On the Convexity of the MSE Region of Single-Antenna Users

On the Con v e xity of the MSE Re gion of Single-Antenna Users Raphae l Hunger and Michael Joham Associate Institute for Signal Processing, T echnische Universit ¨ at M ¨ unchen, 80290 Mu nich, Germa ny T elephon e: +49 8 9 28 9-28 508, F ax: +49 89 28 9-285 04, Email: { hunge r,joham } @t um.de Abstract —W e prov e conv exity of the sum-power constrained mean square error (MSE) region in case of two single-antenna users communicatin g with a mul ti-antenna b ase station. Due to the MSE duality this holds both for the v ector broadcast ch annel and the du al multiple access chan nel. Increasing the nu mber of users to mo re than t wo, we show by means of a simpl e counter- example th at the resulting M SE region is not necessarily con vex any longer , ev en un der the assumption of single-antenna u sers. In conjunction with our former observation that the two u ser MSE region is not necessarily conv ex for two multi-antenn a users, this extends and corrects the hitherto existing notion of the MSE region geometry . I . I N T R O D U C T I O N Up to now , only few contributions on the geometrical structure of the mean square error region exist. In [1], th e authors show that the multi-u ser MIMO M SE region is co n vex under fi xed tra nsmit and receive beamform ing vectors both for li near and nonlin ear prep rocessing. Obviously , a larger set o f MSE tuples can b e achieved b y mea ns of a daptive transmit an d receive b eamform ers. For th is extended setup only the two u ser case has been inv estigated so far . Utilizing matrix inequalities of matrix-conve x function s, the au thors in [2] pr ove th at the two user multi-anten na MSE region cannot exhibit a nonconve x den t between two f easible MSE points con nected by a line segment with − 4 5 ◦ slope. From this ob servation, they claim that the MSE region is convex. For conv exity , howev er , all p ossible slopes would have to b e checked. As a matter of fact, a cha nnel r ealization exh ibiting a nonconvex M SE region with two m ulti-antenn a users has been observed in [3] disproving the co n vexity theorem in [2]. A multi- carrier system wh ere se veral single- antenna users commun icate with a single-anten na ba se station h as been in vestigated in [4]. There, the co mplemen tary MSE region of par allel broad cast c hannels is shown to be not n ecessarily conv ex. Since the system under con sideration in [4] can be recast into a b lock diagona l MIMO broadcast channel, the authors of [4] con clude that the two user multi-anten na MSE region cannot b e co n vex in general which again contrad icts the theor em in [2]. So far , no distinct statements on con vexity of the MSE region d ependin g o n the number of users and antennas per user are available in case o f ad aptive transm it and r eceive beamfo rmers. Some applications for which the geom etry of the MSE region is of in terest ar e for examp le the stream priorization accordin g to buffer states or queue states by m eans of the weighted su m-MSE min imization, cf. [5]. Her e, subop timum transmit and receive filters are der iv ed by repeated ly switching between the downlink and th e dual u plink in comb ination with a g eometric progr am solver for a reason able p ower allocatio n. Balancing is considered in [6] where the weights of a weighted sum-MSE min imization are adapted u ntil cer tain MSE ratios hold. Exp loiting th e rela tionship b etween the deriv ati ve o f th e the mutu al infor mation and the minimu m mean square error, Christensen et al. tackle the weighted sum-rate maximizatio n utilizing r esults fr om a weighted sum- MSE minim ization, see [7]. Howe ver , convexity o f the MSE region is the cr u- cial point for the p roper fu nctionality of above applicatio ns since nonco n vexity may for example prevent c onv ergence o f iterativ e algorithms. Finally , the MSE ε achieved with MMSE receivers is tightly re lated to the max imum SINR via SINR = 1 ε − 1 , (1) and h ence, also to the d ata rate R v ia the simple relation R = − log 2 ε. (2) Summing up, all this clear ly motivates a detailed invest igation. In this p aper, we extend the hitherto existing notion of th e MSE region g eometry . The single antenna case w ith two users is covered in Section II wher eas statements on the con vexity of the M SE region for three or more sing le-antenn a users ar e presented in Section III. Finally , a co njecture on the c onv exity of the multi-a ntenna two user c ase is given in Sectio n IV, and detailed pro ofs for the theorems and c orollaries are attac hed in App endices A – C for the sake of readab ility . I I . C O N V E X I T Y F O R T W O S I N G L E - A N T E N N A U S E R S In th is section we present statements on the geometry of the MSE r egion of two sing le-antenn a users. For this setup, conv exity can always be shown: Theorem II.1 : The MSE r e gion of two sing le-anten na users is conve x both in the multiple- access channel and in th e vector br o adcast chann el. Pr o of: See App endix A. For the m ost im portant part of the boun dary of the two user MSE region ( see Fig . 1) ther e is a f unctiona l relationship ε 2 = g ( ε 1 ) (3) between the two user s’ MSEs ε 1 and ε 2 . If the ch annel vectors h 1 and h 2 describing the tran smission fr om both users to the base station in the du al MA C are not c olinear, the fu nction g is strictly conve x, otherwise, it is affine: Corollary II.1. The function g : ε 1 7→ ε 2 = g ( ε 1 ) describing the efficien t set of the MSE r e gion is strictly conve x if h 1 and h 2 ar e not coline ar . Oth erwise, g is affine. Pr o of: See App endix B. I I I . N O N C O N V E X I T Y E X A M P L E F O R M O R E T H A N T W O S I N G L E - A N T E N N A U S E R S Although the MSE region is con vex for two single-ante nna users, this pro perty may get lost when ad ding an addition al user , even if h e is equipped with only a single a ntenna: Theorem III.1: The thr ee user MS E re gio n of b oth the vector br o adcast channel and the multiple- access channel is not necessarily conve x. Pr o of: For the p roof, we p resent a sim ple example in Ap- pendix C where the line segmen t connecting tw o feasible MSE triples lies outside the MSE region. A further confirmation of Theorem I II.1 resu lts from the o bservation, th at the weighted sum-MSE minimization has mo re th an one lo cal minimum, see Ap pendix C. Nonconve xity implies for e xample that n ot e very p oint of the MSE efficient set can b e achieved by m eans of the weighted sum-MSE minimization tech nique. Balancing a p- proach es b ased on th e weig hted sum-MSE m inimization al- gorithm hence m ay fail to achieve the desired MSE ratios, cf. [3]. Instead they are prone to oscillations. The following theorem covers the case w hen (th ree or ) mo re th an thre e u sers are pr esent in the system: Theorem III.2: The MSE re g ion of more than two u sers may be n onconve x both in the vector br oa dcast channel and in th e multiple-acce ss channel. Pr o of: The three user case has alrea dy been sho wn in Theorem III.1. For mor e than three users, th e MSE region is a mu lti-dimension al ma nifold. Howe ver , setting the p owers of those users to p 4 = . . . = p K = 0 , the intersection of this man- ifold with the K − 3 hyperp lane(s) p i = 0 , i ∈ { 4 , . . . , K } , is again a three- dimension al manif old whic h m ay have the same geometry as the m anifold of the thre e user case. Hence, the MSE regio n may be n onconve x for more tha n three users as well. I V . C O N J E C T U R E O N T H E C O N V E X I T Y O F T H E M U LT I - A N T E N N A T W O U S E R C A S E A co unter-example to con vexity of the MSE region when multi-anten na users are inv olved has b een shown in [3], wh ere two users each eq uipped with two anten nas com municate P S f r a g r e p l a c e m e n t s 1 1 ε 2 ε 1 ε min , 2 ε min , 1 ε 2 = g ( ε 1 ) Fig. 1. MSE ε 2 of user 2 dependin g on MSE ε 1 of user 1 . with a mu lti-antenna base-station. Similarly , th e multi-ca rrier single-anten na system in [4] can be recast into a multi-an tenna MIMO b roadca st channel system where again no nconve xity was ob served. Following th e id ea in the proof of Theo- rem II I.2, the MSE region o f two or m ore th an two users may b e non conve x as soon as two m ulti-anten na users are present. Proving con vexity for the case of one single-anten na user and o ne mu lti-antenn a user turns out to be difficult since a par ametrization of the lower left boundary of the feasible MSE region is not known, points on this bo undary are obtaine d by limits of iterative algo rithms. Nonetheless, extensi ve simulation results b ring us to the co njecture tha t the MSE region of one single antenna user and o ne multi-anten na user is co n vex. A P P E N D I X A P R O O F O F T H E O R E M I I . 1 Because o f the MSE duality between the vector BC [8] and the MA C in [9], [6], [ 10], it suffices to prove conve xity in the MAC whic h is easier to handle. Fig. 1 shows the basic characteristics of the two user MSE region fo r single-anten na users. Here, the MSEs ε 1 and ε 2 of b oth users are upper bound ed b y 1 since MMSE rec eiv ers are assum ed. Allowing for other rec eiv er types does not b ring any reasonable g ain since o nly MSE-p airs where a t least on e entry may lie ab ove 1 would arise. Und er the assum ption of MMSE receivers, the right par t of th e bo undar y of the MSE region is obtain ed when user one does not tr ansmit any da ta to the base station at all and user two varies its transmit power fr om zero to P Tx . Similarly , the u pper p art of the boun dary is reach ed when user two does not transmit at all whereas user one v aries its transmit power fro m zero to P Tx . Eviden tly , the most interesting par t of the bound ary is the lower lef t one, where the sum o f both transmit powers eq uals the maxim um av ailable power P Tx . MSE p airs lying on this bound ary fe ature the fu nctional relationship ε 2 = g ( ε 1 ) , whe re th e d omain an d the image of g are the sets [ ε min , 1 , 1] and [ ε min , 2 , 1] , resp ectiv ely . When less than th e to tal transmit p ower P Tx is consumed, p oints are achieved th at are element of the interior of the MSE region . As a c onclusion , conv exity of the set of feasible MSE points correspo nds to conve xity of the fu nction g relating the MSE ε 1 of user one to the MSE ε 2 of u ser two on the lower left bound ary of th e MSE region. In the following, we show that ∂ 2 ε 2 ∂ ε 2 1 = ∂ 2 g ( ε 1 ) ∂ ε 2 1 ≥ 0 (4) holds wh ich imm ediately implies conve xity of g . Unfortu nately , a direct functio nal relation ship between ε 2 and ε 1 is not available. Instead, the two MSE s ε 1 and ε 2 are param etrized by the transmit power of on e of them, for example b y the transmit power p ∈ D = [0 , P Tx ] of user one: ε 1 = f 1 ( p ) , ε 2 = f 2 ( p ) . W e can con clude that user tw o has to transmit with power P Tx − p in ord er to u tilize the c omplete power budget. In conjunc tion with MMSE receivers, the mean square er ror of user on e read s as ε 1 = f 1 ( p ) = 1 − p h H 1 X − 1 ( p ) h 1 > 0 , (5) with the po siti ve definite covariance m atrix of the r eceiv ed signal X ( p ) = σ 2 η I N + p h 1 h H 1 + ( P Tx − p ) h 2 h H 2 (6) and σ 2 η > 0 rep resents the variance of the noise at every antenna elem ent. Similarly , the M SE of user two is d enoted by ε 2 = f 2 ( p ) = 1 − ( P Tx − p ) h H 2 X − 1 ( p ) h 2 > 0 . (7) Combining (5), (7), and (6), the fun ction f 1 turns o ut to be strictly monoton ically decr easing in p , i.e., ˙ ε 1 := ∂ f 1 ( p ) ∂ p < 0 ∀ p ∈ D , (8) whereas f 2 is strictly mon otonically incre asing in p : ˙ ε 2 := ∂ f 2 ( p ) ∂ p > 0 ∀ p ∈ D . (9) From (8) an d (9), p seudo-c onv exity of g already follows. Before validating (4), we compute th e fir st de riv ati ve: ∂ g ( ε 1 ) ∂ ε 1 = ∂ f 2 ( p ) ∂ p ∂ f 1 ( p ) ∂ p      p = f − 1 1 ( ε 1 ) . (10) Note that f − 1 1 ( ε 1 ) denotes the inv erse fu nction o f f 1 which exists du e to (8). Differentiating (10) again with r espect to ε 1 yields ∂ 2 g ( ε 2 ) ∂ ε 2 1 = ∂ ∂ ε 1 ∂ f 2 ( p ) ∂ p ∂ f 1 ( p ) ∂ p      p = f − 1 1 ( ε 1 ) ! = ∂ ∂ p ∂ f 2 ( p ) ∂ p ∂ f 1 ( p ) ∂ p !      p = f − 1 1 ( ε 1 ) · ∂ f − 1 1 ( ε 1 ) ∂ ε 1 = ¨ ε 2 ˙ ε 1 − ¨ ε 1 ˙ ε 2 ( ˙ ε 1 ) 2      p = f − 1 1 ( ε 1 ) · 1 ˙ ε 1 | p = f − 1 1 ( ε 1 ) = ¨ ε 2 ˙ ε 1 − ¨ ε 1 ˙ ε 2 ( ˙ ε 1 ) 3      p = f − 1 1 ( ε 1 ) . (11) Since f − 1 1 maps from [ ε min , 1 , 1] to D , and since ˙ ε 1 < 0 ho lds ∀ p ∈ D , the function g is conve x iff [see (11) and cond . (4)] ¨ ε 2 ˙ ε 1 − ¨ ε 1 ˙ ε 2 ≤ 0 ⇔ g is conv ex . (12) For notation al brevity , we intr oduce th e two substitutions a i,j = h H i X − 1 ( p ) h j and b i,j = h H i X − 2 ( p ) h j , (13) which satisfy a i,j = a ∗ j,i and b i,j = b ∗ j,i . Makin g use of ∂ X − 1 ( p ) ∂ p = − X − 1 ( p ) ∂ X ( p ) ∂ p X − 1 ( p ) , the first deriv ativ es with respect to p in (8) and (9) can be shown to equ al ˙ ε 1 = − σ 2 η b 1 , 1 − P Tx | a 1 , 2 | 2 , ˙ ε 2 = + σ 2 η b 2 , 2 + P Tx | a 1 , 2 | 2 , (14) respectively . Differentiating (14) again w .r .t. p , we ob tain ¨ ε 1 = 2 σ 2 η [ a 1 , 1 b 1 , 1 − ℜ{ a 1 , 2 b 2 , 1 } ] + 2 P Tx | a 1 , 2 | 2 ( a 1 , 1 − a 2 , 2 ) , ¨ ε 2 = 2 σ 2 η [ a 2 , 2 b 2 , 2 − ℜ{ a 2 , 1 b 1 , 2 } ] + 2 P Tx | a 1 , 2 | 2 ( a 2 , 2 − a 1 , 1 ) . Inserting (14) and the last two equations into (12) and applying ℜ{ a 2 , 1 b 1 , 2 } = ℜ{ a 1 , 2 b 2 , 1 } results in ¨ ε 2 ˙ ε 1 − ¨ ε 1 ˙ ε 2 = 2 σ 2 η P Tx | a 1 , 2 | 2 [2 ℜ{ a 1 , 2 b 2 , 1 } − a 2 , 2 b 1 , 1 − a 1 , 1 b 2 , 2 ] + 2 σ 4 η b 1 , 1 ( ℜ{ a 1 , 2 b 2 , 1 } − a 1 , 1 b 2 , 2 ) + 2 σ 4 η b 2 , 2 ( ℜ{ a 1 , 2 b 2 , 1 } − a 2 , 2 b 1 , 1 ) . (15) In order to prove that (15) is not positi ve to fulfill t he con vexity requirem ent in (12), we will reveal that all three summands in (15) are not p ositiv e. For the first summ and, this turn s out to be very easy: Noticing that a i,i > 0 and b i,i > 0 , th e first summand in (15) is no npositive if 4 ℜ 2 { a 2 , 1 b 1 , 2 } ≤ ( a 2 , 2 b 1 , 1 + a 1 , 1 b 2 , 2 ) 2 . (16) Clearly , we can up per boun d th e real part by the magnitu de and ap ply the Cauchy-Schwarz -ineq uality with (13) to bou nd the mag nitude: 4 ℜ 2 { a 2 , 1 b 1 , 2 } ≤ 4 | a 2 , 1 b 1 , 2 | 2 ≤ 4 a 2 , 2 a 1 , 1 b 1 , 1 b 2 , 2 . (17) V alidating the inequality ( a 2 , 2 b 1 , 1 + a 1 , 1 b 2 , 2 ) 2 ≥ 4 a 2 , 2 a 1 , 1 b 1 , 1 b 2 , 2 ⇔ ( a 2 , 2 b 1 , 1 − a 1 , 1 b 2 , 2 ) 2 ≥ 0 leads in conjunctio n with (17) to the co nclusion that (16) is fulfilled, i.e., the first summand in (15) is nonp ositiv e. Nonpositivity o f the second summand in (15) is r esembled by th e inequ ality ℜ  a 1 , 2 a 1 , 1 b 2 , 1 b 2 , 2  ≤ 1 . (18) T o prove (1 8) we explicitly ha ve to exploit th e stru cture of X ( p ) in (6) wh ich makes the proof lo nger th an the one fo r the first summand. Interestingly , the real part operator i n (18) is redund ant as its argument turns out to be real-valued. App lying the matrix inv ersion lemm a several times, we get a 1 , 2 a 1 , 1 = σ 2 η h H 1 h 2 σ 2 η k h 1 k 2 2 + d ( P Tx − p ) , (19) with the substitution d = k h 1 k 2 2 k h 2 k 2 2 − | h H 1 h 2 | 2 ≥ 0 . (20) Applying several times the matr ix in version lemm a as for the first fr action, the secon d fraction in ( 18) c an be expressed as b 2 , 1 b 2 , 2 = h H 2 h 1  σ 4 η − p ( P Tx − p ) d  σ 4 η k h 2 k 2 2 + dp (2 σ 2 η + p k h 1 k 2 2 ) . (21 ) Multiplying ( 19) by (21) yields the real-valued expr ession b 2 , 1 a 1 , 2 a 1 , 1 b 2 , 2 = σ 6 η | h H 1 h 2 | 2 − c 1 σ 6 η k h 1 k 2 2 k h 2 k 2 2 + c 2 ∈ R with the two substitutions c 1 = σ 2 η | h H 1 h 2 | 2 pd ( P Tx − p ) ≥ 0 , c 2 =  σ 2 η k h 1 k 2 2 + d ( P Tx − p )  dp  2 σ 2 η + p k h 1 k 2 2  + σ 4 η k h 2 k 2 2 d ( P Tx − p ) ≥ 0 . Since both c 1 and c 2 are no nnegative, we find b 2 , 1 a 1 , 2 a 1 , 1 b 2 , 2 ≤ σ 6 η | h H 1 h 2 | 2 σ 6 η k h 1 k 2 2 k h 2 k 2 2 as an upper boun d f rom which (18) directly f ollows du e to the Cau chy-Schwarz -inequa lity . Th us, th e n onpositivity o f th e second summand in (15) is p roven. Finally , the non positivity of the third summan d in (1 5) is shown by th e same reasonin g as for th e second summan d: ℜ  a 1 , 2 a 2 , 2 b 2 , 1 b 1 , 1  ≤ 1 (22 ) is ded uced fr om b 2 , 1 a 1 , 2 a 2 , 2 b 1 , 1 = σ 6 η | h H 1 h 2 | 2 − d 1 σ 6 η k h 1 k 2 2 k h 2 k 2 2 + d 2 ∈ R , where d i follows from c i by interchan ging indices and powers: d 1 = c 1 , d 2 =  σ 2 η k h 2 k 2 2 + dp  d ( P Tx − p )  2 σ 2 η + ( P Tx − p ) k h 2 k 2 2  + σ 4 η k h 1 k 2 2 dp ≥ 0 . As all thre e summa nds in (15) are n onpositive, th e ineq uality in (12) is satisfied and the pro of f or the conv exity of the MSE region is c omplete. A P P E N D I X B P R O O F O F C O R O L L A RY I I . 1 If the inequ ality in (1 2) is strict for all p ∈ D , g is strictly conv ex. Exclud ing equality in (12) therefo re en sures that g is strictly c onv ex. The d ifference in (15) is zero if and o nly if all three summan ds are zero since each summan d is no npositive. In order to let the first summ and vanish, th e Cau chy-Schwarz - inequality in (17) has to be fulfilled with equ ality . T o this end, h 1 and h 2 have to be colin ear wh ich also f ulfills (16) with equality . If both c hannel vectors are colin ear , d = 0 r esults from (20) and the variables c 1 , c 2 , d 1 , and d 2 are zero a s well. Obviously , (18) holds with equality and the last tw o summands in ( 15) vanish. Thus, we have shown th at if the two channel vectors h 1 and h 2 are not colinear, then the f unction g is strictly conv ex. Addition ally , if b oth vectors are co linear, g has cu rvature zero f or all p owers p ∈ D . As a con sequence, g is affine. In the latter case, we have the relatio nship g ( ε 1 ) = − ε 1 | α | 2 + | α | 2 γ k h 1 k 2 2 1 + | α | 2 γ k h 1 k 2 2 + 1 + | α | 2 1 + | α | 2 γ k h 1 k 2 2 , (23 ) where γ = P Tx /σ 2 η denotes the transmit SNR, h 2 = α h 1 , and ε 1 ∈ [ ε min , 1 , 1] with ε min , 1 = 1 1 + γ k h 1 k 2 2 . (24) A P P E N D I X C P R O O F O F T H E O R E M I I I . 1 A nonconv ex three user MSE region ca n fo r exam ple b e obtained b y the chann el matrix H = [ h 1 , h 2 , h 3 ] =  1 0 1 0 1 1  (25) and a tr ansmit power P Tx = 10 . In this case, the base station is equ ipped with N = 2 anten nas, and the c hannel vector h 3 is the sum of h 1 and h 2 . Note that the b ase station has fewer anten nas th an users are present in the system in this special case. Nonconve xity of the MSE region can also be and has been observed when the ch annel vectors of all u sers are linearly in depend ent ( N ≥ K must hold then) . If the MSE region was conv ex, the lin e segment between every two feasible MSE triples would have to b e a subset of th e region. Moreover , the weighted sum -MSE minim ization with arbitr ary nonnegative weig hts w = [ w 1 , . . . , w K ] T ≥ 0 K , w 6 = 0 K may have stationary points fulfilling th e KKT cond itions with only o ne common value o f the weighted min imization. In the following, we show that these co nditions are violated for the channel in (2 5). The weig hted sum-M SE minimiz ation reads as minimize p 1 ,...,p K K X k =1 w k ε k s.t.: K X k =1 p k ≤ P Tx , p k ≥ 0 ∀ k , (26) where p k is the power with which user k tran smits in th e uplink an d the MSE of u ser k re ads as ε k = 1 − p k h H k X − 1 h k (27) with the received signal covariance matrix X = σ 2 η I N + K X ℓ =1 p ℓ h ℓ h H ℓ . (28) The Lagrangian fu nction associated to (26) r eads as L = K X k =1 w k ε k + λ  K X k =1 p k − P Tx  − K X k =1 µ k p k . (29) Note that the Lagrangian multipliers λ and µ 1 , . . . , µ K have to be nonnegative real. If above Lagrangian L has stationa ry points with d ifferent values for L , the und erlying MSE region is not con vex since m ore than one h yperp lane with norma l vector [ w 1 , . . . , w K ] T locally supportin g the MSE region exists. The KKT condition s read as h H k ˇ X − 1 ( w k ˇ X − ˇ S ) ˇ X − 1 h k = ˇ λ − ˇ µ k ∀ k , (30) ˇ p k ≥ 0 ∀ k , (31) ˇ p k ˇ µ k = 0 ∀ k , (32) ˇ µ k ≥ 0 ∀ k , (33) K X k =1 ˇ p k ≤ P Tx , (34) ˇ λ  K X k =1 ˇ p k − P Tx  = 0 , (35) ˇ λ ≥ 0 , (36) with the substitution S = K X ℓ =1 w ℓ p ℓ h ℓ h H ℓ . (37) Note that ch ecked v ariables ˇ ( · ) are those which fu lfill the KKT condition s. Assuming a weight vector w = [0 . 2 2 , 0 . 54 , 0 . 24] T , (38) the weighted sum-MSE minimization ( 26) features two sta- tionary points satisfyin g the KKT co ndition s ( 30)–(36) fo r th e channel vectors ( 25) and a tran smit power P Tx = 10 . The first set of p rimal and dual variables f ulfilling th e KKTs reads as ˇ p (1) = [3 . 6753 , 6 . 324 7 , 0] T , ˇ λ (1) = 0 . 01 01 , ˇ µ 1 (1) = ˇ µ 2 (1) = 0 , ˇ µ 3 (1) = 0 . 0266 , (39) and ach iev es a weigh ted sum-MSE P 3 k =1 w k ˇ ε k (1) = 0 . 3607 8 . The seco nd set of variables reads as ˇ p (2) = [0 , 7 . 0794 , 2 . 9206 ] T , ˇ λ (2) = 0 . 01 15 , ˇ µ 1 (2) = 0 . 00 7 , ˇ µ 2 (2) = ˇ µ 3 (2) = 0 , (40) and obtains a sligh tly la rger metric P 3 k =1 w k ˇ ε k (2) = 0 . 38 28 . The existence o f two KKT p oints with different values alg e- braically proves the non conv exity o f the MSE region. A geometr ical proof is shown in Fig. 2, where the three- user MSE region f or the chann el in (25) is plotted with P Tx = 10 . The two KKT po ints in (3 9) and (4 0) achieve i ndividual MSEs ˇ ε (1) =  ˇ ε 1 (1) , ˇ ε 2 (1) , ˇ ε 3 (1)  T = [0 . 2139 , 0 . 136 5 , 1] T , ˇ ε (2) =  ˇ ε 1 (2) , ˇ ε 2 (2) , ˇ ε 3 (2)  T = [1 , 0 . 1977 , 0 . 2335 ] T . (41) 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 P S f r a g r e p l a c e m e n t s ε 1 ε 2 ε 3 Fig. 2. Exa mple of a noncon vex MSE region for K = 3 users. The line segment conne cting two feasible points lies outside the region. Howe ver , the line segment connectin g ˇ ε (1) and ˇ ε (2) does not completely belong to the MSE r egion, it lies ou tside the region and tou ches the bou ndary of the MSE region at ˇ ε (1) and ˇ ε (2) . Evidently , the MSE region canno t be conv ex. R E F E R E N C E S [1] S. Shi and M. Schubert , “Con ve xity Analysis of the Feasible MSE Regi on of Sum-Power Constrained Mult iuser MIMO Systems, ” in Proc . IEEE Internat. Symp. on P erson al, Indoor and Mobile Radio Commu- nicati ons (PIMRC), Berlin, Germany , Sept. 2005. [2] E. Jorswieck and H. 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