vector u = [ u , . . . ,u K ] = , nonnegativepower amplifier inefficiency µ , and positive static circuit power con-sumption P c . This problem has two operational meanings: With unitweights, it maximizes the EE and, with µ = 0 , P c = 1 , it maximizesthe WSR.The following problem is equivalent to (2) and will be solved bythe developed algorithm: max p c , p ,..., p K , c , γ p ,s, d , e P k ∈K u k ( C k + log(1 + γ p,k )) µ (cid:0) k p c k + P k ∈K k p k k (cid:1) + P c (3a)s.t. √ γ p,k (cid:18)X j ∈K\ k | h Hk p j | + 1 (cid:19) / ≤ h Hk p k (3b) √ s (cid:16)X j ∈K | h H p j | + 1 (cid:17) / ≤ h H p c (3c) √ s (cid:16)X j ∈K | h Hk p j | + 1 (cid:17) / ≤ d k , ∀ k > (3d) ( e k , d k ) ∈ C , ∀ k > (3e) ℜ{ h Hk p k } ≥ , ℑ{ h Hk p k } = 0 (3f) ℜ{ h H p c } ≥ , ℑ{ h H p c } = 0 (3g) ∀ k > d k ≥ , e k = h Hk p c (3h) X k ∈K C k ≤ log(1 + s ) (3i)(2d) and (2e) (3j)with ( e, d ) ∈ C = { e ∈ C , d ∈ R : d ≤ | e |} . (4)A crucial observation is that this problem is a second-order coneprogram (SOCP) for fixed s , γ p , except for constraint (3h). Hence,the nonconvexity of (2) is only due to the SINR expressions andnot due to the beamforming vectors. We will exploit this partialconvexity in the final algorithm to limit the numerical complexity. Proposition 1.
Problems (2) and (3) have the same optimal valueand every solution of (3) also solves (2) .Proof.
Omitted due to space constraints. Use the SOC reformulationfrom [19] for the SINRs, with additional auxiliary variables for themulticast beamformer p c [16].
3. GLOBALLY OPTIMAL BEAMFORMING
Problem (3) is an NP-hard nonconvex optimization problem due tothe multicast beamforming [8] and the power allocation in the privatemessages [9]. Previous global optimization algorithms for similarproblems rely on BB procedures with SOCP bounding [14, 15, 17,18]. However, this either leads to an infinite algorithm or requiresthe additional solution of several SOCPs to obtain a feasible pointin each iteration [14] which is required to obtain a finite algorithm.Moreover, the auxiliary SOCP that is solved in every iteration of theBB procedure is numerically challenging and leads to problems evenwith commercial state-of-the-art solvers like Mosek [20]. This canbe alleviated by the modified auxiliary problem in [14, §2.2.2] butthis approach greatly increases convergence times. Instead, we de-sign an algorithm based on the SIT scheme [21–24] and combineit with a branch reduce and bound (BRB) procedure. The result-ing algorithm is numerically stable, has proven finite convergence,also solves EE maximization, and is the first global optimization al-gorithm specifically designed for RSMA. Practically, it outperformsalgorithms for similar problems as will be verified in Section 4. To better illustrate the core principles of SIT, consider the gen-eral optimization problem max ( x , ξ ) ∈D f ( x , ξ ) s . t . g i ( x , ξ ) ≤ , i = 1 , . . . , n (5)with continuous, real valued functions f, g , . . . , g n and nonemptyfeasible set. Further, assume that f is concave, g , . . . , g n are con-vex in ξ for fixed x , and D is a closed convex set. Depending onthe structure of g , . . . , g n in x , this problem might be quite hard tosolve for BB methods [23, 25]. Instead, consider the problem min ( x , ξ ) ∈D max i { g i ( x , ξ ) } s . t . f ( x , ξ ) ≥ δ (6)that is obtained from (5) by exchanging the objective and constraints.If the optimal value of (6) is less than or equal to zero, the optimalvalue of (5) is greater than or equal to δ . Instead, if the optimalvalue of (6) is greater than zero, the optimal value of (5) is less than δ [22, Prop. 7]. Hence, the optimal solution of (5) can be obtainedby solving a sequence of (6) with increasing δ . Since the feasibleset of (6) is closed and convex, it can be solved much easier by BRBthan (5) [22].The SIT and BRB procedures can be integrated into a singleBRB algorithm that solves (6) with low precision and updates δ whenever a point x k feasible in (5) is encountered that achieves anobjective value f ( x k ) > δ . This BRB procedure relaxes the feasibleset and subsequently partitions it in such a way that upper and lowerbounds on the minimum objective value of (6) can be computed ef-ficiently for each partition element. In particular, we use rectangularsubdivision and define the initial box as M = [ r , s ] = { x : r i ≤ x i ≤ s i } satisfying M ⊇ proj x D . The algorithm subse-quently partitions the relaxed feasible set M into smaller boxes andstores the current partition of M in R k . In iteration k , the algorithmselects a box M k = [ r k , s k ] and bisects it into two new subrectan-gles. For each of these new boxes, a lower bound on the objectivevalue is computed using a bounding function β ( M ) that computes alower bound on the objective value of (6) with additional constraint x ∈ M . If this problem is infeasible, then β ( M ) = ∞ . To ensureconvergence, the bounding needs to be consistent with branching,i.e., β ( M ) has to satisfy β ( M ) − min ( x , ξ ) ∈F , x ∈M max i { g i ( x , ξ ) } → x , y ∈M k x − y k → , (7)and a dual feasible point x k ∈ proj x F ∩ M k is required, where F = { x ∈ D : f ( x ) ≥ δ } is the feasible set of (6).The following lemma is essential to establish the convergenceof the SIT procedure. It follows that it can be incorporated in aBB procedure with pruning criterion β ( M ) < − ε and terminationcriterion > min ξ g ( x k , ξ ) s . t . ( x k , ξ ) ∈ F . Lemma 1 ([24, Prop. 5.9]) . Let ε > be given and define g ( x , ξ ) =max i { g i ( x , ξ ) } . Either g ( x k , ξ ∗ ) < for some k and ( x k , ξ ) ∈F , or β ( M k ) > − ε for some k . In the former case, ( x k , ξ ∗ ) is anonisolated feasible solution of (5) satisfying f ( x k , ξ ∗ ) ≥ δ . In thelatter case, no ε -essential feasible solution ( x , ξ ) of (5) exists suchthat f ( x , ξ ) ≥ δ . Next, we design a suitable bounding procedure that satisfies (7).
The SIT dual should contain all of the problem’s nonconvexity inthe objective function. Following the discussion in Section 2.1, the Although this assumption does not hold for (3), the approach is still ap-plicable since the sole purpose of this assumption is to obtain a convex feasi-ble set in (6). This is also true for outer approximation methods [25]. onconvexity in (3) is due to (3b)–(3e). We obtain the SIT dual as min p c , p ,..., p K , c , γ p ,s, d , e max h √ s (cid:16)X j ∈K | h H p j | + 1 (cid:17) / − h H p c , max k> (cid:26) √ s (cid:16)X j ∈K | h Hk p j | + 1 (cid:17) / − d k (cid:27) , max k ∈K (cid:26) √ γ p,k (cid:18)X j ∈K\ k | h Hk p j | + 1 (cid:19) / − h Hk p k (cid:27) , max k> (cid:8) d k − | e k | (cid:9) i (8a)s.t. P k ∈K u k ( C k + log(1 + γ p,k )) µ (cid:0) k p c k + P k ∈K k p k k (cid:1) + P c ≥ δ (8b)(3f)–(3j) . (8c)Observe that (8b) is equivalent to the SOC X k ∈K u k ( C k + log(1 + γ p,k )) ≥ δ (cid:18) µ (cid:18) k p c k + X k ∈K k p k k (cid:19) + P c (cid:19) since the denominator in (8b) is positive.A bounding function β ( M ) that satisfies (7) is required. First,observe that the objective of (8) is increasing in ( γ p , s ) . Hence, alower bound on [¯ γ p , ¯ γ p ] × [¯ s, ¯ s ] is obtained by setting γ p = ¯ γ p and s = ¯ s in the objective. Next, smoothen the objective of (8)by using the epigraph form with auxiliary variable t , and convertthe pointwise maximum expressions to smooth constraints. Then,the new constraints t ≥ d k − | e k | , for k > , are equivalent to ( e k , d k − t ) ∈ C . This set C is nonconvex. Consistent bounding ofthis set is obtained using argument cuts [16], i.e., introduce auxiliaryvariables α k ∈ [0 , π ] , k > , and add the constraint ∠ e k = α k .The variables α are included in the nonconvex variables handledby the BRB solver. Then, a lower bound on the objective value of(8) over the box [¯ α , ¯ α ] is obtained by replacing the constraints d k ≤| e k | , ∠ e k ∈ [¯ α k , ¯ α k ] , with their convex envelope. For ¯ α k − ¯ α k ≤ π ,this is sin(¯ α k ) ℜ{ e k } − cos(¯ α k ) ℑ{ e k } ≤ (9a) sin(¯ α k ) ℜ{ e k } − cos(¯ α k ) ℑ{ e k } ≥ (9b) a k ℜ{ e k } + b k ℑ{ e k } ≥ ( d k − t )( a k + b k ) (9c)and ( e k , d k ) ∈ C × R otherwise [16, Prop. 1], where a k = (cos(¯ α k ) + cos(¯ α k )) , and b k = (sin(¯ α k ) + sin(¯ α k )) .The resulting bounding problem depends on γ p and s onlythrough to the constraints (2d), (3i), (8b), and ( γ p , s, α ) ∈ M .These can be transformed into affine functions of ( γ p , s ) by sub-stituting s ′ = log(1 + s ) and γ ′ p,k = log(1 + γ p,k ) . Then, theseconstraints are equivalent to X k ∈K u k ( C k + γ p,k ) ≥ δ (cid:16) µ (cid:16) k p c k + X k ∈K k p k k (cid:17) + P c (cid:17) (10a) X k ∈K C k ≤ s, C k ≥ max n , R thk − γ p,k o , ∀ k ∈ K (10b) s ∈ [log(1 + ¯ s ) , log(1 + ¯ s )] (10c) γ p,k ∈ [log(1 + ¯ γ p,k ) , log(1 + ¯ γ p,k )] , ∀ k ∈ K (10d)and the final bounding problem is the SOCP min p c , p ,..., p K , c , γ p ,s, d , e ,t t (11a)s.t. p ¯ γ p,k (cid:18) X j ∈K\ k | h Hk p j | + 1 (cid:19) / ≤ t + h Hk p k (11b) √ ¯ s (cid:16)X j ∈K | h H p j | + 1 (cid:17) / ≤ t + h H p c (11c) √ ¯ s (cid:18) X j ∈K | h Hk p j | + 1 (cid:19) / ≤ t + d k , ∀ k > (11d) ∀ k ∈ I M : (9a)–(9c) (11e)(2e), (3f)–(3h), (10a)–(10d) (11f)where I M = (cid:8) k ∈ K : k > ∧ max ¯ α, ¯ α ∈M | ¯ α k − ¯ α k | ≤ π (cid:9) . The bound β ( M ) takes the optimal value of (11) if it is feasible. Otherwise, β ( M ) = ∞ otherwise. A dual feasible point is obtained from the solution ( γ ⋆p , s ⋆ , e ⋆ , . . . ) of (11) as ( γ kp , s k , α k ) with γ kp,i = 2 γ ⋆p,i − , for i ∈ K , s k = 2 s ⋆ − and α k ∈ proj α M k = [¯ α k , ¯ α k ] . Numerical experiments showthat the obvious choice α ki = ∠ e ⋆i leads to very slow convergence.A much faster alternative is α ki = arg min α ∈{ ¯ α ki , ¯ α ki } | α − ∠ e ⋆i | .This point is primal feasible if the optimal value of min p ,..., p K , p c , c , d , e ,t t s.t. (8b), (3f)–(3j) | γ p = γ kp ,s = s k (12a)(11b)–(11d) | ¯ γ p = γ kp , ¯ s = s k (12b) ∀ i > e i , d i − t ) ∈ C , ∠ e i = α ki (12c)is less than or equal to zero. This is an SOCP since (12c) is affine.Denote the optimal solution of (12) as ( t ∗ , c ∗ , y ∗ ) . It can beshown that the primal objective value of ( c ∗ , y ∗ ) is greater than orequal to δ . This value can be further increased without impairing pri-mal feasibility by updating c ∗ with the solution of the linear program max c P k ∈K u k C k s . t . (2d), (3i), (8b) | y ∗ . The convergence criterion (7) implies that the quality of the bound β ( M ) improves as the diameter of M shrinks. Since tighter boundslead to faster convergence, it is beneficial to reduce the size of M prior to bounding if possible at low computational cost. To ensureconvergence to the global solution, it is important that the reducedbox M ′ ⊆ M still contains all solution candidates.Consider the box M = [¯ γ p , ¯ γ p ] × [¯ s, ¯ s ] × [¯ α , ¯ α ] . Due to mono-tonicity, a necessary condition for the feasibility of (8) over M isthat (2d), (3i), (8b) hold for ¯ γ p , ¯ s, ¯ α . Clearly, (2d) and (3i) can onlyhold if X k ∈I (cid:16) R thk − log(1 + ¯ γ p,k ) (cid:17) − log(1 + ¯ s ) ≤ (13)with I = { k ∈ K : R thk − log(1 + ¯ γ p,k ) > } . Similarly, anecessary condition for (8b) to hold is max k ∈K { u k } log(1 + ¯ s ) + X k ∈K u k log(1 + ¯ γ p,k ) ≥ δW (14)with W = (cid:0) µ (cid:0) min k p c k + P k ∈K min k p k k (cid:1) + P c (cid:1) , wherethe minimum is such that γ p ∈ M . This can be relaxed as min p c ,..., p K k p κ k s . t . ¯ γ p,κ ≤ | h Hκ p κ | . From the Karush-Kuhn-Tucker conditions, the optimal value of this problem is ob-tained as ¯ γ p,κ k h κ k − . Similarly, a lower bound for min k p c k isobtained as ¯ s max k k h k k − . Hence, W = µ (cid:16) ¯ s max k k h k k − + X k ∈K ¯ γ p,k k h k k − (cid:17) + P c . (15)Conditions (13) and (14) can be used to reduce M and as apreliminary feasibility check before bounding. For the reduction,let M ′ = [¯ γ ′ p , ¯ γ ′ p ] × [¯ s ′ , ¯ s ′ ] × [¯ α , ¯ α ] and consider (14). Everydual feasible γ p,κ ∈ M satisfies W δ ≤ U − u κ log(1 + ¯ γ p,κ ) + κ log(1 + γ p,κ ) , where U is the right-hand side of (14). Hence,every dual feasible γ p,κ satisfies γ p,κ ≥ Wδ − Uuκ (1 + ¯ γ p,κ ) − .Similarly, let V be the left-hand side of (13). From this condition, wesee that every dual feasible γ p,κ satisfies γ p,κ ≥ V (1+¯ γ p,κ ) − , for κ ∈ I , and γ p,κ ≥ V + R thκ − , for κ / ∈ I . Thus, the lower boundfor γ p,k can be reduced to ¯ γ ′ p,k = max { ¯ γ p,k , ¯ γ ′′ p,k } without losingfeasible solution candidates, where ¯ γ ′′ p,k = 2 max { Wδ − Uuk , V } (1 +¯ γ p,k ) − if k ∈ I , and max { Wδ − Uuk (1 + ¯ γ p,κ ) , V + R thk } − otherwise. Likewise, the lower bound s can be reduced to ¯ s ′ =max { ¯ s, max (cid:8) Wδ − U max k ∈K{ uk } , V (cid:9) (1 + ¯ s ) − } .Let W ′ be as in (15), evaluated at (¯ s ′ , ¯ γ ′ p ) , and consider (14)again. With a similar argument as before, the upper bound of M ′ canbe reduced to ¯ γ ′ p,k = min (cid:8) ¯ γ p,k , ¯ γ ′ k,p +( δµ ) − k h k k ( U − δW ′ ) (cid:9) and ¯ s ′ = min (cid:8) ¯ s, ¯ s ′ + ( δµ ) − min k k h k k ( U − δW ′ ) (cid:9) . Observethat this reduction procedure may lead to M ′ = ∅ . The complete algorithm is stated in Algorithm 1. It is essentially aBRB procedure [24, 25] that solves the SIT dual of (3) and updatesthe constant δ whenever a primal feasible point is encountered.The initial box in Step 0 is computed as M = [ , ¯ γ p ] × [0 , ¯ s ] × [0 , π ] K − with ¯ γ p,k = P k h k k and ¯ s = min k ∈K P k h k k . Theset R k holds the current partition of the feasible set, δ k is the cur-rent best value adjusted by the tolerance η , and ¯ x k is the current bestsolution (CBS). In Step 1, the next box is selected as M k and bi-sected into P k . These boxes are reduced according to Section 3.3in Step 2. In Step 3, bounds for each reduced box are computed,infeasibility is detected, and dual feasible points are obtained fromthe bounding problem. For each of these points, primal feasibility ischecked in Step 4. If feasible, a feasible point is recovered as in Sec-tion 3.2 and the corresponding primal objective value is computed.If necessary, the CBS and δ k are updated in Step 5. Boxes that can-not contain primal ε -essential feasible solutions are pruned in Step 6.The algorithm is terminated in Step 7. Theorem 1.
Alg. 1 converges in finitely many steps to a ( ε, η ) -optimal solution of (3) or establishes that no such solution exists.Proof. Omitted due to space constraints.
4. NUMERICAL EVALUATION
As most numerical problems of similar state-of-the-art algorithmsarise from the multiple unicast beamforming problem, i.e., where p c = , we evaluate the performance of the algorithm for this case.In particular, we have generated 100 random i.i.d. channel realiza-tions and solved (2) for u k = 1 , µ = 0 , P c = 0 , R thk = 0 , P dB = − , − , . . . , , and K = M ∈ { , , } . This results in700 problem instances per K . As baseline comparison and verifica-tion, we chose the straightforward BB implementation of this prob-lem [14, 15] (“BB”) and its variant with modified bounding problemfrom [14, §2.2.2] (“BB2”). For K = 2 , BB2 stalled in 364 prob-lem instances, while the other algorithms solved all problems. For K = 3 , BB2 stalled 146 times and BB failed 13 × due to numericalproblems of the convex solver. Finally, for K = 4 , BB did not solvea single problem instance due to numerical issues and BB2 stalledin 27 instances. Moreover, Algorithm 1 and BB2 did not solve theproblem withing 60 minutes in 4 and 60 instances, respectively. Av-erage computation times on a single core of an Intel Cascade Lake Algorithm 1
SIT Algorithm for (3)
Step 0 (Initialization)
Set ε, η > . Let k = 1 and R = {M } . Ifan initial feasible solution y = ( p c , . . . , p K ) is available, set δ = η + v (2) | y and initialize ¯ x = ( γ p , s , α ) from (1), s = min k ∈K γ c,k , and α k = ∠ h Hk p c . Otherwise, do not set ¯ x and choose δ = 0 . Step 1 (Branching)
Let M k = [ r k , s k ] ∈ arg min { β ( M ) | M ∈ R k − } . Bisect M k into M − = { x : r j ≤ x j ≤ v j , r i ≤ x i ≤ s i ( i = j ) }M + = { x : v j ≤ x j ≤ s j , r i ≤ x i ≤ s i ( i = j ) } where j k ∈ arg max j s kj − r kj and v k = ( s k + r k ) . Set P k = {M k − , M k + } . Step 2 (Reduction)
Replace each box in P k with M ′ as in Section 3.3. Step 3 (Bounding)
For each reduced box
M ∈ P k , solve (11). If infea-sible, set β ( M ) = ∞ . Otherwise, set β ( M ) to the optimal valueof (11) and obtain a dual feasible point x ( M ) as in Section 3.2. Step 4 (Feasible Point)
For each
M ∈ P k , if β ( M ) ≤ solve (12) for x ( M ) and denote the optimal value as t ( x ( M )) . If t ( x ( M )) ≤ , x ( M ) is primal feasible. Recover x ′ ( M ) from the solution of(12) with γ ′ p , s ′ as in Step 0 and α ′ k = ∠ e ∗ k , k > , where e ∗ is from the optimal solution of (12). Update c ∗ as in Section 3.2and compute the primal objective value f ( M ) . If β ( M ) > or t ( x ( M )) > , set f ( M ) = −∞ . Step 5 (Incumbent)
Let M ′ ∈ arg min { f ( M ) : M ∈ P k } . If f ( M ′ ) > δ k − − η , set ¯ x k = x ′ ( M ′ ) and δ k = f ( M ′ ) + η .Otherwise, set ¯ x k = ¯ x k − and δ k = δ k − . Step 6 (Pruning)
Delete every
M ∈ P k with β ( M ) > − ε and collectthe remaining sets in P ′ k . Set R k = P ′ k ∪ ( R k − \ {M k } ) . Step 7 (Termination)
Terminate if R = ∅ : If ¯ x k is not set, then (3) is ε -essential infeasible; else ¯ x k is an essential ( ε, η ) -optimal solutionof (3). Otherwise, update k ← k + 1 and return to Step 1. K = 2 K = 3 K = 4 Alg. 1 0.175 s / 0.099 s 4.579 s / 1.959 s 334.8 s / 126.3 sBB 0.173 s / 0.091 s 7.605 s / 2.606 s —BB2 42.41 s / 2.380 s 158.5 s / 12.42 s 704.1 s / 265.8 s
Table 1 . Mean / median run times to obtain the optimal solution.Problem instances where not all algorithms converged are ignored.Platinum 9242 CPU are reported in Table 1. It can be observed thatthe proposed Algorithm 1 is more efficient than the two baseline al-gorithms especially when more users are in the system. Moreover,the joint beamforming problem, i.e., with p c = , was solved byAlgorithm 1 for K = 2 with mean and median run times of 942 sand 2786 s. However, 23 instances were not solved within 12 hours.Observe from the discussion in Section 3 that the complexityscales with O (exp(2 K )) in the number of users and polynomiallyin the number of antennas M . Hence, no noticeable changes in thereported run times are to be expected by varying M .
5. CONCLUSIONS
We developed the first global optimization algorithm to solve MISOdownlink beamforming for RSMA with respect to WSR and EEmaximization. This problem is an instance of joint multicast andunicast beamforming and also solves these problems separately. Thealgorithm is numerically stable and outperforms state-of-the-art mul-tiple unicast beamforming algorithms considerably. . REFERENCES [1] B. Clerckx, H. Joudeh, C. Hao, M. Dai, and B. Rassouli, “Ratesplitting for MIMO wireless networks: A promising PHY-layer strategy for LTE evolution,”
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