Asynchronous Iterative Waterfilling for Gaussian Frequency-Selective Interference Channels

Asyn hronous Iterativ e W aterlling for Gaussian F requeny-Seletiv e In terferene Channels Gesualdo Sutari 1 , Daniel P . P alomar 2 , and Sergio Barbarossa 1 E-mail: { sutari, sergio } info om.uniroma1.it, palomarust.hk 1 Dpt. INF OCOM, Univ. of Rome La Sapienza, Via Eudossiana 18, 00184 Rome, Italy . 2 Dpt. of Eletroni and Computer Eng., Hong K ong Univ. of Siene and T e hnology , Hong K ong. Submitted to IEEE T r ansations on Information The ory , August 22, 2006. Revised Septem b er 25, 2007. A epted Jan uary 14, 2008. ∗ Abstrat This pap er onsiders the maximization of information rates for the Gaussian frequeny-seletiv e in terferene  hannel, sub jet to p o w er and sp etral mask onstrain ts on ea h link. T o deriv e deen- tralized solutions that do not require an y o op eration among the users, the optimization problem is form ulated as a stati nono op erativ e game of omplete information. T o a hiev e the so-alled Nash equilibria of the game, w e prop ose a new distributed algorithm alled asyn hronous itera- tiv e w aterlling algorithm. In this algorithm, the users up date their p o w er sp etral densit y in a ompletely distributed and asyn hronous w a y: some users ma y up date their p o w er allo ation more frequen tly than others and they ma y ev en use outdated measuremen ts of the reeiv ed in terferene. The prop osed algorithm represen ts a unied framew ork that enompasses and generalizes all kno wn iterativ e w aterlling algorithms, e.g., sequen tial and sim ultaneous v ersions. The main result of the pap er onsists of a unied set of onditions that guaran tee the global on v erge of the prop osed algorithm to the (unique) Nash equilibrium of the game. Index T erms: Game theory , Gaussian frequeny-seletiv e in terferene  hannel, Nash equilib- rium, totally asyn hronous algorithm, iterativ e w aterlling algorithm. 1 In tro dution and Motiv ation In this pap er w e fo us on the frequeny seletiv e in terferene  hannel with Gaussian noise. The apait y region of the in terferene  hannel is still unkno wn. Only some b ounds are a v ailable (see, e.g., [ 1 , 2℄ for a summary of the kno wn results ab out the Gaussian in terferene  hannel). A pragmati approa h that leads to an a hiev able region or inner b ound of the apait y region is to restrit the system to op erate as a set of indep enden t units, i.e., not allo wing m ultiuser eno ding/deo ding or the use of in terferene anelation te hniques. This a hiev able region is v ery relev an t in pratial systems with limitations on ∗ P art of this w ork w as presen ted in IEEE W orkshop on Signal Pr o  essing A dvan es in Wir eless Communi ations, (SP A W C-2006) , July 2-5, 2006, and in Information The ory and Appli ations (IT A) W orkshop , Jan. 29 - F eb. 2, 2007. This w ork w as supp orted b y the SURF A CE pro jet funded b y the Europ ean Comm unit y under Con trat IST-4-027187- STP-SURF A CE. 1 the deo der omplexit y and simpliit y of the system. With this assumption, m ultiuser in terferene is treated as noise and the transmission strategy for ea h user is simply his p o w er allo ation. The system design redues then to nding the optim um P o w er Sp etral Densit y (PSD) for ea h user, aording to a sp eied p erformane metri. Within this on text, existing w orks [ 4℄ − [15℄ onsidered the maximization of the information rates of all the links, sub jet to transmit p o w er and (p ossibly) sp etral mask onstrain ts on ea h link. The latter onstrain ts are esp eially motiv ated in adaptiv e senarios, e.g., ognitiv e radio, where previously allo ated sp etral bands ma y b e reused, but pro vided that the generated in terferene falls b elo w sp ei- ed thresholds [3℄. In [4, 5 ℄, a en tralized approa h based on dualit y theory w as prop osed to ompute, under te hnial onditions, the largest a hiev able rate region of the system (i.e., the P areto-optimal set of the a hiev able rates). Our in terest, in this pap er, is fo used on nding distributed algorithms with no en tralized on trol and no o op eration among the users. Hene, w e ast the system design under the on v enien t framew ork of game theory . In partiular, w e form ulate the rate maximization problem as a strategi non-o op erativ e game of omplete information, where ev ery link is a pla y er that omp etes against the others b y  ho osing the sp etral p o w er allo ation that maximizes his o wn information rate. An equilibrium for the whole system is rea hed when ev ery pla y er is unilaterally optim um, i.e., when, giv en the urren t strategies of the others, an y  hange in his o wn strategy w ould result in a rate loss. This equilibrium onstitutes the elebrated notion of Nash Equilibrium (NE) [17 ℄. The Nash equilibria of the rate maximization game an b e rea hed using Gaussian signaling and a prop er PSD from ea h user [9℄ − [12℄. T o obtain the optimal PSD of the users, Y u, Ginis, and Cio prop osed the se quential Iterativ e W aterFilling Algorithm (IWF A) [6 ℄ in the on text of DSL systems, mo deled as a Gaussian frequeny-seletiv e in terferene  hannel. The algorithm is an instane of the Gauss-Seidel s heme [18℄: the users maximize their o wn information rates se quential ly (one after the other), aording to a xed up dating order. Ea h user p erforms the single-user w aterlling solution giv en the in terferene generated b y the others as additiv e (olored) noise. The most app ealing features of the sequen tial IWF A are its lo w-omplexit y and distributed nature. In fat, to ompute the w aterlling solution, ea h user only needs to measure the noise-plus-in terferene PSD, without requiring sp ei kno wledge of the p o w er allo ations and the  hannel transfer funtions of all other users. The on v ergene of the sequen tial IWF A has b een studied in a n um b er of w orks [ 7℄, [11℄ − [15℄, ea h time obtaining milder on v ergene onditions. Ho w ev er, despite its app ealing prop erties, the sequen tial IWF A ma y suer from slo w on v ergene if the n um b er of users in the net w ork is large, just b eause of the sequen tial up dating strategy . In addition, the algorithm requires some form of en tral s heduling to determine the order in whi h users up date their PSD. T o o v erome the dra wba k of slo w sp eed of on v ergene, the simultane ous IWF A w as prop osed in [9, 11, 12℄. The sim ultaneous IWF A is an instane of the Jaobi s heme [18℄: at ea h iteration, the users up date their o wn strategies simultane ously , still aording to the w aterlling solution, but using the in terferene generated b y the others in the pr evious iteration. The sim ultaneous IWF A w as sho wn to 2 on v erge to the unique NE of the rate maximization game faster than the sequen tial IWF A and under w eak er onditions on the m ultiuser in terferene than those giv en in [6, 7℄, [13℄ − [15 ℄ for the sequen tial IWF A. F urthermore, dieren tly from [6, 7, 13, 15 ℄, the algorithm as prop osed in [11 ℄ tak es expliitly in to aoun t the sp etral masks onstrain ts. Ho w ev er, the sim ultaneous IWF A still requires some form of syn hronization, as all the users need to b e up dated sim ultaneously . Clearly , in a real net w ork with man y users, the syn hronization requiremen t of b oth sequen tial and sim ultaneous IWF As go es against the non-o op eration priniple and it migh t b e unaeptable. This pap er generalizes the existing results for the sequen tial and sim ultaneous IWF As and dev elops a unied framew ork based on the so-alled asynhr onous IWF A, that falls within the lass of totally asyn hronous s hemes of [18 ℄. In this more general algorithm, all users still up date their p o w er allo a- tions aording to the w aterlling solution, but the up dates an b e p erformed in a total ly asynhr onous w a y (in the sense of [18℄). This means that some users ma y up date their p o w er allo ations mor e fr e- quently than others and they ma y ev en use an outdate d measuremen t of the in terferene aused from the others. These features mak e the asyn hronous IWF A app ealing for all pratial senarios, either wired or wireless, as it strongly relaxes the need for o ordinating the users' up dating s hedule. The main on tribution of this pap er is to deriv e suien t onditions for the global on v ergene of the asyn hronous IWF A to the (unique) NE of the rate maximization game. In terestingly , our on v ergene onditions are sho wn to b e indep enden t of the users' up date s hedule. Hene, they represen t a unied set of onditions enompassing all existing algorithms, either syn hronous or asyn hronous, that an b e seen as sp eial ases of our asyn hronous IWF A. Our onditions also imply that b oth sequen tial and sim ultaneous algorithms are robust to situations where some users ma y fail to follo w their up dating s hedule. Finally , w e sho w that our suien t onditions for the on v ergene of the asyn hronous IWF A oinide with those giv en reen tly in [11 ℄ for the on v ergene of the (syn hronous) sequen tial and sim ultaneous IWF As, and are larger than onditions obtained in [ 6 , 7℄, [13 ℄ − [15℄ for the on v ergene of the sequen tial IWF A in the absene of sp etral mask onstrain ts. The pap er is organized as follo ws. Setion 2 pro vides the system mo del and form ulates the opti- mization problem as a strategi non-o op erativ e game. Setion 3 on tains the main result of the pap er: the desription of the prop osed asyn hronous IWF A along with its on v ergene prop erties. Setion 4 reo v ers the sequen tial and sim ultaneous IWF As as sp eial ases of the asyn hronous IWF A and then, as a b y pro dut, it pro vides a unied set of on v ergene onditions for b oth algorithms. Finally , Setion 5 dra ws some onlusions. 2 System Mo del and Problem F orm ulation In this setion w e larify the assumptions and the onstrain ts underlying the system mo del and w e form ulate the optimization problem addressed in this pap er expliitly . 3 2.1 System mo del W e onsider a Gaussian frequeny-seletiv e in terferene  hannel omp osed b y m ultiple links. Sine our goal is to nd distributed algorithms that require neither a en tralized on trol nor a o ordination among the users, w e fo us on transmission te hniques where in terferene anelation is not p ossible and m ultiuser in terferene is treated b y ea h reeiv er as additiv e olored noise. The  hannel frequeny- seletivit y is handled, with no loss of optimalit y , adopting a m ultiarrier transmission strategy . 1 Giv en the ab o v e system mo del, w e mak e the follo wing assumptions: A.1 Ea h  hannel  hanges suien tly slo wly to b e onsidered xed during the whole transmission, so that the information theoreti results are meaningful; A.2 The  hannel from ea h soure to its o wn destination is kno wn to the in tended reeiv er, but not to the other terminals; ea h reeiv er is also assumed to measure with no errors the o v erall PSD of the noise plus in terferenes generated b y the other users. Based on this information, ea h reeiv er omputes the optimal p o w er allo ation aross the frequeny bins for its o wn transmitter and feeds it ba k to its transmitter through a lo w bit rate (error-free) feedba k  hannel. 2 A.3 All the users are blo  k-syn hronized with an unertain t y at most equal to the yli prex length. This imp oses a minim um length of the yli prex that will dep end on the maxim um  hannel dela y spread. W e onsider the follo wing onstrain ts: Co.1 Maxim um o v erall transmit p o w er for ea h user: E n k s q k 2 2 o = N X k =1 ¯ p q ( k ) ≤ N P q , q = 1 , . . . , Q , (1) where s q on tains the N sym b ols transmitted b y user q on the N arriers, ¯ p q ( k ) , E n | s q ( k ) | 2 o denotes the p o w er allo ated b y user q o v er arrier k , and P q is p o w er in units of energy p er transmitted sym b ol. Co.2 Sp etral mask onstrain ts: E n | s q ( k ) | 2 o = ¯ p q ( k ) ≤ ¯ p max q ( k ) , k = 1 , . . . , N , q = 1 , . . . , Q , (2) where ¯ p max q ( k ) represen ts the maxim um p o w er that is allo w ed to b e allo ated on the k -th frequeny bin from the q -th user. Constrain ts lik e ( 2 ) are imp osed to limit the amoun t of in terferene generated b y ea h transmitter o v er pre-sp eied bands. The main goal of this pap er is to obtain the optimal v etor p o w er allo ation p q , ( p q (1) , . . . , p q ( N )) , for ea h user, aording to the optimalit y riterion in tro dued in the next setion. 1 Multiarrier transmission is a apait y-lossless strategy for suien tly large blo  k length [21 , 22 ℄. 2 In pratie, b oth measuremen ts and feedba k are inevitably aeted b y errors. This senario an b e studied b y extending our form ulation to games with partial information [23 , 24 ℄, but this go es b ey ond the sop e of the presen t pap er. 4 2.2 Problem form ulation as a game W e onsider a strategi non-o op erativ e game [23 , 24 ℄, in whi h the pla y ers are the links and the pa y o funtions are the information rates on ea h link: Ea h pla y er omp etes rationally 3 against the others b y  ho osing the strategy that maximizes his o wn rate, giv en onstrain ts Co.1 and Co.2 . A NE of the game is rea hed when ev ery user, giv en the strategy prole of the others, do es not get an y rate inrease b y  hanging his o wn strategy . Using the signal mo del desrib ed in Setion 2.1, the a hiev able rate for ea h pla y er q is omputed as the maxim um information rate on the q -th link, assuming al l the other r e  eive d signals as additive  olor e d noise . It is straigh tforw ard to see that a (pure or mixed strategy) NE is obtained if ea h user transmits using Gaussian signaling, with a prop er PSD. In fat, for ea h user, giv en that all other users use Gaussian o deb o oks, the optimal o deb o ok maximizing m utual information is also Gaussian [21℄. 4 Hene, the maxim um a hiev able rate for the q -th user is giv en b y [21℄ R q = 1 N N X k =1 log (1 + sinr q ( k )) , (3) with sinr q ( k ) denoting the Signal-to-In terferene plus Noise Ratio (SINR) on the k -th arrier for the q -th link: sinr q ( k ) ,   ¯ H q q ( k )   2 ¯ p q ( k ) / d γ q q σ 2 q ( k ) + P r 6 = q   ¯ H q r ( k )   2 ¯ p r ( k ) / d γ r q , | H q q ( k ) | 2 p q ( k ) σ 2 q ( k ) + P r 6 = q | H q r ( k ) | 2 p r ( k ) , (4) where ¯ H q r ( k ) denotes the frequeny-resp onse of the  hannel b et w een soure r and destination q ex- luding the path-loss d γ q r with exp onen t γ and d q r is the distane b et w een soure r and destination q ; σ 2 q ( k ) is the v ariane of the zero mean irularly symmetri omplex Gaussian noise at reeiv er q o v er the arrier k ; and for the on v eniene of notation, w e ha v e in tro dued the normalized quan tities H q r ( k ) , ¯ H q r ( k ) p P r /d γ q r and p q ( k ) , ¯ p q ( k ) / P q . Observ e that in the ase of pratial o ding s hemes, where only nite order onstellations an b e used, w e an use the gap appro ximation analysis [ 25, 26 ℄ and write the n um b er of bits transmit- ted o v er the N substreams from the q -th soure still as in ( 3) but replaing | H q q ( k ) | 2 in (4 ) with | H q q ( k ) | 2 / Γ q , where Γ q ≥ 1 is the gap. The gap dep ends only on the family of onstellation and on P e,q ; for M -QAM onstellations, for example, if the sym b ol error probabilit y is appro ximated b y P e,q ( sinr q ( k )) ≈ 4 Q  p 3 sinr q ( k ) / ( M − 1)  , the resulting gap is Γ q = ( Q − 1 ( P e,q / 4)) 2 / 3 [25, 26 ℄. In summary , w e ha v e a game with the follo wing struture: G = { Ω , { P q } q ∈ Ω , { R q } q ∈ Ω } , (5) 3 The rationalit y assumption means that ea h user will nev er  hose a stritly dominated strategy . A strategy prole x q is stritly dominated b y z q if Φ q ( x q , y − q ) < Φ q ( z q , y − q ) , for a giv en admissible y − q , ( y 1 , . . . , y q − 1 , y q +1 , . . . , y Q ) , where Φ q denotes the pa y o funtion of pla y er q . 4 Observ e that, in general, Nash equilibria a hiev able using arbitrary non-Gaussian o des ma y exist. In this pap er, w e fo us only on transmission using Gaussian o deb o oks. 5 where Ω , { 1 , 2 , . . . , Q } denotes the set of the Q ativ e links, P q is the set of admissible (normalized) p o w er allo ation strategies, aross the N a v ailable arriers, for the q -th pla y er, dened as 5 P q , ( p q ∈ R N : 1 N N X k =1 p q ( k ) = 1 , 0 ≤ p q ( k ) ≤ p max q ( k ) , k = 1 , . . . , N ) , (6) with p max q ( k ) , p max q ( k ) / P q and R q is the pa y o funtion of the q -th pla y er, dened in ( 3). The optimal strategy for the q -th pla y er, giv en the p o w er allo ation of the others, is then the solution to the follo wing maximization problem 6 maximize p q 1 N N X k =1 log (1 + sinr q ( k )) sub ject to p q ∈ P q , ∀ q ∈ Ω (7) where sinr q ( k ) and P q and are giv en in (4 ) and (6 ), resp etiv ely . Note that, for ea h q , the maxim um in (7) is tak en o v er p q , for a xe d p − q , ( p 1 , . . . , p q − 1 , p q +1 , . . . , p Q ) . The solutions of (7) are the w ell-kno wn Nash Equilibria, whi h are formally dened as follo ws. Denition 1 A (pur e) str ate gy pr ole p ⋆ =  p ∗ 1 , . . . , p ∗ Q  ∈ P 1 × . . . × P Q is a Nash Equilibrium of the game G in ( 5 ) if R q ( p ⋆ q , p ⋆ − q ) ≥ R q ( p q , p ⋆ − q ) , ∀ p q ∈ P q , ∀ q ∈ Ω . (8) Observ e that, for the pa y o funtions dened in (3 ), w e an indeed limit ourselv es to adopt pure strategies w.l.o.g., as w e did in (5), sine ev ery NE of the game is pro v ed to b e a hiev able using pure strategies in [10℄. A ording to (7), all the (pure) Nash equilibria of the game, if they exist, m ust satisfy the w aterlling solution for e ah user, i.e., the follo wing system of nonline ar equations: p ⋆ q = WF q  p ⋆ 1 , . . . , p ⋆ q − 1 , p ⋆ q +1 , . . . , p ⋆ Q  = WF q ( p ⋆ − q ) , ∀ q ∈ Ω , (9) with the w aterlling op erator WF q ( · ) dened as [ WF q ( p − q )] k , " µ q − σ 2 q ( k ) + P r 6 = q | H q r ( k ) | 2 p r ( k ) | H q q ( k ) | 2 # p max q ( k ) 0 , k = 1 , . . . , N , (10) where [ x ] b a denotes the Eulidean pro jetion of x on to the in terv al [ a, b ] . 7 The w ater-lev el µ q is  hosen to satisfy the p o w er onstrain t (1 / N ) P N k =1 p ⋆ q ( k ) = 1 . 5 In order to a v oid the trivial solution p ⋆ q ( k ) = p max q ( k ) for all k , P N k =1 p max q ( k ) > N is assumed for all q ∈ Ω . F urthermore, in the feasible strategy set of ea h pla y er, w e an replae, without loss of generalit y , the original ine quality p o w er onstrain t (1 / N ) P N k =1 p q ( k ) ≤ 1 , with equalit y , sine, at the optim um, this onstrain t m ust b e satised with equalit y . 6 In the optimization problem in (7), an y ona v e stritly inreasing funtion of the rate an b e equiv alen tly onsidered as pa y o funtion of ea h pla y er. The optimal solutions of this new set of problems oinide with those of (7). 7 The Eulidean pro jetion [ x ] b a , with a ≤ b , is dened as follo ws: [ x ] b a = a , if x ≤ a , [ x ] b a = x , if a < x < b , and [ x ] b a = b , if x ≥ b . 6 Observ e that in the absene of sp etral mask onstrain ts (i.e., when p max q ( k ) = + ∞ , ∀ q , ∀ k ), the Nash equilibria of game G are giv en b y the lassial sim ultaneous w aterlling solutions [6, 7℄, where WF q ( · ) in (9) is still obtained from (10 ) simply setting p max q ( k ) = + ∞ , ∀ q , ∀ k. In terestingly , the presene of sp etral mask onstrain ts do es not aet the existene of a pure-strategies NE of game G , as stated in the follo wing. Prop osition 1 The game G in ( 5) always admits at le ast one pur e-str ate gy NE, for any set of hannel r e alizations, p ower and sp e tr al mask  onstr aints. Pro of. The pro of omes from standard results of game theory [23, 24 ℄ and it is giv en in [10℄. In general, the game G ma y admit m ultiple equilibria, dep ending on the lev el of m ultiuser in terfer- ene [10℄. In the forthoming setions, w e pro vide suien t onditions ensuring the uniqueness of the NE and w e address the problem of ho w to rea h su h an equilibrium in a totally distributed w a y . 3 Asyn hronous Iterativ e W aterlling T o rea h the NE of game G , w e prop ose a totally asyn hronous distributed iterativ e w aterlling pro e- dure, whi h w e name asyn hronous Iterativ e W aterFilling Algorithm. The prop osed algorithm an b e seen as an instane of the totally asyn hronous s heme of [18 ℄: all the users maximize their o wn rate in a total ly asynhr onous w a y . More sp eially , some users are allo w ed to up date their strategy more frequen tly than the others, and they migh t p erform their up dates using outdate d information ab out the in terferene aused from the others. What w e sho w is that the asyn hronous IWF A on v erges to a stable NE of game G , whihever the up dating she dule is , under rather mild onditions on the m ultiuser in terferene. In terestingly , these onditions are also suien t to guaran tee the uniqueness of the NE. T o pro vide a formal desription of the asyn hronous IWF A, w e need to in tro due some preliminary denitions. W e assume, without an y loss of generalit y , that the set of times at whi h one or more users up date their strategies is the disrete set T = N + = { 0 , 1 , 2 , . . . } . Let p ( n ) q denote the v etor p o w er allo ation of user q at the n -th iteration, and let T q ⊆ T denote the set of times n at whi h user q up dates his p o w er v etor p ( n ) q (th us, implying that, at time n / ∈ T q , p ( n ) q is left un hanged). Let τ q r ( n ) denote the most reen t time at whi h the in terferene from user r is p ereiv ed b y user q at the n -th iteration (observ e that τ q r ( n ) satises 0 ≤ τ q r ( n ) ≤ n ). Hene, if user q up dates its p o w er allo ation at the n -th iteration, then it w aterlls, aording to ( 10 ), the in terferene lev el aused b y the p o w er allo ations of the others: p ( τ q ( n )) − q ,  p ( τ q 1 ( n )) 1 , . . . , p ( τ q q − 1 ( n )) q − 1 , p ( τ q q +1 ( n )) q +1 , . . . , p ( τ q Q ( n )) Q  . (11) The o v erall system is said to b e totally asyn hronous if the follo wing w eak assumptions are satised for ea h q [18 ℄: A1) 0 ≤ τ q r ( n ) ≤ n ; A2) lim k →∞ τ q r ( n k ) = + ∞ ; and A3) |T q | = ∞ ; where { n k } is a sequene of elemen ts in T q that tends to innit y . Assumptions A1 − A3 are standard in asyn hronous on v ergene theory [18℄, and they are fullled in an y pratial implemen tation. In fat, A1 simply 7 indiates that, at an y giv en iteration n , ea h user q an use only the in terferene v etors p ( τ q ( n )) − q allo ated b y the other users in the previous iterations (to preserv e ausalit y). Assumption A2 states that, for an y giv en iteration index n k , the v alues of the omp onen ts of p ( τ q ( n )) − q in (11 ) generated prior to n k , are not used in the up dates of p ( n ) q , when n b eomes suien tly larger than n k ; whi h guaran tees that old information is ev en tually purged from the system. Finally , assumption A3 indiates that no user fails to up date its o wn strategy as time n go es on. Giv en game G , let D min q ⊆ { 1 , · · · , N } denote the set { 1 , . . . , N } (p ossibly) depriv ed of the arrier indies that user q w ould nev er use as the b est resp onse set to any strategies adopted b y the other users [10℄: D min q ,  k ∈ { 1 , . . . , N } : ∃ p − q ∈ P − q su h that [ WF q ( p − q )] k 6 = 0  , (12) with WF q ( · ) dened in (10) and P − q , P 1 × · · · × P q − 1 × P q +1 × · · · × P Q . In [10℄, an iterativ e pro edure to obtain a set D q su h that D min q ⊆ D q ⊆ { 1 , · · · , N } is giv en. Let the matrix S max ∈ R Q × Q + b e dened as [ S max ] q r ,      max k ∈D q ∩D r | ¯ H q r ( k ) | 2 | ¯ H q q ( k ) | 2 d γ q q d γ q r P r P q , if r 6 = q , 0 , otherwise, (13) with the on v en tion that the maxim um in (13) is zero if D q ∩ D r is empt y . In (13), ea h set D q an b e  hosen as an y subset of { 1 , · · · , N } su h that D min q ⊆ D q ⊆ { 1 , · · · , N } , with D min q dened in (12 ). Using the ab o v e notation, the asyn hronous IWF A is desrib ed in Algorithm 1 (where N it denotes the n um b er of iterations). Algorithm 1: Asyn hronous Iterativ e W aterlling Algorithm Set n = 0 and p (0) q = an y feasible p o w er allo ation; for n = 0 : N it , p ( n +1) q =      WF q  p ( τ q 1 ( n )) 1 , . . . , p ( τ q q − 1 ( n )) q − 1 , p ( τ q q +1 ( n )) q +1 , . . . , p ( τ q Q ( n )) Q  , if n ∈ T q , p ( n ) q , otherwise ; ∀ q ∈ Ω (14) end The on v ergene of the algorithm is guaran teed under the follo wing suien t onditions. Theorem 1 Assume that the fol lowing  ondition is satise d: ρ ( S max ) < 1 , (C1) wher e S max is dene d in ( 13 ) and ρ ( S max ) denotes the sp e tr al r adius 8 of S max . Then, as N it → ∞ , 8 The sp etral radius ρ ( S ) of the matrix S is dened as ρ ( S ) = max {| λ | : λ ∈ eig ( S ) } , with eig ( S ) denoting the set of eigen v alues of S [27 ℄. 8 the asynhr onous IWF A desrib e d in A lgorithm 1  onver ges to the unique NE of game G , for any set of fe asible initial  onditions and up dating she dule. Pro of. The pro of onsists in sho wing that, under (C1), onditions of the Asyn hronous Con v ergene Theorem in [18℄ are satised. A k ey p oin t in the pro of is giv en b y the follo wing prop ert y of the m ultiuser w aterlling mapping WF ( p ) = ( WF q ( p − q )) q ∈ Ω based on the in terpretation of the w aterlling solution (10) as a prop er pro jetor [11℄:    WF ( p (1) ) − WF ( p (2) )    ≤ β    p (1) − p (2)    , ∀ p (1) , p (2) ∈ P , (15) where k·k is a prop er v etor norm and β is a p ositiv e onstan t, whi h is less than 1 if ondition ( C1 ) is satised. See App endix A for the details. T o giv e additional insigh t in to the ph ysial in terpretation of the on v ergene onditions of Algorithm 1, w e pro vide the follo wing orollary of Theorem 1. Corollary 1 A suient  ondition for ( C1) in The or em 1 is given by one of the two fol lowing set of  onditions: 1 w q X r 6 = q max k ∈D r ∩D q  | ¯ H q r ( k ) | 2 | ¯ H q q ( k ) | 2  d γ q q d γ q r P r P q w r < 1 , ∀ q ∈ Ω , (C2) 1 w r X q 6 = r max k ∈D r ∩D q  | ¯ H q r ( k ) | 2 | ¯ H q q ( k ) | 2  d γ q q d γ q r P r P q w q < 1 , ∀ r ∈ Ω , (C3) wher e w , [ w 1 , . . . , w Q ] T is any p ositive ve tor. 9 Note that, aording to the denition of D q in (13 ), one an alw a ys  ho ose D q = { 1 , . . . , N } in (C1 ) and (C2)-(C3 ). Ho w ev er, less stringen t onditions are obtained b y remo ving unneessary arriers, i.e., the arriers that, for the giv en p o w er budget and in terferene lev els, are nev er going to b e used. Reall that, if nite order onstellations are used, Theorem 1 is still v alid using the gap-appro ximation metho d [25 , 26℄ as p oin ted out in Setion 2.2 . It is suien t to replae ea h | H q q ( k ) | 2 in (C1) with | H q q ( k ) | 2 / Γ q . Remark 1 - Global on v ergene and robustness of the algorithm : Ev en though the rate maximization game in (7 ) and the onsequen t w aterlling mapping (10 ) are nonlinear, ondition (C1 ) guaran tees the glob al on v ergene of the asyn hronous IWF A. Observ e that Algorithm 1 on tains as sp eial ases a plethora of algorithms, ea h one obtained b y a p ossible  hoie of the s heduling of the users in the up dating pro edure (i.e., the parameters { τ q r ( n ) } and {T q } ). The imp ortan t result stated in Theorem 1 is that all the algorithms resulting as sp eial ases of the asyn hronous IWF A are guaran teed to rea h the unique NE of the game, under the same set of on v ergene onditions (pro vided 9 The optimal p ositiv e v etor w in (C2 )-(C3) is giv en b y the solution of a geometri programming, as sho wn in [11 , Corollary 5℄ 9 that A1 − A3 are satised), sine ondition (C1 ) do es not dep end on the partiular  hoie of {T q } and { τ q r ( n ) } . Remark 2 - Ph ysial in terpretation of on v ergene onditions: As exp eted, the on v ergene of the asyn hronous IWF A and the uniqueness of NE are ensured if the in terferers are suien tly far apart from the destinations. In fat, from (C2)-(C3 ) one infers that, for an y giv en set of  hannel realizations and p o w er onstrain ts, there exists a distane b ey ond whi h the on v ergene of the asyn hronous IWF A (and the uniqueness of NE) is guaran teed, orresp onding to the maxim um lev el of in terferene that ma y b e tolerated b y ea h reeiv er [as quan tied, e.g., in (C2 )℄ or that ma y b e generated b y ea h transmitter [as quan tied, e.g., in (C3 )℄. But the most in teresting result oming from (C1 ) and (C2 )-(C3 ) is that the on v ergene of the asyn hronous IWF A is robust against the w orst normalized  hannels | H q r ( k ) | 2 / | H q q ( k ) | 2 ; in fat, the sub  hannels orresp onding to the highest ratios | H q r ( k ) | 2 / | H q q ( k ) | 2 (and, in partiular, the sub  hannels where | H q q ( k ) | 2 is v anishing) do not neessarily aet the on v ergene of the algorithm, as their arrier indies ma y not b elong to the set D q . Remark 3 - Distributed nature of the algorithm: Sine the asyn hronous IWF A is based on the w aterlling solution (10), it an b e implemen ted in a distributed w a y , where ea h user, to maximize his o wn rate, only needs to measure the PSD of the o v erall in terferene-plus-noise and w aterll o v er it. More in terestingly , aording to the asyn hronous s heme, the users ma y up date their strategies using a p oten tially outdated v ersion of the in terferene. F urthermore, some users are ev en allo w ed to up date their p o w er allo ation more often than others, without aeting the on v ergene of the algorithm. These features strongly relax the onstrain ts on the syn hronization of the users' up dates with resp et to those imp osed, for example, b y the sim ultaneous or sequen tial up dating s hemes. W e an generalize the asyn hronous IWF A giv en in Algorithm 1 b y in tro duing a memory in the up dating pro ess, as giv en in Algorithm 2. W e all this new algorithm smo othe d asyn hronous IWF A. Algorithm 2: Smo othed Asyn hronous Iterativ e W aterlling Algorithm Set n = 0 and p (0) q = an y feasible p o w er allo ation and α q ∈ [0 , 1) , ∀ q ∈ Ω ; for n = 0 : N it , p ( n +1) q =    α q p ( n ) q + (1 − α q ) WF q  p ( τ q ( n )) − q  , if n ∈ T q , p ( n ) q , otherwise , ∀ q ∈ Ω; (16) end Ea h fator α q ∈ [0 , 1) in Algorithm 2 an b e in terpreted as a forgetting fator: The larger α q is, the longer the memory of the algorithm is. 10 In terestingly the  hoie of α q 's do es not aet the 10 In this pap er, w e are only onsidering onstan t  hannels. Nev ertheless, in a time-v arying senario, (16) ould b e used to smo oth the utuations due to the  hannel v ariations. In su h a ase, if the  hannel is xed or highly stationary , it is on v enien t to tak e α q lose to 1 ; on v ersely , if the  hannel is rapidly v arying, it is b etter to tak e a small α q . 10 on v ergene apabilit y of the algorithm (although it ma y aet the sp eed of on v ergene [11 ℄), as pro v ed in the follo wing. Theorem 2 Assume that  ondition of The or em 1 is satise d. Then, as N it → ∞ , the smo othe d asynhr onous IWF A desrib e d in A lgorithm 2  onver ges to the unique NE of game G , for any set of fe asible initial  onditions, up dating she dule, and { α q } q ∈ Ω , with α q ∈ [0 , 1) , ∀ q ∈ Ω . Pro of. See App endix A. Remark 4 - Asyn hronous IWF A in the presene of in terarrier in terferene: The prop osed AIWF A an b e extended to the ase where the transmission b y the dieren t users on tains time and frequeny syn hronization osets. In [19 , 20 ℄ w e sho w ed that the Asyn hronous IWF A is robust against the in terarrier in terferene due to time and/or frequeny osets among the links and w e deriv ed suien t onditions guaran teeing its on v ergene in the presene of su h time/frequeny misalignmen ts. 4 T w o Sp eial Cases: Sequen tial and Sim ultaneous IWF As In this setion, w e sp eialize our asyn hronous IWF A to t w o sp eial ases: the se quential and the simultane ous IWF As. As a b y-pro dut of the prop osed unied framew ork, w e sho w that b oth algorithms on v erge to the NE under the same suien t onditions, that are larger than onditions obtained for the on v ergene of the sequen tial IWF A in [6, 7℄, [13℄, [15℄ (without onsidering the sp etral mask onstrain ts) and [14 ℄ (inluding the sp etral mask onstrain ts). Sequen tial Iterativ e W aterlling: The sequen tial IWF A is an instane of the Gauss-Seidel s heme b y whi h ea h user is sequen tially up dated [18 ℄ based on the w aterlling mapping (10 ). In fat, in sequen tial IWF A ea h pla y er, sequen tially and aording to a xed order, maximizes his o wn rate (3), p erforming the single-user w aterlling solution in (10 ), giv en the others as in terferene. This s heme an also b e seen as a partiular ase of the general asyn hronous IWF A with the follo wing parameters: T q = { kQ + q , k ∈ N + } = { q , Q + q , 2 Q + q , . . . } and τ q r ( n ) = n, ∀ r , q . Using this settings in Algorithm 1, the sequen tial IWF A an b e written in ompat form as in Algorithm 3. Algorithm 3: Sequen tial Iterativ e W aterlling Algorithm Set n = 0 and p (0) q = an y feasible p o w er allo ation; for n = 0 : N it , p ( n +1) q =    WF q  p ( n ) − q  , if ( n + 1) mo d Q = q , p ( n ) q , otherwise , ∀ q ∈ Ω; (17) end 11 Sim ultaneous Iterativ e W aterlling: The sim ultaneous IWF A an b e in terpreted as the syn-  hronous Jaobi instane of the asyn hronous IWF A. In fat, in the sim ultaneous IWF A, all the users up date their o wn o v ariane matrix simultane ously at ea h iteration, p erforming the single user w ater- lling solution (10), giv en the in terferene generated b y the other users in the pr evious iteration. This is a partiular ase of the asyn hronous IWF A in Algorithm 1 with the follo wing parameters: T q = N + , and τ q r ( n ) = n, ∀ r, q , whi h leads to Algorithm 4. Algorithm 4: Sim ultaneous Iterativ e W aterlling Algorithm Set n = 0 and p (0) q = an y feasible p o w er allo ation; for n = 0 : N it , p ( n +1) q = WF q  p ( n ) − q  , ∀ q ∈ Ω; (18) end By diret pro dut of our unied framew ork, in v oking Theorem 1 w e obtain the follo wing unied set of on v ergene onditions for b oth sequen tial and sim ultaneous IWF As [11 ℄. Theorem 3 Assume that  ondition (C1 ) of The or em 1 is satise d. Then, as N it → ∞ , the se quential and simultane ous IWF As, desrib e d in A lgorithm 3 and 4, r esp e tively,  onver ge ge ometri al ly to the unique NE of game G for any set of fe asible initial  onditions and up dating she dule. Remark 5 - Algorithm robustness: It follo ws form Theorem 3 that sligh t v ariations of the sequen tial or sim ultaneous IWF As that fall within the unied framew ork of the asyn hronous IWF A, are still guaran teed to on v erge, under the ondition in Theorem 1. This means that, using for example Algorithm 3 , if the order in the users' up dates  hanges during time, or some user skips some up date, or he uses an outdated v ersion of the in terferene PSD, this do es not aet the on v ergene of the algorithm. What is aeted is only the on v ergene time. Moreo v er, as for the smo othed asyn hronous IWF A, also for the sequen tial and sim ultaneous IWF As desrib ed in Algorithm 3 and 4, resp etiv ely , one an in tro due a memory in the up dating pro ess [11 ℄, still guaran teeing on v ergene under onditions of Theorem 3. Remark 6 - Comparison with previous on v ergene onditions: Algorithm 3 generalizes the w ell-kno wn sequen tial iterativ e w aterlling algorithm prop osed b y Y u et al. in [6℄ to the ase where the sp etral mask onstrain ts are expliitly tak en in to aoun t. In fat, the algorithm in [6℄ an b e obtained as a sp eial ase of Algorithm 3 , b y remo ving the sp etral mask onstrain ts in ea h set P q in (6 ), (i.e. setting p max q ( k ) = + ∞ , ∀ k , q ), so that the w aterlling op erator in ( 10) b eomes the lassial w aterlling solution [21 ℄, i.e., WF q ( p − q ) = ( µ q 1 N − insr q ) + , where ( x ) + = max (0 , x ) and insr q , [ insr q (1) , . . . , insr q ( N )] T , with insr q ( k ) = ( σ 2 w q ( k ) + P r 6 = q | H q r ( k ) | 2 p r ( k )) / | H q q ( k ) | 2 . The on v ergene of the sequen tial IWF A has b een studied in a n um b er of w orks, either in the absene 12 [6, 7, 8 , 13 , 15℄ or in the presene [ 11 , 14℄ of sp etral mask onstrain ts. In terestingly , onditions in [6, 7, 8, 13, 15℄ and [14 ℄ imply our ondition (C1 ), whi h is more relaxed as sho wn next. Let Υ ,  I − S max lo w  − 1 S max upp , (19) with S max lo w and S max upp denoting the stritly lo w er and stritly upp er triangular part of the matrix S max , resp etiv ely , and S max is dened similar to S max in (13 ), but taking the maxim um o v er the whole set { 1 , . . . , N } . The relationship b et w een (suien t) onditions for the on v ergene of sequen tial IWF A as deriv ed in [6 , 7, 8 , 13, 14, 15 ℄ and ondition (C1 ) 11 is giv en in the follo wing orollary of Theorem 3. Corollary 2 Suient  onditions for (C1 ) in The or em 1 ar e [6℄ − [8 ℄ 12 max k ∈{ 1 ,...,N }  | ¯ H r q ( k ) | 2 | ¯ H q q ( k ) | 2  d α q q d α r q P r P q < 1 Q − 1 , ∀ r , q ∈ Ω , q 6 = r, (C4) or [13℄ max k ∈{ 1 ,...,N }  | ¯ H r q ( k ) | 2 | ¯ H q q ( k ) | 2  d α q q d α r q P r P q < 1 2 Q − 3 , ∀ r , q ∈ Ω , q 6 = r , (C5) or [14℄ ρ ( Υ ) < 1 , (C6) wher e Υ is dene d in (19). Pro of. See App endix B. Sine the on v ergene onditions in Corollary 2 dep end on the  hannel realizations  ¯ H q r ( k )  and on the distanes { d q r } , there is a nonzero probabilit y that they are not satised for a giv en  hannel realization, dra wn from a giv en probabilit y spae. T o ompare the range of v alidit y of our onditions vs. the onditions a v ailable in the literature, w e tested them o v er a set of  hannel impulse resp onses generated as v etors omp osed of L = 6 i.i.d. omplex Gaussian random v ariables with zero mean and unit v ariane (m ultipath Ra yleigh fading mo del). Ea h user transmits o v er a set of N = 16 sub arriers. W e onsider a m ultiell ellular net w ork as depited in Figure 1(a), omp osed b y 7 (regular) hexagonal ells, sharing the same band. Hene, the transmissions from dieren t ells t ypially in terfere with ea h other. F or the simpliit y of represen tation, w e assume that in ea h ell there is only one ativ e link, orresp onding to the transmission from the base station (BS) to a mobile terminal (MT) plaed at a orner of the ell. A ording to this geometry , ea h MT reeiv es a useful signal that is omparable, in a v erage sense, with the in terferene signal transmitted b y the BSs of t w o adjaen t ells. The o v erall net w ork is mo deled as a set of sev en wideband in terferene  hannels. In Figure 1(b) , w e plot the 11 Reall that ondition (C1) guaran tees also the on v ergene of the more general asyn hronous IWF A, as stated in Theorem 1. 12 In [6 ℄, the authors deriv ed onditions ( C4 ) for a game omp osed b y Q = 2 users and in the absene of sp etral mask onstrain ts. 13 2 BS 4 BS 3 BS 5 BS 6 BS 7 BS 1 BS 2 MT x 1 MT 3 MT 4 MT 5 MT 6 MT 7 MT r (a) Multiell ellular system 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Convergence Probability (C1) (C6) (C4) r (b) Probabilit y of (C1), (C4 ) and (C6) v ersus r . Figure 1: Probabilit y of (C1), (C4) and (C6) v ersus r [subplot (b)℄ for a 7 ell (do wnlink) ellular system [subplot (a)℄; Q = 7 , N = 16 , γ = 2 . 5 , P q = P r , Γ q = 1 , P q /σ 2 q = 7 dB, ∀ r , q ∈ Ω , and w = 1 . probabilit y that onditions (C1 ), (C4 ) and (C6) are satised v ersus the (normalized) distane r ∈ [0 , 1) (see Figure 1(a) ), b et w een ea h MT and his BS (assumed to b e equal for all the MT/BS pairs). W e tested our onditions onsidering the set D q , obtained using the algorithm desrib ed in [10 ℄. As exp eted, the probabilit y of guaran teeing on v ergene inreases as ea h MT approa hes his BS (i.e., r → 1 ). What is w orth while notiing is that our ondition ( C1 ) signian tly enlarges (C4 ) and (C6), sine the probabilit y that (C1) is satised is alw a ys m u h larger than ( C4 ) and (C6 ). Remark 7 - Sequen tial v ersus sim ultaneous IWF A: Sine the sim ultaneous IWF A in Algorithm 4 is still based on the w aterlling solution (10), it k eeps the most app ealing features of the sequen- tial IWF A, namely its lo w-omplexit y and distributed nature. In addition, it allo ws all the users to up date their optimal p o w er allo ation simultane ously . Hene, the sim ultaneous IWF A is faster than the sequen tial IWF A, esp eially if the n um b er of ativ e users in the net w ork is large. A quan titativ e omparison b et w een the sequen tial and sim ultaneous IWF As, in terms of on v ergene sp eed, is giv en in [11 ℄. In [19 ℄, w e also pro vided a losed form expression of the error estimates as a funtion of the iteration index, obtained b y the sequen tial and sim ultaneous IWF As. 5 Conlusion In this pap er, w e ha v e studied the maximization of the information rates for the Gaussian frequeny- seletiv e in terferene  hannel, giv en onstrain ts on the transmit p o w er and the sp etral masks on ea h link. W e ha v e form ulated the optimization problem as a strategi nono op erativ e game and w e ha v e prop osed a no v el, totally asyn hronous iterativ e distributed algorithm, named asyn hronous IWF A, to rea h the Nash equilibria of the game. This algorithm on tains as sp eial ases the w ell-kno wn sequen tial IWF A and the reen tly prop osed sim ultaneous IWF A, where the users up date their strategies 14 sequen tially and sim ultaneously , resp etiv ely . The main adv an tage of the prop osed algorithm is that no rigid s heduling in the up dates of the users is required: Users are allo w ed to  ho ose their o wn strategies whenev er they w an t and some users ma y ev en use an outdated v ersion of the measured m ultiuser in terferene. This relaxes the o ordination requiremen ts among the users signian tly . Finally , w e ha v e pro vided the onditions ensuring the global on v ergene of the asyn hronous IWF A to the unique NE of the game. In terestingly , our on v ergene onditions do not dep end on the sp ei up dating s heduling p erformed b y the users and, hene, they represen t a unied set of on v ergene onditions for all the algorithms that an b e seen as sp eial ases of the asyn hronous IWF A. App endix A Pro of of Theorems 1 and 2 W e start with some denitions and in termediate results that will b e instrumen tal to pro v e Theorems 1 and 2. Prop erties of the w aterlling mapping. F or te hnial reasons, w e dene the admissible set P e = P e 1 × · · · × P e Q ⊆ P , where P e q , { p q ∈ P q with p q ( k ) = 0 ∀ k / ∈ D q } , (20) is the subset of P q on taining all the feasible p o w er allo ations of user q , with zero p o w er o v er the arriers that user q w ould nev er use in an y of its w aterlling solutions (10 ), against an y admissible strategy of the others. Observ e that the game do es not  hange if w e use P e instead of the original P . F or an y giv en { α q } q ∈ Ω , with α q ∈ [0 , 1) , let T ( p ) = ( T q ( p )) q ∈ Ω : P e 7→ P e b e the mapping dened, for ea h q , as T q ( p ) , α q p q + (1 − α q ) WF q ( p − q ) , p ∈ P e , (21) where WF q ( p − q ) : P e − q 7→ P e q is the w aterlling op erator dened in (10). Observ e that all the Nash equilibria of game G orresp ond to the xed p oin ts in P e of the mapping T in (21 ). Hene, the existene of at least one xed p oin t for T is guaran teed b y Prop osition 1. Giv en T in (21 ) and some w , [ w q , . . . , w Q ] T > 0 , let k·k w 2 , blo  k denote the (v etor) blo  k-maxim um norm , dened as [18 ℄ k T ( p ) k w 2 , blo  k , max q ∈ Ω   T q ( p )   2 w q , (22) where k·k 2 is the Eulidean norm. Let k·k w ∞ , v e b e the ve tor w eigh ted maxim um norm , dened as [27℄ k x k w ∞ , v e , max q ∈ Ω | x q | w q , w > 0 , x ∈ R Q , (23) and let k·k w ∞ , mat denote the matrix norm indued b y k·k w ∞ , v e , dened as [27℄ k A k w ∞ , mat , max q 1 w q Q X r =1 [ A ] q r w r , A ∈ R Q × Q . (24) 15 Finally , w e in tro due the matrix S max α , dened as S max α , D α + ( I − D α ) S max , with D α , diag( α q . . . α Q ) , (25) where S max is dened in (13). The blo  k-on tration prop ert y of mapping T in (21) is giv en in the follo wing theorem that omes diretly from [11, Prop osition 2℄. Theorem 4 (Con tration prop ert y of mapping T ) Given w , [ w 1 , . . . , w Q ] T > 0 and { α q } q ∈ Ω , with α q ∈ [0 , 1) , the mapping T dene d in (21) satises    T ( p (1) ) − T ( p (2) )    w 2 , blo ck ≤ k S max α k w ∞ , mat    p (1) − p (2)    w 2 , blo ck , ∀ p (1) , p (2) ∈ P e , (26) wher e k·k w 2 , blo k , k·k w ∞ , mat and S max α ar e dene d in (22), (24) and (25), r esp e tively. If k S max α k w ∞ , mat < 1 , then mapping T is a blo k- ontr ation with mo dulus k S max α k w ∞ , mat . Asyn hronous on v ergene theorem [18 ℄. Let X 1 , X 2 , . . . , X Q b e giv en sets, and let X b e their Cartesian pro dut, i.e., X = X 1 × X 2 × . . . × X Q . (27) Let f q : X 7→ X q b e a giv en v etor funtion and let f : X 7→ X b e the mapping from X to X , dened as f ( x ) =  f 1 ( x ) , . . . , f Q ( x )  , and assumed to admit a xed p oin t x ⋆ = f ( x ⋆ ) . Consider the follo wing distributed asyn hronous iterativ e algorithm to rea h x ⋆ x ( n +1) q =      f q  x ( τ q 1 ( n )) 1 , . . . , x ( τ q Q ( n )) Q  , if n ∈ T q , x ( n ) q , otherwise, , ∀ q ∈ Ω; (28) with 0 ≤ τ q r ( n ) ≤ n and T q denoting the set of times n at whi h x ( n ) q is up dated and satisfying A1 − A3 of Setion 3. Assume that: C.1 ( Nesting Condition ) There exists a sequene of nonempt y sets {X ( n ) } with . . . ⊂ X ( n + 1) ⊂ X ( n ) ⊂ . . . ⊂ X , (29) satisfying the next t w o onditions. C.2 ( Synhr onous Conver gen e Condition ) f ( x ) ∈X ( n + 1) , ∀ n, and x ∈ X ( n ) . (30) F urthermore, if { y ( n ) } is a sequene su h that y ( n ) ∈X ( n ) , for ev ery n, then ev ery limit p oin t of { y ( n ) } is a xed p oin t of f ( · ) . C.3 ( Box Condition ) F or ev ery n there exist sets X q ( n ) ⊂ X q su h that X ( n ) = X 1 ( n ) × . . . × X Q ( n ) . (31) 16 Then w e ha v e the follo wing . Theorem 5 ([18, Prop osition 2.1℄) If the Synhr onous Conver gen e Condition (30) and the Box Condition (31) ar e satise d, and the starting p oint x (0) ,  x (0) 1 , . . . , x (0) Q  of the algorithm (28) b elongs to X (0) , then every limit p oint of { x ( n ) } given by ( 28) is a xe d p oint of f ( · ) . W e are no w ready to pro v e Theorems 1 and 2 through the follo wing t w o steps: Step 1. W e rst sho w that the asyn hronous IWF A in Algorithms 1 and 2 is an instane of the totally asyn hronous iterativ e algorithm in ( 28). Then, using Theorem 4, w e deriv e suien t onditions for C.1 - C.3 . Step 2. In v oking Theorem 5 , w e omplete the pro of sho wing that the asyn hronous IWF A on v erges to the unique NE of G from an y starting p oin t, pro vided that ondition (C1) is satised. Step 1. It is straigh tforw ard to see that the asyn hronous IWF A oinides with the algorithm giv en in (28), under the follo wing iden tiations x q ⇔ p q , x ⋆ q ⇔ p ⋆ q , X q ⇔ P e q , f q ( x ) ⇔ T q ( p ) , ∀ q ∈ Ω , (32) X ⇔ P e = P e 1 × . . . × P e Q , where P e q and T q ( p ) are dened in (20 ) and (21 ), resp etiv ely . Observ e that, to study the on v ergene of the asyn hronous IWF A, there is no loss of generalit y in onsidering the mapping T dened in P e ⊂ P instead of P , sine all the p oin ts pro dued b y the algorithm (exept p ossibly the initial p oin t, whi h do es not aet the on v ergene of the algorithm in the subsequen t iterations) as w ell as the Nash equilibria of the game are onned, b y denition, in P e . W e onsider no w onditions C.1 - C.3 separately . C.1 ( Neste d Condition ) Let p ⋆ =  p ⋆ 1 , . . . , p ⋆ Q  ∈ P e b e a xed p oin t of T in (21) (or, equiv alen tly of f q in (28)) and p (0) =  p (0) 1 , . . . , p (0) Q  ∈ P e b e an y starting p oin t of the asyn hronous IWF A. Using the blo  k-maxim um norm k·k w 2 , blo  k as dened in (22 ), where w = [ w 1 , . . . , w q ] T is an y p ositiv e v etor, w e dene the set X ( n ) in (29) as X ( n ) = n p ∈ P e : k p − p ⋆ k w 2 , blo  k ≤ β n k p (0) − p ⋆ k w 2 , blo  k o ⊂ P e , (33) with β = β ( w , S max α ) , k S max α k w ∞ , (34) and S max α dened in (25). It follo ws from (33 ) that if β n +1 k p (0) − p ⋆ k w 2 , blo  k < β n k p (0) − p ⋆ k w 2 , blo  k , ∀ n = 0 , 1 , ..., (35) then w e obtain the desired result, i.e., X ( n + 1) ⊂ X ( n ) ⊂ P e , ∀ n = 0 , 1 , .... 17 A neessary and suien t ondition for (35 ) is β < 1 . (36) W e will assume in the follo wing that ( 36 ) is satised. C.2 ( Synhr onous Conver gen e Condition ) Let p ( n ) ∈ X ( n ) . Then, from (33), it m ust b e that    p ( n ) − p ⋆    w 2 , blo  k ≤ β n k p (0) − p ⋆ k w 2 , blo  k . (37) Let p ( n +1) = T ( p ( n ) ) . Then, w e ha v e    p ( n +1) − p ⋆    w 2 , blo  k =    T ( p ( n ) ) − p ⋆    w 2 , blo  k ≤ β    p ( n ) − p ⋆    w 2 , blo  k ≤ β n +1 k p (0) − p ⋆ k w 2 , blo  k , (38) where the rst and the seond inequalities follo w from Theorem 4 (using (26 ) with p ⋆ = T ( p ⋆ ) ) and (37), resp etiv ely . Hene, p ( n +1) ∈ X ( n + 1) , as required in (30). F urthermore, sine lim n →∞    p ( n ) − p ⋆    w 2 , blo  k = 0 , with p ( n ) ∈ X ( n ) , ∀ n, the sequene { p ( n ) } generated from p (0) b y the mapping T using the sim ultaneous up dating s heme in (30) m ust on v erge to p ⋆ . Moreo v er, it follo ws from (36) and Theorem 4 that the xed p oin t p ⋆ of T is unique (implied from the fat that the mapping T is a blo  k-on tration [18, Prop osition 1.1℄). C.3 ( Box Condition ) F or ev ery n, the set X ( n ) in (33 ) an b e deomp osed as X ( n ) = X 1 ( n ) × . . . × X Q ( n ) , with X q ( n ) = ( p q ∈ P e q :   p q − p ⋆ q   2 w q ≤ β n k p (0) − p ⋆ k w 2 , blo  k ) ⊂ P e q , ∀ q ∈ Ω . (39) Step 2. Under (36), Theorem 5 is satised if the starting p oin t p (0) of the asyn hronous IWF A is su h that p (0) ∈ P e . The asyn hronous IWF A, as giv en in Algorithm 1 and Algorithm 2, is allo w ed to start from an y arbitrary p oin t p (0) in P . Ho w ev er, after the rst iteration from all the users, the asyn hronous IWF A pro vides a p oin t in P e , for an y p (0) ∈ P . Hene, under (36 ), the asyn hronous IWF A satises Theorem 5, after the rst iteration, whi h still guaran tees that ev ery limit p oin t of the sequene generated b y the asyn hronous IWF A is a NE of the game G . Sine ondition (36 ) is also suien t for the uniqueness of the NE [reall that, under (36), the mapping T in (21 ) is a blo  k-on tration℄, the asyn hronous IWF A m ust on v erge to this unique NE. T o omplete the pro of, w e just need to sho w that (C1 ) is equiv alen t to (36). Sine in (36 ) ea h α q ∈ [0 , 1) , w e ha v e β = k S max α k w ∞ < 1 ⇔ k S max k w ∞ < 1 . (40) Sine S max is a nonnegativ e matrix, there exists a p ositiv e v etor w su h that [18 , Corollary 6.1℄ k S max k w ∞ < 1 ⇔ ρ ( S max ) < 1 . (41) 18 Sine the on v ergene of the asyn hronous IWF A is guaran teed under (36), for an y giv en w > 0 , w e an  ho ose w = w and use ( 41 ). Conditions (C2)-(C3 ) in Corollary 1 an b e obtained as follo ws. Using [18, Prop osition 6.2e℄ ρ ( S max ) ≤ k S max k w ∞ , ∀ w > 0 , a suien t ondition for the ⇒ diretion in (41 ) is k S max k w ∞ < 1 , for some giv en w > 0 ; whi h pro vides ( C2). Condition (C3 ) is obtained similarly , still using ( 41 ) and ρ ( S max ) = ρ  S max T  . B Pro of of Corollary 2 Sine onditions (C2 )-(C3 ) imply (C1 ) (Corollary 1), the suieny of (C4) for (C1 ) follo ws diretly setting, in (C2), P q = P r for all q , r , D q = D r = { 1 , . . . , N } , w = 1 , and using the follo wing upp er b ound | ¯ H r q ( k ) | 2 / | ¯ H q q ( k ) | 2 ≤ max r 6 = q  | ¯ H r q ( k ) | 2 / | ¯ H q q ( k ) | 2  . Observ e that ondition (C5 ) is stronger than (C4), and th us implies ( C1 ). W e pro v e no w that ondition (C6) is stronger than (C1). T o this end, it is suien t to sho w that ρ ( Υ ) < 1 ⇔ ρ  S max  < 1 , where S max is dened after (19), sine S max ≤ S max leads to ρ ( S max ) ≤ ρ  S max  < 1 [28, Corollary 2.2.22℄. 13 W e rst in tro due the follo wing in termediate denition and result [28℄. Denition 2 A matrix A ∈ R n × n is said to b e a Z -matrix if its o-diagonal entries ar e al l non-p ositive. A matrix A ∈ R n × n is said to b e a P -matrix if al l its prinip al minors ar e p ositive. A Z -matrix that is also P is  al le d a K -matrix. Lemma 1 ([28, Lemma 5 . 3 . 14 ℄) L et A ∈ R n × n b e a K -matrix and B a nonne gative matrix. Then ρ ( A − 1 B ) < 1 if and only if A − B is a K -matrix. A ording to Denition 2, I − S max lo w is a Z -matrix. Sine all prinipal minors of I − S max lo w are equal to one (reall that I − S max lo w is a lo w er triangular matrix with all ones on the main diagonal), I − S max lo w is also a P -matrix, and th us a K -matrix. 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