Lagrange Multiplier Optimization in a Nutshell

Transcription

1 Lagrange Multiplier Optimization for Optimal Spectrum Balancing of DSL with Logarithmic Complexity Amir R. Forouzan and Marc Moonen Dept. of Electrical EngineeringESAT-SISTA), Katholiee Universiteit Leuven, Leuven, 3001, Belgium Abstract Lagrange dual optimizationldo) technique is a powerful tool for solving constrained optimization problems in and is generally considered to be optimal in the literature. LDO relaxes a constrained problem into an unconstrained dual problem using Lagrange multipliers. To solve the dual problem, the optimal value of the Lagrange multipliers should be found. The Lagrange multipliers are usually determined in an iterative process and reducing the number of iterations is of crucial importance to obtain systems with manageable computational complexity.inthispaper,weshowthatfortheldotobeoptimal in optimal spectrum balancing of DSL, the joint rate and power regionjrpr) should be strictly convex. Moreover, we propose a new LDO based algorithm with two advantages. Firstly, the computational complexity of the algorithm is logarithmic in the desired precision. Secondly, the algorithm can be used to find the optimal solution even when the JRPR is not strictly convex. Index Terms Convex optimization, digital subscriber line DSL), dual decomposition, dynamic spectrum management DSM), non-convex optimization, resource allocation. I. INTRODUCTION Lagrange dual optimizationldo) techniques have attracted a lot of attention for solving constrained optimization problems in various fields of communications[1] [6]. The most famous problem in this category is the optimal spectrum balancing OSB)ofDSL[1] 1.TheOSBproblemisstatedasfollows: maximize R 1 subjectto R n R n) min foralln;2 n N, and 1b) P n P max n) foralln;1 n N, 1a) 1c) 1 Inthispaper,weconcentrateontheOSBproblem,however,ourresults can be generalized to a wide range of separable optimization problems in MIMO OFDM systems [2], communications in fading channels with quantized states[3], cognitive radios[4], joint routing and resource allocation [5], power allocation in the vector broadcast channels[6], etc. This research wor was carried out at the ESAT Laboratory of Katholiee Universiteit Leuven, in the frame of K.U.Leuven Research Council CoE EF/05/006 Optimization in EngineeringOPTEC), Concerted Research Action GOA-MaNet, The Belgian Programme on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy Office IUAP P6/04 Dynamical systems, control and optimization DYSCO) , Research Project IBBT, Research Project FWO nr.g Design and evaluation of DSL systems with common mode signal exploitation, and IWT Project PHysical layer and Access Node TEchnology Revolutions: enabling the next generation broadband networ PHANTER). The scientific responsibility is assumed by its authors. wherer n andp n arethebit-rateandaggregatetransmitpower of user n, R n) min and Pn) max are the minimum required bitrate and the maximum aggregate transmit power for user n, and N isthetotalnumberofusers.therateregionrr) isdefinedasthesetofallachievable N-dimensionalN-D) vectors R 1,...,R N ) T whichsatisfy1c).tosolve1)using LDO, we maximize the following Lagrangian[1]: L N N w n R n λ n P n 2) where w n 0and λ n 0arecalledtheweightfactorand Lagrangemultiplierforuser n.thesolutionisfoundbyan iterativeprocessinwhich w n and λ n areupdateduntilall oftheconstraintsin1b)and1c)aresatisfied.thenumber of iterations plays an important role in the computational complexity of the technique and several techniques have been proposed to reduce the number of iterations including the bisection search[1], sub-gradient ascend[2], and step-adaptive sub-gradient ascent[7]. The number of iterations required by these algorithms is at least quadratically proportional to the inverse of the desired precision. In [8], an improved dual decomposition approach has been proposed for which the number of required iterations is proportional to the inverse of the desired precision. This means that the average number ofiterationstoachieveaprecisionof1%isafactorof)100. TheoptimalityoftheLDOforsolvingOSBhasbeenshown in[2],[9],[10]basicallybyarguingthatthedualitygapis zero when the number of tones is large. Despite these results, finding a set of weight factors and Lagrange multipliers to solve1) precisely is usually a tedious job and sometimes impossibleinpractice[8].infact,in[11]weshowedthat insomecasestheldofailstofindallpointsontherr when there are no power constraints. This problem occurs whentherrisnotstrictlyconvex.insomeextremecases,the OSB solution provides significantly smaller bit-rates for some oftheusersthanthatcanbeobtainedintheoryorevenby suboptimal techniques such as static spectrum management. To resolve this problem, iterative facet dividingifda) algorithm hasbeenproposedin[11].ithasbeenshownthattheifdais capable of finding any point on the RR very closely. Moreover, the number of iterations is proportional to the logarithm of the inverse of the desired precision, meaning that to achieve aprecisionof1%weonlyneedafactorof) log iterations /11/$ IEEE

2 Theresultsin[11]arelimitedtothecaseswherethereareno power constraints. Bac to the general case, these results bring twoeyissuestoattention.thefirstissueiswhetherthere exist cases for which LDO is incapable of approaching points ontheboundaryoftherr.statedinotherwords,arethere anyscenariosinwhichtherrisnotstrictlyconvexinthe general case? The second issue is whether we can generalize IFDA to find Lagrange multipliers jointly with weight factors inordertotaeadvantageoftherobustnessandspeedofthe technique. In this paper, we address these issues. This paper is organized as follows. The multiuser DSL transmissionsystemisdescribedinsec.ii.insec.iii,we extendtheifdatothegeneralcasewherethereexistssome power constraints. Simulation results are presented in Sec. IV, andfinallythepaperisconcludedinsec.v. II. SYSTEM DESCRIPTION We assume N discrete multi-tonedmt) DSL users with disjoint upstreamus) and downstreamds) bands transmitting over Ktones.Inordertoobtain KparallelMIMOchannels, weassumethatallusersaredmtsymbolsynchronizedatthe receiverside.thebit-rateofuser nperdmtsymbolis R n = K =1 b n) 3) where b n) isthenumberofbitsloadedtotone foruser n obtained by ) } b n) = min b max, log Γ SNRn), 4) where b max isthemaximumnumberofbitsthatcanbeloaded toeachtone, denotesthefloorfunction, Γisthesignal-tonoiseratioSNR)gap,and SNR n) isthesnrattone of user n.thesnrisobtainedby SNR n) = s n) gn,n) σ n) + N m=1;m n sm) g n,m), 5) where s n) isthetransmitpsdofuser nattone, σ n) is the PSDofthe n-threceiver snoiseattone,and g n,m) is the channel s power gain from transmitter m to receiver n at tone.indslsystems,thetransmitpsdofeachuserisbounded byaregulatorypsdmas,i.e., s n) s n),mas.theaggregate transmitpowerofuser nis f P n where f isthedmttone spacing and P n K =1 s n). 6) ThetotaltransmitpowerthatcanbesentunderthePSDmas is f P n) mas,where P n) mas K =1 s n),mas. 7) By substituting3) and6) into2), the problem is decoupled into K parallel per-tone maximization problems, i.e., maximize b,s } L ;1 K, 8) where L N w nb n) N λ ns n) is the per-tone Lagrangianontone.Theper-toneproblemisnotconvexin the general case and is solved by exhaustive search. However, nearly optimal algorithms exist for solving the per-tone problemwithpolynomialcomplexityin N.From5),theelementwiseminimumpowervector s = s 1),...,sN) ) Ttoload ) Tontone thebit-loadingvector b = b 1),...,bN) is obtained by[1],[11] s = D ΓB C ) 1 ΓB σ, 9) } where D = diag,...,g N,N), B = diag 2 b } g 1,1) I N, C = G D [G ] n,m = g n,m) ), and ) T.Thebit-andpower-loadingvector σ = σ 1),...,σN) Tiscalledachievableif b T,s) T b and s satisfy9)and 0 b n) b max and 0 s n) s n),masfor 1 n N. III. GENERALIZED IFDA The IFDA as proposed in[11] is implemented using geometricaloperationsontherr.toextendifdatothegeneralcase where the objective and constraint functions are defined on the bit rates as well as transmit powers, we consider the concept ofjointrateandpowerregionjrpr).thejrprontone, Φ,isthesetofallachievablebit-andpower-loadingvectors b T,s T ) T.TheJRPRoveralltonesis Φ = Φ1 Φ K where denotes the Minowsi sum. The Minowsi sum of twosets Aand Bisdefinedby A B a+b a A b B}. 10) If we loo at R n and P n as independent variables and L as a constant, the Lagrangian L = N w nr n N λ np n can be interpreted as ahyperplane 1 Hinthe 2N-Dspacewithanormalvector ν w 1,w 2,...,w N, λ 1, λ 2,..., λ N ) T.Ifweset L = N L max where L max max b,s ; } w nr n N λ np n, then H indicates a supporting hyperplane to Φ. Now ) consider the line l : t,r 2) min,...,rn) min,p1) max,...,p max N), where t is an independent variable. Assuming that the problem is feasibleandallconstraintsin1b)and1c)arebinding 2,the solutionislocatedon l.moreover,whentheboundaryofthe convexhullof Φisin Φ,thesolutionisontheboundary of Φ. Therefore, the solution is located at the intersection of landtheboundaryof Φandasupportinghyperplaneto Φ 1 Weuseafew N-Dgeometricalobjectsinthispapersuchashyperplane, line, simplex, and facet. Please refer to[12] for their definitions. 2 A constraint is called binding if it is met at equality in the solution, otherwise, it is unbinding or slac.

3 The algorithm is terminated when the required precision isachievedorwhenthelastcalculatedpointislocatedon thelastfacet.thesecondcasecouldonlyhappenwhenthe JRPR is not strictly convex. In this case, traditional LDO based algorithms would fail to find the solution. However, insec.iii-c,weproposeamixingalgorithmwhichfindsthe solution using the bit- and power-loadings associated with the vertices of the last facet. Fig.1. AnillustrationofoneiterationoftheIFDAontheJRPRfora single-user scenario. canbefoundatthesolution.figure1showsthejrprthe shadedarea)and lthedottedline)forasingleusercase 1. Inthefollowingweusethisfiguretoexplainouralgorithm. Considertwopoints Aand BontheboundaryoftheJRPR suchthat lintersects ABthelinesegmentbetween Aand B) at p sol.notethat p sol islocatedontheboundaryof Φonthe arcbetweenthepoints Aand B.Assume ν AB = ν 1,ν 2 ) T is thenormalvectorto ABwhere ν 1 > 0.Considerline AB, thesupportinghyperplaneto Φparallelto AB,andlet C = R C,P C ) T denotethepointatwhich AB supports Φ.Since AB ischaracterizedby ν 1 R 1 + ν 2 P 1 = ν 1 R C + ν 2 P C, C canbereachedbymaximizingthelagrangian L = w 1 R 1 λ 1 P 1 with w 1 = ν 1 and λ 1 = ν 2.Nowconsidertheline segments AC and BC. Since l intersects AB, it intersects either ACor BC.InFig.1, lhasintersected AC.Asitcan beseen,thesolutionpoint p sol isalsobetween Aand Conthe boundaryof Φ.Asitcanberealized,startingfrom AB,we havefoundashorterlinesegment ACintersectedby lwith its ends located on the boundary of Φ. Therefore, by applying this procedure iteratively and assuming that the convex hull ofjrprisstrictlyconvex,wewouldfindtwopointsineach iteration which eventually converge together at the solution. Thealgorithmcanbereadilygeneralizedto N > 1.For N > 1,thepoints Aand Barereplacedby 2Nstartingpoints p 1 to p 2N.Thestartingpointsindicateafacet Fin 2Nspace whichbyassumptionisintersectedby l 2.Tooperateeach iteration,thenormalvector ν = ν 1,...,ν 2N ) T with ν 1 > 0) to F is calculated and the Lagrangian is maximized by setting w 1 = ν 1,..., w N = ν N, λ 1 = ν N+1,..., λ N = ν 2N toobtainanewpoint q = R 1,...,R N,P 1,...,P N ) T onthe boundaryofφ.pointqformsafacetwithany2n 1selection ofpointsfrom p 1 to p 2N.Let F n denotethefacetindicatedby thepoints q, p 1,..., p n 1, p n+1,..., p 2N.Since lintersects F,itintersectsoneofthefacets F 1 to F 2N aswell.let F m denotetheintersectedfacet.thentheverticesof F m,i.e., q, p 1,..., p m 1, p m+1,..., p 2N,arethestartingpointsforthe next iteration of the algorithm. 1 TheJRPRcannotbeillustratedfor N > 1,asithasatleast4dimensions in that case. 2 WewillproposeanalgorithmforfindingthestartingpointsinSec.III-B. A. Unbinding Power Constraints and Negative Lagrange Multipliers Usually, the weight factors and Lagrange multipliers associated with inequality constraints are considered to be nonnegative. The IFDA sets the weight factors and Lagrange multipliers according to the elements of the normal vector to a facet which may not necessarily satisfy this condition. More explicitly,as ν 1 = w 1 isenforcedtobepositive, ν 2 = w 2 to ν N + w N should be non-negative, and ν N+1 = λ 1 to ν 2N = λ N should be non-positive for all iterations. Our simulation results show that sometimes oneor more) elements)in ν N+1 to ν 2N taepositivevaluesleadingtoone or more) negative Lagrange multipliers). This happens when apowerconstraintin1c)isnotbindingortheoptimalpoint isclosetoaregionwhereapowerconstraintisnotbinding). Toresolvethisissue, λ n arelettotaenegativevaluesby modifyingthelagrangian Landper-toneLagrangians L as follows: N N L w n R n λ n Pn, 11) L N w n b n) N λ n s n), 12) Pn ; λ n 0 where Pn P max; n) and s n λ n < 0 sn ; λ n 0 s n) max; λ n < 0. When λ n is positive, L and L reduce to L and L. However, when λ n is negative, the power of user n is virtually assumed to be equal to the maximum value in the range 3. However, the actual power sent by user n is independently set during the exhaustive search for finding the optimal point and can be smaller. This avoids bit rate loss for other users by experiencing less crosstalinducedby s n).notethatthepoint q calculated in each iteration of the IFDA is redefined as well by q = R 1,...,R N, P 1,..., P ) T. N From the geometrical point of view this modification is equal to adding all 2 N 1 projections of Φ on the N hyperplanes P n = P n) masandtheirintersectionsto Φ.Bythis modification l intersects Φ even if the power constraints are not binding. 3 Inthegeneralcase,wemightnothaveaPSDmaslimitation.However, duetopracticallimitations,thetransmitpoweroneachtoneislimitedtoa maximumvaluewhichcanbeusedinsteadof s n),mas in12).

4 B. The Expanding Algorithm for Finding the Starting Points The generalized IFDA discussed above requires 2N starting points p 1 to p 2N atthestart-up.hereweproposeanalgorithm which finds the starting points for the algorithm. If the problem is feasible then the point p 0,R 2) min,...,rn) min,p1) T max,...,p max) N) lislocatedonthe boundaryoftheconvexhullofthejrpr.thealgorithmwors bymainga2n-dsimplex S expan whichisiterativelyexpandedtoward p.since lpassesthrough p,itisguaranteed that lintersects S expan afterafewiterations.thealgorithm, called the expanding algorithm, is explained in more details in the following: 1) First 2N distinct points r 1 to r 2N are calculated by solving the problem at 2N different arbitrarily e.g. randomly)selectedvectorsof w 1,...,w N,λ 1,...,λ N ). Let F denotethefacetindicatedby r 1 to r 2N. 2) If lintersects F,then r 1 to r 2N canbeusedasthe starting points. Otherwise, we go to Step 3. 3) A new point is calculated by solving the problem at w 1,...,w N,λ 1,...,λ N ) T = ν,where ν isanormal vectorto F.Thesignof νshouldbeselectedsuchthat itsdirectionistowards p.let S expan denotethesimplex identifiedbythevertices p and r 1 to r 2N. 4) If l intersects S expan then it intersects two facets of S expan. The vertices of either of these facets can be usedasthestartingpoints p 1 to p 2N,however,forfaster convergence, the facet intersected by l at a greater value of t= R 1 )ispreferable. 5) If ldoesnotintersect S expan,thenwecanfindatleast onefacetof S expan,namely F 1,forwhich p islocated ononesideofitandtheremainingverticesof S expan arelocatedontheothersideofit.nowwereplace r 1 to r 2N bytheverticesof F 1 and F by F 1 andreturnto Step 3. C. The Mixing Algorithm When the solution is located on a nonstrictly-convex region of the JRPR boundary, it is usually not achievable by a particular set of weight factors and Lagrange multipliers. Fortunately, the generalized IFDA provides us with 2N points surrounding the solution. By using the per-tone bit- and power-loading vectors associated with these points in a mixed fashion, we can reach thesolutionverycloselywhenthenumberoftonesislarge. Assumethefinalpoints p 1 to p 2N indicatefacet F final on theboundaryofthejrprandlet b,m = b,m,...,b,m ) T and s,m = s,m,...,s,m ) T denote the bit- and powerloading vectors associated with p m on tone. Let λ n,m denote the n-th Lagrange multiplier associated with the m- s 1),m,..., sn),m thpoint.wedefine s,m = s n),m, λ n,m 0 s n),mas, λ n,m < 0.Notethat p m = ) Twhere s n),m = K =1 [b,m] 1, K =1 [ s,m] N ) T,..., K =1 [b,m] N, K =1 [ s,m] 1,..., where [x] n denotesthe n-thelementofvector x.let p sol = Algorithm 1: The Mixing Algorithm /* STEP 1 */ Set p mix p 1 and µ 1for1 K; /* STEP 2 */ repeat for =1...Kdo ) T; p mix p mix b T,µ, s T,µ µ index min m dist p mix p mix + p sol,p mix + ) T; b T,µ, s T,µ ) } T b T,m, s T,m ; until no improvements can be made; /* STEP 3 */ repeat Pictworandomtoneindices 1, 2 ; 1 2 ; p mix p mix ) T ) T; b T 1,µ1, s T 1,µ1 b T 2,µ 2, s T 2,µ 2 µ 1,µ 2 ) indexmin dist p sol,p mix m 1,m 2) ) T ) } T + b T 1,m1, s T 1,m1 + b T 2,m2, s T 2,m2 ; p mix p mix ) T ) T. + b T 1,µ1, s T 1,µ1 + b T 2,µ 2, s T 2,µ 2 until the desired precision achieved or the maximum number of iterations is reached; Tdenotetheintersection R 1) min,...,rn) min,p1) max,...,p max) N) pointof land F final.let µ [1 : 2N]indicatethepoint whosebit-andpower-loadingswillbeusedontone.the sequence µ 1 to µ K indicateapointon F final calculatedby K p mix = =1 [b,µ ],..., K =1 [b,µ ] N, K =1 [ s,µ ] 1,..., K =1 [ s,µ ] N ) T.Thegoalistooptimize µ1 to µ K such that p mix islocatedcloseto p sol withintherequiredprecision. Algorithm1canbeusedforthispurpose. Thealgorithmworsasfollows:Inthefirststepweassume that p mix isequalto p 1 andwesetallindices µ equalto 1. Inthesecondstep,werepeatedlytestalltones andfindthe point µ whichminimizesthenormalizedeuclideandistance of p mix and p sol ifitsassociatedbit-andpower-loadingsis selectedontone.thenormalizedeuclideandistanceof p mix and p sol iscalculatedbynormalizingthefirst Ndimensions by R 1) minto RN) minandthesecond Ndimensionsby P1) maxto P max.finally,inthethirdstep,wepictworandomtones N) 1 and 2 andfindthepoints µ 1 and µ 2 minimizingthe distanceof p mix to p sol iftheirbit-andpower-loadingsare selectedon 1 and 2,respectively.Thisprocedureisrepeated until we reach the desired precision or the maximum number of iterations.

5 TABLE I SIMULATION PARAMETERS PARAMETER VALUE Crosstal model ANSI standard 1% worst-case model[13] without considering FSAN power sum rule Bandplan and PSD mas VDSL 997 DS:.138-3, MHz, US: 3-5.1, MHz)[14] VDSL2E17 B7-9 DS:.138-3, , MHz, US: 3-5.1, , MHz)[15] Cable type 26 AWG[13] Noise White noise,-140 dbm/hz Tonespacing, f Hz Symbolrate, f s 4Hz b max 15 b min 2 SNRgap, Γ 12.0dB Per tone PSD 3dB granularity IV. SIMULATION RESULTS For the first simulation, we consider a two-user US VDSL scenario in which two equal length loops are transmitting overa250m26-awgcable.weset P max n) =.2 P n) mas = mWandR 2) min =52Mbps.Othersimulationparameters arelistedintablei. The expanding algorithm converges after two iterations, and then the generalized IFDA reaches the optimal point in 32 iterationswithprecision Theverticesofthefinalfacet are p 1 = 34.91Mbps,69.01Mbps,.79mW,1.61mW) T, p 2 = 81.22Mbps,22.71Mbps,2.27mW,.14mW) T, p 3 = 22.71Mbps,81.22Mbps,.14mW,2.27mW) T,and p 4 = 58.44Mbps,45.49Mbps,1.20mW,1.21mW) T. The corresponding vectors of weight factors and Lagrange multipliersω m w 1,m,...,w N,m,λ 1,m,...,λ N,m ) T )are ω 1 = , , , ) T, ω 2 = , , , ) T, ω 3 = , , , ) T,and ω 4 = , , , ) T. As it can be seen, the obtained points are located relatively far away from each other on the boundary of the JRPR. However, the vectors of weight factors and Lagrange multipliers are very close to each other. This clearly shows that theboundaryofthejrprisnotstrictlyconvexandisflatat leasttoaprecisionof 10 4.Asaresult,averysmallchange in the weight factors or Lagrange multipliers causes a large change in the obtained solution which explains why traditional algorithms for tuning the weight factors and Lagrange multipliers fail to find the solution. Line l intersects the facet indicatedbyp 1 top 4 atp sol = Mbps, Mbps, mW, mW) T. By mixing the bit- and powerloadingsof p 1 to p 4 usingtheproposedmixingalgorithm, weobtainp mix = Mbps, Mbps, mW, mW) T.Asitcanbeseen, p mix isconsiderablyclose to p sol. Figure 2 compares the convergence properties of the proposed algorithm with the step-adaptive subgradient method Fig. 2. Convergence of the proposed algorithm compared to the step-adaptive subgradient algorithm for a two-user US VDSL scenario. proposed in[7]. Four curves have been plotted in this figure. The solid curve shows thenormalized Euclidean) distance of the resulting point to the target point for the subgradient algorithm vs. the number of iterations. The original stepadaptive subgradient algorithm described in [7] wors by finding the Lagrange multipliers after fixing the weight factors astherrisunnownbeforesolvingtheproblem.thismeans that the weight factors have to be optimized separately leading to considerably higher number of iterations. Here, we now thelocationofthesolutionontherr,andwehaveused it to find the weight factors and the Lagrange multipliers simultaneously. The other three curves in Fig. 2 show the results for the generalized IFDA. The dotted curve shows the distance of the last obtained point in each iteration with thesolution.thegreencurveshowsthedistanceofthelast obtained point with the corresponding facet. Finally, the dashdotted curve shows the distance of the solution with the intersection point of l and the last facet. The subgradient algorithm converges in about 80 iterations. The minimum achieved distance to the solution for the algorithmis.2185.thedistanceofthepointsobtainedbythe generalizedifdato lisonthesameorderofmagnitude. However,asitcanbeseen,thedistanceoftheintersection point to the solution decreases with an almost constant slope in the logarithmic scale meaning that the required number of iterationsislogarithmicintheinverseofthedesiredprecision 1. These observations also imply that the solution is located on afacetontheboundaryofthejrpr.asaresult,thesolution is not achievable by tuning the weight factors and Lagrange multipliers.thatisbecauseonlyoneofthepointslocatedona facetofthejrprcanbereachedbysettingtheweightfactors and Lagrange multipliers according to the normal vector of 1 Wehaveanalyticallyprovedthattherequirednumberofiterationsforthe algorithm is logarithmic in the inverse of the desired precision which is not includedhereduetolacofspace.

6 thatfacet[11].ontheotherhand,thegeneralizedifdais capable of reaching the point as it mixes the bit-loadings and politeness values for 2N points located around the target point onthefinalfacet. Asforthesecondsimulation,weconsideracaseinwhich the power constraints are not binding for a user. This scenario is a near-far two-user US VDSL2 scenario with one user locatedat100mandtheotheruserlocatedat500mfromthe centraloffice.forthisscenario,weset P max n) =.2 P n) mas and R 2) min = 49 Mbps,where Pn) mas = mw.thedesired precision is 1%. By executing the algorithm, the vertices of thefinalfacetareobtainedas p 1 = Mbps,53.388Mbps,.36mW,13.28mW) T, p 2 = Mbps,46.074Mbps,.36mW,4.31mW) T, p 3 = Mbps,48.268Mbps,.36mW,8.26mW) T, and p 4 = Mbps,46.806Mbps,35.90mW,4.67mW) T. The corresponding vectors of weight factors and Lagrange multipliers are ω 1 = , , , ) T, ω 2 = , , , ) T, ω 3 = , , , ) T,and ω 4 = , , , ) T. Asitcanbeseen,thefirstLagrangemultiplierforthefourth pointi.e.,thethirdelementof ω 4 )isnegative,whichhas resultedinafullvirtualtransmitpowerforthefirstuserthe thirdelementof p 4 ).Usingthemixingalgorithm,weobtain p mix = 101.0Mbps,49.0Mbps,.36mW,7.18mW) T, whichindicatesthat R 2 = R 2) min, P 1 < P max,and 1) P 2 = P max. 2) As it can be seen, the algorithm has successfully identified the unbinding constraint the power constraint for the first user) and the binding constraints the bit rate and power constraints for the second user) and is capable of solving theproblemforthecasesinwhichsomeconstraintsarenot binding without any prior nowledge about the binding and unbinding constraints. REFERENCES [1] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen, Optimal multiuser spectrum balancing for digital subscriber lines, IEEETrans.Commun.,vol.54,no.5,pp ,May2006. [2] W. Yu and R. Lui, Dual methods for nonconvex spectrum optimization of multicarrier systems, IEEE Trans. Commun., vol. 54, no. 7, pp , Jul [3] M. Mohseni, R. Zhang, and J. Cioffi, Optimized transmission for fading multiple-access and broadcast channels with multiple antennas, IEEE J.Select.AreasCommun.,vol.24,no.8,pp ,Aug [4] A. Marques, X. Wang, and G. Giannais, Optimal stochastic dual resource allocation for cognitive radios based on quantized CSI, in IEEE Int l. Conf. on Acoustics, Speech,& Signal Processing, ICASSP 08, Las Vegas, NA, Mar. 2008, pp [5] L. Xiao, M. Johansson, and S. Boyd, Simultaneous routing and resource allocation via dual decomposition, IEEE Trans. Commun., vol. 52, no.7,pp ,Jul [6] W. Yu, Sum-capacity computation for the gaussian vector broadcast channel via dual decomposition, IEEE Trans. Inform. Theory, vol. 52, no.2,pp ,Feb [7] P. Tsiaflais, J. Vangorp, M. Moonen, and J. Verlinden, A low complexity optimal spectrum balancing algorithm for digital subscriber lines, Signal Process, vol. 87, no. 7, pp , Jul [8] P. Tsiaflais, I. Necoara, J. Suyens, and M. Moonen, Improved dual decomposition based optimization for DSL dynamic spectrum management, IEEE Trans. Signal Processing, vol. 58, no. 4, pp , Apr [9] Z.-Q. Luo and S. Zhang, Dynamic spectrum management: complexity and duality, IEEE J. Select. Topics Signal Processing, vol. 2, no. 1, pp , Feb [10], Duality gap estimation and polynomial time approximation for optimal spectrum management, IEEE Trans. Signal Processing, vol. 57, no. 7, pp , Jul [11] A. R. Forouzan, Optimal spectrum management of DSL with nonstrictly convex rate region, IEEE Trans. Signal Processing, vol. 57, no.7,pp ,Jul [12] E. W. Weisstein, Mathword, A Wolfram web resource. [Online]. Available: [13] Spectrum management for loop transmission systems, ANSI Standard T , Feb [14] ETSI, Transmission and MultiplexingTM); Access transmission systems on metallic access cables; Very High Speed Digital Subscriber LineVDSL); Part I: Functional Requirements, ETSI Std. TS , Rev. V.1.3.1, [15] ITU-T G.993.2; Amendment 1, Very high speed digital subscriber line transceivers 2VDSL2), Geneva, Switzerland, Apr V. CONCLUSION Dual optimization techniques are optimal for spectrum balancing of DSL, merely when the JRPR is strictly convex. WhentheJRPRisnotstrictlyconvex,wemaynotbeable tofindthedesiredsolutionbytuningtheweightfactorsand Lagrange multipliers. We proposed a new algorithm which findstheoptimalsolutionbymixingthesolutionofthedual problem for 2N points surrounding the target point. The algorithm consists of three parts: the expanding algorithm which finds the starting points, the generalized IFDA, and the mixing algorithm. The algorithm is capable of finding the optimal solution without any prior nowledge about binding and unbinding power constraints. Our simulation results show that the number of iterations required for the algorithm to converge is proportional to the logarithm of the inverse of the desired precision.