Worst-Case Robust Multiuser Transmit Beamforming Using Semidefinite Relaxation: Duality and Implications

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1 Worst-Case Robust Multuser Transmt Beamforg Usng Semdefnte Relaxaton: Dualty and mplcatons Tsung-Hu Chang, Wng-Kn Ma, and Chong-Yung Ch nsttute of Commun. Eng. & Dept. of Elect. Eng. Natonal Tsng Hua Unversty, Hsnchu, Tawan E-mal: Department of Electronc Engneerng The Chnese Unversty of Hong Kong, Shatn, N.T., Hong Kong E-mal: Abstract Ths paper studes a downlnk multuser transmt beamforg desgn under sphercal channel uncertantes, usng a worst-case robust formulaton. Ths robust desgn problem s nonconvex. Recently, a convex approxmaton formulaton based on semdefnte relaxaton (SDR) has been proposed to handle the problem. Curously, smulaton results have consstently ndcated that SDR can attan the global optmum of the robust desgn problem. Ths paper ntends to provde some theoretcal nsghts nto ths mportant emprcal fndng. Our man result s a dual representaton of the SDR formulaton, whch reveals an nterestng lnkage to a dfferent robust desgn problem, and the possblty of SDR optmalty.. NTRODUCTON Ths paper focuses on a standard wreless multuser uncast system where a multple-antenna transmtter broadcasts ndependent data streams to multple sngle-antenna recevers usng transmt beamforg [1. n ths context, the effcacy of beamforg desgns reles on knowledge of the channel state nformaton (CS) of all the recevers. However, the transmtter often has some uncertantes on the CS, due to ssues such as fnte-length tranng and fnte-rate feedback [2. CS uncertantes at the transmtter can result n sgnfcant performance outage, f not taken nto consderaton n the beamforg desgns. The CS uncertanty problem has motvated consderable research endeavors n robust transmt beamforg desgn technques. Ths ncludes the chance constraned robust desgns [3, [4, where the CS uncertantes are modeled as random varables, and the worst-case robust desgns [5 [8, where the CS uncertantes are modeled as bounded unknowns wthn a predetered, small error set. Our problem of nterest s the worst-case sgnal-tonterference-plus-nose rato (SNR) constraned robust transmt beamforg desgn problem under sphercally bounded CS uncertantes, whch has drawn much nterest recently [6 [8. Presently avalable beamforg solutons for ths worstcase robust problem are based on approxmaton methods, ether restrcton [6, [7 or relaxaton [8, and t s now not clear whether the worst-case robust problem can be optmally (and effcently) solved. However, smulatons seem to have provded the answer to the latter the semdefnte relaxaton (SDR) method [8. SDR s a convex relaxaton technque for a certan class of hard (nonconvex) optmzaton problems, and has recently ganed popularty owng to ts wde scope of applcablty [9, [10. For a general applcaton, SDR s consdered a suboptmal solver; however, for the worstcase robust beamforg problem, smulaton results have ndcated that SDR should to be a globally optmal solver, whch s a rather surprsng emprcal fndng. As such, beng able to provde a theoretcal analyss provng whether SDR s optmal would be of much sgnfcance. A recent result [11 has partally addressed ths open queston, where the SDR optmalty under suffcently small error rad s analyzed. Ths paper ntends to address the mystery of SDR optmalty n worst-case robust transmt beamforg optmzaton usng a dfferent analyss approach. We show that the worst-case robust problem has a close relatonshp to a dfferent robust beamforg problem, n form of - optmzaton. n partcular, we prove that ther SDR problems are dual, or equvalent, to each other. Ths new, ntrgung, dualty relatonshp provdes a new perspectve and useful nsghts explanng the optmalty of SDR. n partcular, we wll gve a condton under whch SDR provdes globally optmal solutons to the worst-case robust problem.. SGNAL MODEL AND BACKGROUND Consder a wreless downlnk system where a transmtter, equpped wth N t antennas, wants to communcate wth K sngle-antenna recevers usng transmt beamforg. The problem formulaton follows a standard uncast settng [1: Let h C Nt denote the channel vector of recever, andlet w C Nt be the assocated beamforg vector for recever. The SNR of recever s gven by h H SNR (w 1,...,w K, h )= w 2 K k hh w, (1) k 2 + σ 2 where σ 2 > 0 s the nose power at recever, for all =1,...,K. Our goal s to desgn the beamforg vectors {w } K such that each recever acheves a desred SNR level. Conventonally, transmt beamforg desgns requre full channel state nformaton (CS) at the transmtter;.e., knowledge of {h } K. n wreless communcatons, however, t /11/$ EEE 1579 Aslomar 2011

2 s dffcult for the transmtter to acqure accurate CS, due to mperfect channel estmaton and fnte rate feedback [2. Hence there are channel uncertantes at the transmtter;.e., h = h + e, =1,...,K, (2) where h denotes the channel estmate avalable at the transmtter, and e C Nt represents the channel uncertanty. n ths work, we focus on sphercally bounded channel uncertantes: e 2 r 2, =1,...,K, (3) where denotes the Eucldean norm, and r > 0 s the radus of the uncertanty ball. We study the followng worstcase robust beamforg desgn [6, [8: w C N t,,...,k w 2 (4a) s.t. SNR (w 1,...,w K, h + e ) γ e 2 r 2, =1,...,K, (4b) where γ > 0 s the SNR requrement of recever, whch must be fulflled even under worst possble CS uncertantes. The challenge of solvng the worst-case robust problem (4) les n the worst-case SNR constrants n (4b), each of whch corresponds to an nfnte number of nonconvex quadratc constrants. As mentoned, there are several approxmaton methods for managng problem (4) [6 [8, and here we focus on the SDR method [8. The development of SDR conssts of two steps. The frst step, whch s standard (see, e.g. [10), s to substtute W = w w H, k =1,...,K, nto (4b), and then replace W = w w H by W 0 (.e., W beng postve semdefnte (PSD)) to obtan a relaxed problem W H N t,,...,k Tr(W ) s.t. ( h + e ) H 1 W γ k ( h + e ) σ 2 (5a) e 2 r 2, k =1,...,K, (5b) W 1,...,W K 0, (5c) where H Nt s the set of all N t by N t Hermtan matrces, and Tr(W ) denotes the trace of W. The motvaton of ths step s to lnearze the nonconvex constrants. The second step s to turn (5b) to fnte numbers of constrants, thereby enablng effcent mplementatons. By applyng S-lemma (see [12) to (5b), we obtan the followng SDR formulaton of (4): W H N t,λ R,...,K Tr(W ) (6a) s.t. Ψ (W 1,...,W K,λ ) 0, =1,...,K, (6b) W 1,...,W K 0, λ 1,...,λ K 0, where the matrx functons Ψ (W 1,...,W K,λ ) are defned as [ Ψ (W 1,...,W K,λ ) 1 h H W [ γ h k [ λ σ 2 λ r 2, =1,...,K, (7) where s the N t by N t dentty matrx. Note that the SDR problem (6) s a semdefnte program (SDP), whch s convex and tractable. The SDR problem (6) s methodologcally an approxmaton to the worst-case robust problem (4) because the ranks of W are not constraned. However, f the optmal soluton of the SDR problem (6), denoted by (W1,...,W K ), s of rank one;.e., W = w (w )H for all =1,...,K, then t can be verfed that (w1,...,w K ) s a globally optmal soluton to the worst-case robust formulaton (4). Rather surprsngly, t s found through smulatons that SDR yelds rank-one soluton automatcally, and t happens seegly all the tme [8, [11 (see also [4). Our endeavor n the subsequent secton s to provde a dual formulaton of the SDR problem (6) that may shed lght nto ths emprcal fndng. Before we proceed to the man result, let us present some smulaton results to further strengthen the motvaton of the rased analyss problem. Specfcally, we benchmark the SDR method aganst other concurrent approxmaton methods, namely, the robust SOCP-based method n [6, and the MMSEbased SDP method n [7. The smulaton settngs are: N t =4, K = 4, γ γ 1 = = γ K,σ1 2 = = σk 2 = 0.1, r r 1 = = r K =0.1, and( h 1,..., h K ) beng ndependent and dentcally dstrbuted complex Gaussan random varables wth zero mean and unt varance. The result s shown n Fg. 1, where we see that the SDR method outperforms the other two methods. Moreover, we should emphasze that the SDR method yelded rank-one soluton n all the trals ran.. DUALTY OF WORST-CASE ROBUST SDR Consder the followng - optmzaton problem w 2 e C N t,,...,k w C N t,,...,k s.t. SNR (w 1,...,w K, h + e ) γ, =1,...,K, s.t. e 2 r 2, =1,...,K. (8) At frst look, problem (8) s dfferent from the worst-case robust problem n (4). n (8), the nner mzaton s a standard non-robust beamforg desgn problem [1 whch fnds the most power effcent desgn gven a presumed CS { h + e } K. The outer mzaton, however, targets to fnd a worst set of CS uncertantes {e } K that mzes the nner-mum transmt power. We should also note that problem (8) has a flavor of two-player zero-sum game. We are partcularly nterested n applyng SDR to (8). Lke SDR for the worst-case robust problem, we replace each 1580

3 Average transmsson power (db) Robust SOCP va structured SDP [6 MMSE-based SDP [7 SDR n (5) γ (db) Fg. 1: Smulaton results of average transmsson power versus target SNR γ, for uncertanty radus r =0.1. w w H wth a PSD matrx W, and each [e H 1[e H 1 H wth a PSD matrx V, to obtan the followng problem Tr(W ) W H N t s.t. Tr 1 W R σ 2 γ, V H N t +1,,...,K k =1,...,K, W 1,...,W K 0. s.t. Tr(V ) (1 + r 2 ), =1,...,K, [V Nt+1 =1, =1,...,K, V 1,...,V K 0, (9) where [V Nt+1 s the (N t +1,N t +1)th entry of V and R = [ h [ H V h, =1,...,K. An mportant observaton of problem (9) s that there always exsts a rank-one soluton for the nner mzaton of problem (9): Fact 1 [1 Consder the followng SDP: W H N t,,...,k Tr(W ) (10) s.t. Tr 1 W γ k W 0, =1,...,K, R σ 2, =1,...,K, where R 1,...,R K 0. Suppose that (10) s feasble. Then there exsts an optmal soluton (W1,...,WK ) for whch rank(w )=1for all. Fact 1 mples that the SDR of (W 1,...,W K ) s always tght for the - SDR problem (9). Fact 1 rases an ntrgung queston What s the relatonshp between the - SDR problem (9) and the robust SDR problem (6)? f the optmal solutons of (W 1,...,W K ) of the two problems are dentcal, then Fact 1 mmedately mples that (6) has a rank-one optmal soluton and hence SDR s tght to (6) as well. A. Man Result t turns out that problems (9) and (6) are strongly connected: Proposton 1 Suppose that problem (6) s feasble. Then problems (9) and (6) attan the same optmal objectve value. Moreover, f (W1,...,WK,λ 1,...,λ K ) s an optmal soluton of problem (6), then there exsts (V1,...,V K ) such that (V1,...,VK, W 1,...,WK ) s an outer-nner soluton of problem (9). As the man contrbuton of ths paper, Proposton 1 provdes a soluton correspondence between problems (9) and (6), showng that problem (9) s actually a dual representaton of problem (6). To prove that problems (9) and (6) attan the same optmal objectve value, we show that the Lagrangan dual of problem (6) s equvalent to the Lagrangan dual of problem (9). The former can be shown to be A H N t +1,,...,K σ 2 [A Nt+1 (11) s.t. Y (A 1,...,A K ) 0, =1,...,K, Tr(A ) (1 + r 2 )[A Nt+1, =1,...,K, A 1,...,A K 0, where A 1,...,A K H Nt+1 are the (Lagrangan) dual varables assocated wth constrants (6b), and Y (A 1,...,A K ) 1 [ [ h A γ + h H [ [ hk Ak h H, =1,...,K. (12) k Now let us consder the Lagrangan dual of the nner mzaton problem of (9), whch can be shown to be μ σ 2 (13) μ 1,...,μ K 0 s.t. μ γ R + μ k R 0, =1,...,K, where μ 1,...,μ K are the dual varables assocated wth the trace nequalty constrants of the nner problem of (9). Replacng the nner problem of (9) wth ts dual (13), we obtan the followng problem μ σ 2 μ 0,,...,K (14) V H N t +1,,...,K s.t. Y (μ 1 V 1,...,μ K V K ) 0, =1,...,K, s.t. Tr(V ) (1 + r 2 ), =1,...,K, [V Nt+1 =1, =1,...,K, V 1,...,V K 0. Snce strong dualty holds for the nner parts of (9) and (14), the two problems have the same optmal objectve value. 1581

4 One may observe a connecton between (11) and (14): [A Nt+1 = μ, A = μ V, =1,...,K. (15) n fact, (11) and (14) are equvalent problems, as we show n Appendx the followng lemma: Lemma 1 f (A 1,...,A K ) s an optmal soluton of (11), then (V1,...,V K )=(A 1 /[A 1 N t+1,...,a K /[A K N t+1), (μ 1,...,μ K)=([A 1 Nt+1,...,[A K Nt+1) (16) s an optmal outer-nner soluton par of (14). f (V1,...,V K,μ 1,...,μ K ) s an optmal outer-nner soluton of (14), then(a 1,...,A K )=(μ 1 V 1,...,μ K V K ) s optmal to (11). Lemma 1 shows that (V1,...,V K ) of (14) only dffers from (A 1,...,A K ) of (11) up to a postve scalar. Hence, (14) and (11) attan the same optmal objectve value, mplyng that (9) and (6) attan the same optmal objectve value. By Lemma 1, one can further show that (W1,...,W K ), the optmal prmal soluton of (6), s also optmal to (9). The detaled proof s presented n Appendx. B. mplcaton and Concludng Remark To show that the robust SDR problem (6) has a rank-one soluton, we stll need to prove that the optmal (W 1,...,W K ) of (9) s also optmal to (6). Now, let us assume: Condton 1 The optmal soluton of the nner mzaton of problem (9), (W1,...,W K ), s unque. Condton 1 s consdered mld; by numercal experence, Condton 1 s found to hold all the tme. Under Condton 1, we can nfer from Fact 1 and Proposton 1 that the SDR problem (6) has a rank-one soluton. Hence, we conclude that Clam 1 Under Condton 1, the SDR problem (6) solves the worst-cast robust problem (4) optmally. Our analyss above narrows down the SDR optmalty queston to the proof of unque rank-one soluton of the nner mzaton problem of (9). As a future research drecton, t would be nterestng to nvestgate suffcent condtons under whch Condton 1 holds true. V. APPENDX KKT condtons of (6) The KKT condtons of (6) and (11) can be shown to be W 1,...,W K 0,λ 1,...,λ K 0, A 1,...,A K 0, (17a) Ψ (W 1,...,W K,λ ) 0, =1,...,K, (17b) Y (A 1,...,A K ) 0, =1,...,K, (17c) Ψ (W 1,...,W K,λ )A = 0, =1,...,K, (17d) Y (A 1,...,A K )W = 0, =1,...,K, (17e) Tr(A ) (1 + r 2 )[A Nt+1, =1,...,K, (17f) ( Tr(A ) (1 + r 2 )[A ) Nt+1 λ =0, =1,...,K, (17g) where Ψ ( ) and Y ( ) are defned n (7) and (12), respectvely. Proof of Lemma 1: Lemma 1 can be easly proved by nspecton of (14) and (11). What remans s to show that μ > 0 and [A N t+1 > 0 for all = 1,...,K. The former has been proved n [13, Proposton 4.2; whle the latter can be proved as follows. One can observe from (17a) and (17f) that [A N t+1 = 0 results n A = 0. n ths case, Y (A 1,...,A K ) n (12) s postve defnte,.e., Y (A 1,...,A K ) 0. By the complementary slackness (17e), ths leads to the prmal soluton W = 0, whch however volates (17b) [see (7) due to σ 2 > 0. Proof of Proposton 1: Here we prove that (W1,...,WK ), the optmal prmal soluton of (6), s also optmal to (9). By Lemma 1 whch shows that (A 1/[A 1 Nt+1,...,A K /[A K N t+1) s an optmal outer mzer of (9), t suffces to show that (W1,...,W K ) s optmal to the followng problem W 1,...,W K 0 Tr(W ) (18) s.t. Tr 1 W γ σ 2 [A Nt+1, =1,...,K. [ h A [ h H Ths can be shown by exang that (W 1,...,W K ) satsfes the KKT condtons of (18), whch are gven as follows: W 1,...,W K 0, μ 1,...,μ K 0, (19a) Y (μ 1 (A 1 /[A 1 N t+1),...,μ K (A K /[A K N t+1)) 0, (19b) Y (μ 1 (A 1/[A 1 Nt+1),...,μ K (A K/[A K Nt+1)) W = 0, (19c) Tr 1 W [ [ γ h A = h H σ 2 [A N t+1, (19d) for =1,...,K. Snce (W1,...,WK,λ 1,...,λ K ) and (A 1,...,A K ) satsfy the KKT condtons n (17a), (17c) and (17e), (W1,...,WK ) and (μ 1,...,μ K ) ([A 1 N t+1,...,[a K N t+1) satsfy (19a), (19b) and (19c). To show that (W1,...,W K ) also fulflls (19d), let us consder an alternatve representaton of (6): Lemma 2 Problem (6) can be equvalently expressed as the followng problem W 0,,...,K s.t. Tr(W ) Tr 1 W V V γ σ 2, =1,...,K, (20a) [ [ h V h H (20b) 1582

5 where V = {V H Nt+1 Tr(V ) (1 + r 2 ), [V Nt+1 = 1, V 0}. t s easy to verty that, for (W1,...,WK ), Tr 1 K W γ Wk V V [ h V [ h H = σ 2, =1,...,K, (21).e., the nequalty constrants n (20b) are all actve for the optmal soluton (W1,...,WK ). Hence, to show that (19d) s also fulflled by (W1,...,W K ), t s suffcent to prove that A /[A N t+1 = arg Tr 1 K W V 0 γ W [ [ k h V h H s.t. Tr(V ) (1 + r 2 ), (22a) [V Nt+1 =1, (22b) for all = 1,...,K. Let ξ and τ be the dual varables assocated wth the constrants n (22a) and (22b), respectvely, and defne Ψ (W1,...,WK,ξ,τ ) [ h H 1 W γ W k [ [ h ξ ξ + τ The KKT condtons of the mzaton problem n (22) can be obtaned as Tr(V ) (1 + r 2 ), [V Nt+1 =1,V 0,ξ 0,τ R, Ψ (W1,...,W K,ξ,τ ) 0, Ψ (W1,...,WK,ξ,τ ) V = 0, ( ξ Tr(V ) (1 + r 2 )) =0, τ ([V Nt+1 1) = 0. (23a) (23b) (23c) (23d) For each {1,...,K}, letv = A /[A N t+1, ξ = λ, τ = σ 2 (1 + r 2)λ. t follows from the KKT condtons n (17a), (17b), (17d) and (17g) that (V,ξ,τ ) satsfes all the condtons n (23). Thus (22) s true for all =1,...,K. The proof s then completed. Proof of Lemma 2: t suffces to show that (6b) s equvalent to (20b). Note that (6b) s equvalent to e 2 r 2 ( h + e ) H 1 W ( h + e ) γ σ 2, =1,...,K. (24) (the equvalence s owng to the S-Lemma; see [8, [11). Note that the mzaton problem on the left-hand sde of (24) may not be convex wth respect to (e 1,...,e K ) because the matrx ( 1 γ W K k ) may not be postve semdefnte. Nevertheless, SDR can be appled. Through the same procedure as n obtanng (9), one can obtan the SDR problem of the mzaton problem n (24) as Tr 1 W [ [ γ h V h H. (25) V V Whle (25) s obtaned by relaxaton of the rank of V,theSDR problem (25) s actually tght and optmal to the mzaton problem n (24); see [14, Lemma 3.1. We thus obtan (20b) by substtutng (25) nto (24). V. ACKNOWLEDGEMENTS Ths work s supported n part by Natonal Scence Councl, R.O.C., under Grant NSC E MY3, by a General Research Fund of Hong Kong Research Grant Councl (CUHK ), and by a Drect Grant awarded by the Chnese Unversty of Hong Kong (Project Code ). REFERENCES [1 M. Bengtsson and B. Ottersten, Optmal and suboptmal transmt beamforg, Chapter 18 n Handbook of Antennas n Wreless Communcatons, L. C. Godara, Ed., CRC Press, Aug [2 D. J. Love, R. W. Heath, V. K. N. Lau, D. Gesbert, B. D. Rao, and M. Andrews, An overvew of lmted feedback n wreless communcaton systems, EEE Journal on Sel. Areas n Comm., vol. 26, pp , Oct [3 M. B. Shenouda and T. N. Davdson, Probablstcally-constraned approaches to the desgn of the multple antenna downlnk, n Proc. EEE Aslomar Conf. Sgnals, Systems and Computers, Pacfc Grove, Oct , 2008, pp [4 K.-Y. Wang, A. M.-C. So, T.-H. Chang, W.-K. Ma, and C.-Y. Ch, Outage constraned robust transmt optmzaton for multuser MSO downlnks: Tractable approxmatons by conc optmzaton, submtted to EEE Trans. Sgnal Process., 2011, abrdged versons publshed n EUSPCO 2010 and CASSP [5 M. B. Shenouda and T. N. Davdson, Nonlnear and lnear broadcastng wth QoS requrements: Tractable approaches for bounded channel uncertantes, EEE Trans. Sgnal Process., vol. 57, no. 5, pp , May [6, Convex conc formulatons of robust downlnk precoder desgns wth qualty of servce constrants, EEE J. Sel. Topcs n Sgnal Process., vol. 1, pp , Dec [7 N. Vu cć and H. Boche, Robust QoS-constraned optmzaton of downlnk multuser MSO systems, EEE Trans. Sgnal Process., vol. 57, pp , Feb [8 G. Zheng, K.-K. Wong, and T.-S. Ng, Robust lnear MMO n the downlnk: A worst-case optmzaton wth ellpsodal uncertanty regons, EURASP Journal on Advances n Sgnal Process., vol. 2008, pp. 1 15, June 2008, Artcle D [9 Z.-Q. Luo and T.-H. Chang, SDP relaxaton of homogeneous quadratc optmzaton: Approxmaton bounds and applcatons, Chapter 4 n Convex Optmzaton n Sgnal Processng and Communcatons, D.P. Palomar and Y. Eldar, Eds., UK: Cambrdge Unversty, [10 Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, Semdefnte relaxaton of quadratc optmzaton problems, EEE Sgnal Process. Mag., pp , May [11 E. Song, Q. Sh, M. Sanjab, R. Sun, and Z.-Q. Luo, Robust SNRconstraned MSO downlnk beamforg: When s semdefnte programg relaxaton tght? n Proc. EEE CASSP, Prague, Czech, May 22-27, 2011, pp [12 S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge, UK: Cambrdge Unversty Press, [13 Y. Huang and D. P. Palomar, A dual perspectve on separable semdefnte programg wth applcatons to optmal downlnk beamforg, EEE Trans. Sgnal Process., vol. 58, no. 8, pp , August [14, Rank-constraned separable semdefnte program wth applcatons to optmal beamforg, EEE Trans. Sgnal Process., vol. 58, no. 2, pp , Feb