Parallel and Distributed Methods for Nonconvex Optimization--Part II: Applications

0.49 MB·

by  Gesualdo Scutari

1
Parallel and Distributed Methods for Nonconvex
Optimization−Part II: Applications
Gesualdo Scutari, Francisco Facchinei, Lorenzo Lampariello, Peiran Song, and Stefania Sardellitti
Abstract—InPartIofthispaper,weproposedandanalyzeda both depending on the current iterate xν; and X(xν) is the
novelalgorithmicframeworkfortheminimizationofanonconvex feasible set of (2). Denoting by xˆ(xν) the unique solution of
(smooth) objective function, subject to nonconvex constraints, (2),themainiterateofthealgorithmreads[3]:givenxν ∈X,
based on inner convex approximations. This Part II is devoted
to the application of the framework to some resource allocation xν+1 =xν +γν (xˆ(xν)−xν), ν ≥0, (3)
problemsincommunicationnetworks.Inparticular,weconsider
two non-trivial case-study applications, namely: (generalizations were {γν} is a step-size sequence. The proposed scheme
6 of) i) the rate profile maximization in MIMO interference represents a gamut of new algorithms, each of them corre-
1 broadcast networks; and the ii) the max-min fair multicast spondingtoaspecificchoiceofthesurrogatefunctionsU˜ and
multigroup beamforming problem in a multi-cell environment.
0 g˜, and the step-size sequence {γν}. Several choices offering
We develop a new class of algorithms enjoying the following j
2
distinctivefeatures:i)theyaredistributedacrossthebasestations great flexibility to control iteration complexity, communica-

n (withlimitedsignaling)andleadtosubproblemswhosesolutions tion overhead and convergence speed, while all guaranteeing
a are computable in closed form; and ii) differently from current convergence are discussed in Part I of the paper, see [3,
J relaxation-based schemes (e.g., semidefinite relaxation), they are
Th. 1]. Quite interestingly, the scheme leads to new efficient
5 proved to always converge tod-stationary solutionsof theafore-
distributed algorithms even when customized to solve well-
1 mentioned class of nonconvex problems. Numerical results show
that the proposed (distributed) schemes achieve larger worst- researched problems. Some examples include power control

] caserates(resp.signal-to-noiseinterferenceratios)thanstate-of- problems in cellular systems [4, 5, 6, 7], MIMO relay op-
T the-art centralized ones whilehaving comparable computational timization [8], dynamic spectrum management in DSL sys-
I complexity.
tems[9,10],sum-ratemaximization,proportional-fairnessand
.
s
max-minoptimizationofSISO/MISO/MIMOad-hocnetworks
c I. INTRODUCTION
[ [11, 12, 13, 14, 15, 16, 17, 18], robust optimization of CR
In Part I of this paper [3], we proposed a novel general
networks [12, 19, 20, 21], transmit beamforming design for
1 algorithmic framework for the minimization of a nonconvex
multiple co-channel multicast groups [22, 23, 24, 25], and
v (smooth) objective function U : K → R subject to convex
9 cross-layer design of wireless networks [26, 27, 28].
constraints K and nonconvex smooth ones g (x) ≤ 0, with
5 j Among the problems mentioned above, in this Part II,
0 gj : K→R, we focus as case-study on two important resource allocation
4 min U(x)
x designs (and their generalizations) that are representative of
0
1. s.t. gj(x)≤0, j = 1,...,m ,X. (1) scoatmioenkneeytwchoarkllse,nngaems peloys:ed1)btyhethreatneexptr-ogfielneermataixoinmciozamtimonuniin-
0 x∈K )
MIMO interference broadcast networks; and 2) the max-min
6
Building on the idea of inner convex approximation, our fair multicast multigroup beamforming problem in multi-cell
1
: approach consists in solving a sequence of strongly convex systems. The interference management problem as in 1) has
v
inner approximations of (1) in the form: given xν ∈X, become a compelling task in 5G densely deployed multi-cell
i
X networks, where interference seriously limits the achievable
xˆ(xν)=argmin U˜(x;xν)
r rate performance if not properly managed. Multicast beam-
a x
formingasin2)isapartoftheEvolvedMultimediaBroadcast
s.t. g˜ (x;xν)≤0, j = 1,...,m
j ,X(xν), Multicast Service in the Long-Term Evolution standard for
x∈K ) efficientaudioandvideostreamingandhasreceivedincreasing
(2) attention by the research community, due to the proliferation
where U˜(x;xν) is a strongly convex surrogate function of of multimedia services in next-generation wireless networks.
U(x)andg˜j(x;xν)isanupperconvexapproximantofgj(x), Building on the framework developed in Part I, we propose
a new class of convex approximation-based algorithms for
Scutari is with the School of Industrial Engineering and the Cy-
the aforementioned two problems enjoying several desirable
ber Center (Discovery Park), Purdue University, West-Lafayette, IN,
USA; email: [email protected]. Facchinei and Lampariello features. First, they provide better performance guarantees
are with the Dept. of Computer, Control, and Management Engi- than ad-hoc state-of-the art schemes (e.g., [18, 22, 29]),
neering, University of Rome “La Sapienza”, Rome, Italy; emails:
boththeoreticallyandnumerically.Specifically,ouralgorithms
<facchinei, lampariello>@diag.uniroma1.it. Song is with
the School of Information and Communication Engineering, Beijing In- achieve (d-)stationary solutions of the problems under con-
formation Science and Technology University, Beijing, China; email: siderations, whereascurrentrelaxation-basedalgorithms(e.g.,
[email protected],
semidefinite relaxation) either convergejust to feasible points
Electronics and Telecommunications, University of Rome “La Sapienza”,
Rome,Italy;email:[email protected]. or to stationary points of a related problem, which are not
The work of Scutari was supported by the USA National Science Foun- provedto be stationaryfor the originalproblems.Second,our
dation under Grants CMS 1218717, CCF 1564044, and CAREER Award
schemes represent the first class of distributed algorithms in
1254739.TheworkofFacchineiwaspartiallysupportedbytheMIURproject
PLATINO,Grantn.PON01_01007.Partofthisworkappearedin[1,2]. the literature for such problems: at each iteration, a convex
BS2
BS1 2
pBwraaosybel-(eSwmtaitthiioslnissmoil(tveBedSd,ss)i,wgnhaainlcidhngtnhaaumtusoranilsglysthodelevccaoebmlllesp)o.insMesoaraecdorivostesrsri,butthhteee (cid:33)M1T 11 MT21 MT3M1T41 MT1… 2 BSK MT22 MT32 (cid:33)2
solution of the subproblems is computable in closed form by
eachBS. We remarkthattheproposedparallelanddistributed
dmeuclotim-tpieorssitnioentwaocrkrosasrchthiteecctuerlelss wnahteurreailnlyhimghat-cshpeesedsmwodireerdn (cid:33)K MT1K MT2K MT3K
links are dedicated to coordination and data exchange among Figure1:Interference Broadcast network.
BSs. Third, our algorithms are quite flexible and applicable
with BS k is not allowed to transmit; and iii) soft-shaping
also to generalizations of the original formulations 1) and 2).
Ik tr(GHQ G ) ≤ β and peak-power constraints
For instance, i) one can readily add additional constraints, i=1 k ik k k
Ik λ (GHQ G ) ≤ β¯ , which limit respectively the
includinginterferenceconstraints,nullconstraints,per-antenna Pi=1 max k ik k k
totalaverageandpeakaveragepowerradiatedalongtherange
peak and average power constraints, and quality of service
P
space of matrix G , where λ (A) denotes the maximum
constraints;and/orii) onecan considerseveralotherobjective k max
eigenvalue of the Hermitian matrix A.
functions(ratherthanthemax-minfairness),suchasweighted
Treatingtheintra-cellandinter-cellinterferenceateachMT
users’ sum-rate or rates’ weighted geometric mean;all of this
as noise, the maximum achievable rate of each user i is
without affecting the convergenceof the resulting algorithms. k
This is a major improvement on current solution methods, R (Q),logdet I+H Q HH R¯ (Q )−1 (5)
ik ikk ik ikk ik −ik
whichareinsteadrigidad-hocschemesthatarenotapplicable
to other (even slightly different) formulations. whereHikl ∈CMik×(cid:0)Tl representsthechannelmatrixbe(cid:1)tween
BSl andMTi ;R¯ (Q ),σ2 I+ H Q HH +
The rest of the paper is organized as follows. Sec.II k ik −ik ik j6=i ikk jk ikk
focusesontherateprofilemaximizationinMIMOinterference l6=k Ijl=1HiklQjlHHikl is the covaPriance matrix of the
broadcastnetworksanditsgeneralizations:afterreviewingthe Gaussian thermal noise (assumed to be white w.l.o.g., other-
P P
state of the art (cf. Sec.II-B), we propose centralized and wiseonecanalwayspre-whitenthechannelmatrices)plusthe
distributedschemesalongwiththeirconvergencepropertiesin intra-cell (second term) and inter-cell (last term) interference;
Sec. II-C and Sec. II-D, respectively,while some experiments and we denoted Q−ik , (Qjl)(j,l)6=(i,k) and Q , (Qk)Kk=1.
arereportedinSec.II-E.Sec.IIIstudiesthemax-minfairmul- Accordingly, with slight abuse of notation, we will write
ticastmultigroupbeamformingprobleminmulti-cellsystems: Q∈Q to mean Qk ∈Qk for all k ∈KBS.
the state of the art is summarized in Sec.III-B; centralized Rate profile maximization. Max-min fairness has long been
algorithmsbasedonalternativeconvexificationsareintroduced consideredanimportantdesigncriterionforwirelessnetworks.
in Sec.III-C; whereas distributed schemes are presented in Here we introduce the following more general rate-profile
Sec.III-D; finally, Sec.III-E presents some numerical results. maximization: given the profile α , (α ) , with each
ik ik∈I
Conclusions are drawn in Sec.IV. α >0 and α =1, let
ik ik∈I ik
R (Q)
II. INTERFERENCE BROADCAST NETWORKS maxP U(Q), min ik
A. System model Q ik∈I αik (P)
s.t. Q∈Q.
Consider the Interference Broadcast Channel (IBC), mod-
Special instances of this formulation have been proved to be
eling a cellular system composed of K cells; each cell k ∈
NP-hard (see, e.g., [18]); therefore in the following our focus
K ,{1,...,K}, contains one Base Station (BS) equipped
BS is on computing efficiently (d-)stationary solutions of P.
with T transmit antennas and serving I Mobile Terminals
k k
(MTs); see Fig. 1. We denote by ik the i-th user in cell k, Definition 1 (d-stationarity). Given Q and D , (Dk)Kk=1,
equippedwithMik antennas;thesetofusersin cellk andthe withDk ,(Dik)Ii=k1andDik ∈CTk×Tk,letU′(Q;D)denote
setofalltheusersaredenotedbyI ,{i : 1≤i≤I }and the directional derivative of U at Q ∈ Q in the direction D,
k k k
I , {i : i ∈ I and k ∈ K }, respectively. We denote defined as
k k k BS
by Qk , (Qik)Ii=k1 the tuple of covariance matrices of the U′(Q;D),limU(Q+tD)−U(Q).
signals transmitted by each BS k to the Ik users in the cell, t↓0 t
with each Qik ∈ CTk×Tk being the covariance matrix of the A tuple Q⋆ is a d-stationary solution of P if
information symbols of user i .
k
Each BS k is subject to power constraints in the form U′(Q⋆;Q−Q⋆)≤0, ∀Q∈Q. (6)
Of course, (local/global) optimal solutions of P satisfy (6).
Q ∈W : Q (cid:23)0, ∀i ∈I ,
Qk , k k iIkk tr(Q k)≤Pk , (4) Equivalent smooth reformulation: To compute d-stationary
(cid:26) i=1 ik k (cid:27) solutions of the nonconvex and nonsmooth problem P, we
where the set Wk, assumedPto be closed and convex (with preliminarilyrewrite P in an equivalentsmooth (still noncon-
nonempty relative interior) captures possibly additional con- vex) form: introducing the slack variables R≥0, we have
straints, such as: i) per-antenna limits Ik [Q ] ≤ α ,
with t = 1,...,T ; ii) null constraints Ui=H1 ikIkttQ =t max R
0, where Uk is ak given matrix whoPse ckolumin=s1 coinktain Qs,R.t≥. 0 Q∈Q (Ps)
P
the directions (angle, time-slot, or frequency bands) along
R (Q)≥α R, ∀i ∈I.
ik ik k
3
We denoteby Z thefeasible setof P . ProblemsP andP going to introduce satisfy Assumptions 2 and 3 (or 3 and 4)
s s
are equivalent in the following sense. in [3], implying [together with Proposition 2(b)] convergence
Proposition 2. Given P and P , the following hold: of the our algorithms.
s
(i) Every feasiblepointof P isregular[i.e.,itsatisfies
s C. Centralized solution method
theMangasarian-FromovitzConstraintQualification
The nonconvexity of Problem P is due to the nonconvex
(MFCQ)]; s
rate constraints R (Q) ≥ α R. Exploiting the concave-
(ii) Q⋆ is a d-stationary solution of P if and only if ik ik
convex structure of the rate function
there exists R⋆ such that (Q⋆,R⋆) is a stationary
solution of Ps. Rik(Q)=fi+k(Q)−fi−k(Q−ik), (7)
Proof: See supporting material. where f+(Q)=logdet R¯ (Q )+H Q HH and
Proposition 2 opens the way to the computation of d- fi−k(Q−iikk) = logdet(R¯(cid:0)ik(Qik−ik−)i)kare coinkkcaviekfuinkckt(cid:1)ions, a
stationary solutions of P while designing algorithms for the tightconcavelowerboundofRik(Q)(satisfyingAssumptions
smooth(nonconvex)formulationP . We are notawareof any 2-4in[3])isnaturallyobtainedbyretainingin(7)theconcave
s
algorithm with provableconvergenceto d-stationarysolutions partf+ andlinearizingtheconvexfunction−f−,whichleads
ik ik
of P, as documented next. to the following rate approximation functions: given Qν ,
(Qν ) , with each Qν (cid:23)0,
B. Related works ik ik∈I ik
R (Q)≥R (Q;Qν),f+(Q)−f˜−(Q ;Qν) (8)
Several resource allocation problems have been studied in ik ik ik ik −ik
the literature for the vector Gaussian Interference Channel with
(IC), modeling multiuser interference networks. Some repre- f˜−(Q ;Qeν)
ik −ik
sentative examples corresponding to different design criteria ,f−(Qν )+ Π− (Qν ),Q −Qν , (9)
are:i)thesum-ratemaximizationproblem[11,12,13,21];ii) ik −ik ikjl −ik jl jl
the minimization of the transmit power subject to QoS con- (j,lX)6=(i,k(cid:10)) (cid:11)
straints [7, 27]; iii) the weighted Mean-Square-Error (MSE) where hA,Bi , Re{tr(AHB)}, and we denoted by
manidniimv)izthateiornateanpdrotfihelemopinti-mmiazxatMionSEovfearirMneIsSsOd/eSsIiMgnO[3[104,,3116];, Πcoin−kjujlg(aQteν−gikr)ad,ien∇tQof∗jlffi−−k(wQ.rν−.ti.kQ) = H[3Hi4k]l.R¯−ik1(Qν−ik)Hikl the
17, 32] and MIMO (single-stream) [15, 33] ICs. Since the IC Given Zν , (Qν,Riνk) ∈ Z anjdl τ , τ > 0, the convex
R Q
is a special case of the IBC model, algorithmsin [14, 15, 17]
approximation of problem P [cf. (2)] reads
s
along with their convergence analysis cannot be applied to
τ
ProRbelleamtedPtso(PanrodbtlheumsPP)i.sMthoeremovaexr-,mthineyfaairreneaslsl fcoernmtruallaiztieodn. Zˆ(Zν), aQrg,Rm≥a0x R− 2R(R−Rν)2−τQkQ−Qνk2F
recently considered in [18]. There are however several differ- s.t. nQ∈Q o
ences between [18] and our approach. First, the formulation R (Q;Qν)≥α R, ∀i ∈I,
in [18] is a special case of Problem P, corresponding to ik ik k (10)
eaqdudaitlioαnaikl caonndstrsatainntdsaWrd p).owHeerncceo,ntshteraainlgsor(iit.he.m, winith[1o8u]tatnhde wadhdeerde ttohemtahkeeqiutasdtrreoantigclytecrmonsveinx (thseeeo[b3j,ecAtisvseumfupnticotnionB1a]r)e;
itsconvergenceanalysisdkonotapplytoP.Second,eveninthe and we denoted by Zˆ(Zν) the unique solution of (10). Sta-
simplified setting considered in [18], the algorithm therein is tionarysolutionsofProblemPs canbethencomputedsolving
notprovedtoconvergetoa(d-)stationarysolutionofthemax- the sequence of convexifiedproblems(10) via (3); the formal
min fairness formulation,but to critical points of an auxiliary description of the scheme is given in Algorithm 1, whose
smooth problem that however might not be stationary for convergenceis stated in Theorem 3; the proof of the theorem
the original problem. Our algorithmic framework instead is follows readily from Proposition 2 and [3, Th.2].
guaranteedto convergeto stationarysolutionsof P and thus,
s Algorithm 1: NOVA Algorithm for Problem P (and P).
by Proposition 2, to d-stationary solutions of P. Third, the s
algorithm in [18] is centralized. Data: τR,τQ >0, Z0 ,(Q0,R0) ∈ Z; set ν =0.
TheanalysisoftheliteratureshowsthatProblemP (andPs) (S.1) If Zν is a stationary solution of (Ps): STOP.
remainsunexploredinitsgenerality.Themaincontributionof (S.2) Compute Zˆ(Zν) by (10).
this section is to propose the first (distributed) algorithmic (S.3)SetZν+1 =Zν+γν(Zˆ(Zν)−Zν)forsomeγν ∈(0,1].
framework with provable convergence to d-stationary solu- (S.4) ν ←ν+1 and go to step (S.1).
tions of P. More specifically, building on the iNner cOnVex
Approximation (NOVA) framework developed in Part I [3], Theorem3. Let{Zν ,(Qν,Rν)}bethesequencegenerated
we propose next three alternative convexifications of the by Algorithm 1. Choose any τ ,τ > 0, and the step-
R Q
nonconvex constraints in P , which lead to different convex size sequence {γν} such that γν ∈ (0,1], γν → 0, and
s
subproblems [cf. (2)] and algorithms [cf. (3)]. We start in γν = +∞. Then {Zν} is bounded and every of its
ν
Sec. II-C with a centralized instance, while two alternative limit points Z∞ , (Q∞,R∞) is a stationary solution of
P
distributed implementations are derived in Sec. II-D. We Problem P . Therefore, Q∞ is d-stationary for Problem P.
s
remark from the outset that all the convexifications we are Furthermore, if the algorithm does not stop after a finite
4
onfumUb,earndofthstuespsR,∞no>ne0o.f the Q∞ above is a local minimum fi+k(Yik)−fi−k(Q−ik)≥, fR+ik((YQ−)ik−,Yf−ik;(QQν) ;Qν).
bik ik ik −ik
Theorem 3 offers some flexibility in the choice of free
The approximation of Problem P′ becomes [cf.(2)]: given a
parameters (τ ,τ ) and {γν}, while guaranteeing conver- s e
R Q feasible Wν ,(Qν,Rν,Yν), and any τ ,τ ,τ >0,
gence of Algorithm 1. Some effective choices for {γν} are Q R Y
dthisactuAsslegdoriinthPmart1Id[o3e].sNnootteraelmsoaitnhattrtahpepethdeoinremR∞gua=ran0te,eas Wˆ (Wν), Qa,rRg≥m0a,xY R− τ2R (R−Rν)2
“degenerate” stationary solution of Ps (the global minimizer n−τQkQ−Qνk2F −τYkY−Yνk2F
ofPs andP),atwhichsomeusersdonotreceiveanyservice. s.t. Q∈Q, o
Algorithm 1 is centralized because (10) cannot be decom-
R (Q , Y ;Qν)≥α R, ∀i ∈I,
posed across the base stations. This is due to the lack of ik −ik ik ik k
separability of the rate constraints in (10): fi+k(Q) depends 0b(cid:22)Yik (cid:22)Iik(Q), ∀ik ∈I. (11)
on the covariancematrices of all the users. We introducenext
To compute d-stationary solutions of Problem P via (11), we
an alternative valid convex approximation of the nonconvex
can invokeAlgorithm 1 wherein the Z variables[resp.Zˆ(Zν)
rate constraints leading to distributed schemes.
inStep2]arereplacedbytheWones[resp.Wˆ (Wν),defined
D. Distributed implementation in (11)]. Convergence of this new algorithm is still given
by Theorem 3. The difference with the centralized approach
A centralized implementation might not be appealing in
in Sec.II-C is that now, thanks to the additively separable
heterogeneous multi-cell systems, where global information
structureof the objectiveand constraintfunctionsin (11), one
is not available at each BS. Distributing the computationover
can compute Wˆ in a distributed way, by leveraging standard
the cells as well as alleviating the communication overhead
dual decomposition techniques; which is shown next.
among the BSs is thus mandatory. This subsection addresses
With W , (Q,R,Y), denoting λ , (λ ) and Ω ,
this issue, and it is devoted to the design of a distributed ik ik∈I
(Ω ) the multipliers associated to the rate constraints
algorithm convergingto d-stationary solutions of Problem P. ik ik∈I
R (Q , Y ;Qν) ≥ α R and slack variable constraints
By keeping the concave part f+(Q) unaltered, the approx- ik −ik ik ik
imation R in (8) has the desireidk property of preserving the Yik (cid:22) Iik(Q) respectively, let us define the (partial) La-
ik gbrangian of (11) as
structure of the original constraint function R as much as
ik
possible.eHowever, the structure of Rik is not suited to be L(W,λ,Ω;Wν),LR(R,λ;Rν)+ Kk=1LQk(Qk,λ,Ω;Qν)
decomposed across the users due to the nonadditive coupling + K L (Y ,λ,Ω;Yν)
among the variables Q in f+(Q). To cope with this issue, k=1 Yk kP
the proposed idea is toikintroduikce in P slack variables whose where P
τ
purpose is to decouple in each fi+k the covariance matrix Qik LR(R,λ;Rν),−R+ 2R(R−Rν)2+λTαR;
of user i from those of the other users−the interference
k
term R¯ik(Q−ik). More specifically, introducing the slack LQk(Qk,λ,Ω;Qν),τQkQk−Qνkk2F + ik∈Ikλikfi−k(Qν−ik)
variables Y , (Y ) , with Y = (Y ) , and
setting I (Q) , kKk∈KBSIl H Qk HH , wikeikc∈aInk write + ik∈Ik (j,l)6=(i,k) λjlΠ−jliPk(Qν−jl),Qik −Qνik
ik l=1 j=1 ikl jl ikl
fi+k(Q) = fi+k(Yik)P, withPfi+k(Yik) , logdet(σi2kI+Yik) −Pik∈IkP jl∈IHHj(cid:10)lkΩjlHjlk,Qik ; (cid:11)
and Yik =Iik(Q). Then, in view of (7), Problem Ps can be P DP E
rewritten in the following equivalent form: L (Y ,λ,Ω;Yν),τ kY −Yνk2 + hΩ ,Y i
max R Yk k Y k k F ik∈Ik ik ik
Q,Rs≥.t0.,Y Q∈Q, (P′) − ik∈Ikλiklogdet(σi2kI+PYik),
fi+k(Yik)−fi−k(Q−ik)≥αikR, ∀ik ∈I, s wL(itWh α,λ,,Ω;(WαikνP))ikl∈eaId.sTtohethaedfdoiltliovweliyngsedpeacroamblpeossittriuocntuorfethoef
0(cid:22)Y (cid:22)I (Q), ∀i ∈I.
ik ik k dual function:
Thenextpropositionstatesthe formalconnectionbetweenP′ D(λ,Ω;Wν), minL (R,λ;Rν)
s R
and P , whose proof is omitted because of space limitations. R≥0
s + K min L (Q ,λ,Ω;Qν)
Proposition 4. Given Ps and Ps′, the following hold: k=1Qk∈Qk Qk k
(i) MEvFeCryQf;easible point (Q,R,Y) of Ps′ satisfies the +PPKk=1(Yik(cid:23)m0i)nik∈IkLYk(Yk,λ,Ω;(Y12ν)).
(ii) (Q⋆,R⋆) is a stationary solution of Ps if and The unique solutions of the above optimization prob-
only if there exists Y⋆ such that (Q⋆,R⋆,Y⋆) is lems are denoted by R⋆(λ;Rν) = argmin L (R,λ;Rν),
a stationary solution of Ps′. Q⋆k(λ,Ω;Qν) = argminQk∈QkLQk(QRk,≥λ0,ΩR;Qν), and
It follows from Propositions 2 and 4 that, to compute d- Y⋆(λ,Ω;Yν) = argmin L (Y ,λ,Ω;Yν).
stationary solutions of P, we can focus w.l.o.g on P′. Using Wek shownextthatR⋆ andY(Y⋆ihk(cid:23)av0e)iak∈cIlkoseYdkformk expression,
s k
(9), we can minorize the left-hand side of the rate constraints andsodoesQ⋆,whenthefeasiblesetsQ containonlypower
k k
in P′ as budget constraints [i.e., there is no set W in (4)], a fact that
s k
5
will be tacitly assumed hereafter (the quite standard proof of Algorithm 2, whose convergence is stated in Theorem 8 (the
Lemma 5 is omitted because of space limitations). proofofthetheoremfollowsfromclassicalargumentsandthus
Lemma 5 (Closed form of R⋆(λ;Rν)). The optimal solution is omitted). In Algorithm 2, PH (•) denotes the orthogonal
+
R⋆(λ;Rν) has the following expression projection onto the set of complex positive semidefinite (and
thus Hermitian) matrices.
λTα−1
R⋆(λ;Rν)= Rν − , (13)
τ
" R # Algorithm 2 Distributed dual algorithm solving (18)
+
where [x] ,max(0,x) (applied component-wise). Data: λ0 , (λ0 ) ≥ 0, Ω0 , (Ω0 (cid:23) 0) , Wν =
+ ik ik∈I ik ik∈I
(Qν,Rν,Yν). Set n=0.
Lemma 6 (ClosedformofQ⋆(λ,Ω;Qν)). LetUν Dν UνH
k ik ik ik (S.1) If λn and Ωn satisfy a suitable termination criterion:
be the eigenvalue/eigenvector decomposition of 2τ Qν −
Q ik STOP.
λ Π− Qν + HH Ω H with
DtPioνin(kj,l)Q6=,⋆k(i(,kλ)D,Ωiajlg;Q(djννilk)ik).(cid:0),L−et(jQl(cid:1)u⋆isk(λp,aPΩrti;jtQlio∈nIν))tihkjle∈kIko.jpltTimhjaleknl seoalcuh- (Y(SSi⋆..k32(λ))nUC,pΩodmnatp;eYuλteν)aen[acdcfh.Ω(1Ra3c⋆)c-((oλ1rn5d;)in]R.gνt)o, Q⋆ik(λn,Ωn;Qν), and
Q⋆ (λ,Ω;Qν) has the following waterfilling-like expression
[wiekomit the dependence on (λ,Ω;Qν)] λnik+1 = λnik +βn∇λikD(λ,Ω;Wν) + , ∀i ∈I
Q⋆ik =UνikDiag (cid:20)dνik2τ−Qξk⋆(cid:21)+!UνikH, ∀ik ∈Ik, (14) forΩsonikm+1e=βnP>Hh+0(cid:16).Ωnik +βn∇Ω∗ikD(λ,Ω;Wiν)(cid:17) k
where ξ⋆ ≥ 0 is the water-level chosen to satisfy the power (S.4) n←n+1, go back to (S.1).
k
constraint Ik tr(Q⋆ ) ≤ P , which can be found either
ik∈Ik ik k
exactly by the finite-step hypothesis method (see Algorithm4,
P Theorem 8. Let {(λn,Ωn)} be the sequence generated
AppendixA) or approximately via bisection on the interval
0, tr Diag(dν ) /(T I ) . by Algorithm 2. Choose the step-size sequence {βn}
ik∈Ik ik k k such that βn > 0, βn → 0, βn = +∞,
Proof: See Appendix A. n
(cid:2) P (cid:0) (cid:1) (cid:3) and (βn)2 < ∞. Then, {(λn,Ωn)} converges
bLeemthmeeaig7en(Cvalolusee/deifgoernmveocftoYrdk⋆e(λco,mΩp;oYsiνti)o)n. oLfe2tτVYiνYkDiνkνi−kVΩiννikHk, t{o(R⋆a(Pλnn;sRolνu)t,ioQn⋆(λofn,Ω(1n8;)Q. ν)T,Yhe⋆re(fλorneP,,Ωnt;hYeν))s}equence
with Dνik = Diag(dνik). Let us partition the optimal solu- converges to the unique solution of (11).
tion Y⋆(λ,Ω;Yν) , (Y⋆ (λ ,Ω ;Yν)) . Then, each
Y⋆ (λk,Ω ;Yν) has thiekfolilkowinigk expresiski∈oInk[we omit the Algorithm 2 can be implemented in a fairly distributed
deipkendieknceikon (λ ,Ω ;Yν)] way across the BSs, with limited communication overhead.
ik ik More specifically, given the current value of (λ,Ω), each
Y⋆ =Vν Diag yν VνH, ∀i ∈I , (15)
ik ik ik ik k k BS k updates individually the covariance matrices Qk =
with 1 (cid:0)1 (cid:1) (Qik)ik∈Ik of the users in the cell k as well as the slack
yiνk , (cid:20)−2(cid:18)σi2k − 2τYdνik(cid:19) (16) svoalruiatibolness YQk⋆ik(=λn(,YΩink;)iQk∈νI)k,[cbf.y(c1o4m)]pauntidngYti⋆hke(λcnlo,sΩedn;fYorνm)
1 1 2 2λ [cf.(15)].TheupdateoftheR-variable[usingR⋆(λ;Rν)]and
+ σ2 + dν + ik 1 , the multipliers (λ,Ω) require some coordination among the
2s(cid:18) ik 2τY ik(cid:19) τY  BSs: it can be either carried out by a BS header or locally by
+
where 1 denotes the vector of all ones.  alltheBSsifaconsensus-likeschemeisemployedinorderto
Proof: See Appendix B. obtainlocallytheinformationrequiredtocomputeR⋆(λ;Rν)
Finally, note that, since L(W,λ,Ω;Wν) is strongly con- and the gradients in (17).
vex for any given λ ≥ 0 and Ω , (Ω (cid:23)0) , the The dual problem (18) can be also solved using a second
dual function D(λ,Ω;Wν) in (12) is (R-)idkifferenitki∈abIle on order-basedscheme,whichis expectedinpracticeto befaster
RI+× ik∈ICMik×Mik, with (conjugate) gradient given by than a gradient-based one. It is sufficient to replace Step 3
of Algorithm 2 with the following updating rules for the
∇ QD(λ,Ω;Wν)= α R⋆−R Q⋆ , Y⋆ ;Qν
λik ik ik −ik ik multipliers:
∇Ω∗ikD(λ,Ω;Wν)=Yi⋆k − jl∈bIH(cid:0)iklQ⋆jlHHikl, (cid:1)(17) λn+1 =λn+sn [λˆn+1]+−λn , (19)
where R⋆, Q⋆ik, and Yi⋆k are giPven by (13), (14), and (15), Ωn+1 =Ωn+sn(cid:16) PH+(Ωˆn+1)−(cid:17)Ωn ,
respectively.Also,itcanbeshownthatthedualfunctionisC2,
(cid:16) (cid:17)
with Lipschitz continuous (augmented) Hessian with respect where λˆn+1 and Ωˆn+1 can be computed by Λˆn+1 ,
to Wν. Then, the dual problem [λˆn+1,vec(Ωˆn+1)T]T, with vec(Ωˆ) , (vec(Ωˆ) ) , which
max D(λ,Ω;Wν) (18) is updated according to ik ik∈I
λ≥0
Ω=(Ωik(cid:23)0)ik∈I Λˆn+1 ,Λˆn+(∇2 D(λn,Ωn;Wν))−1
can be solved using either first or second order methods. A λ,vec(Ω∗) (20)
gradient-based scheme with diminishing step-size is given in ·∇ D(λn,Ωn;Wν).
λ,vec(Ω∗)
6
3
Max−Min WMMSE 1 Slack−based decomposition 1 decentralized Newton dual method
Hz)2.5 WNSJOMBVMRAS (Eproposed approach) 00..89 Centralized algorithm 00..89 centralized
s/s/ 2 GSJBR 0.7 0.7
ate (nat1.5 AS ¯nR00..56 kr 00..56
R
m 0.4 0.4
mu 1 0.3 0.3
ni
Mi0.5 0.20 50 I1t0e0rationind1e50xn 200 0.20 10 20 Ite3r0ationin40dexk 50 60
Figure 3: Minimum rate vs. iterations. Left plot: Centralized algorithm vs.
00 5 10 15 20 25 30 Distributed, first order; Right plot: Centralized algorithm vs. Distributed,
SNR (dB) secondorder.
Figure 2: Minimum rate vs. SNR: The proposed algorithm reaches better
solutions thanthestateoftheart. ∼45%) for our algorithm and Max-Min WMMSE; ∼50%
In Appendix C we provide the explicit expression of the (resp. ∼60%) for GSJBR; and ∼80% (resp. ∼80%) for AS.
(augmented) Hessian matrices and gradients in (20). Between Algorithm 1 and the Max-Min WMMSE [18], the
former provides better solutions, both in terms of minimum
E. Numerical results
rate(cf.Fig.2)andsum-rate(notreported).Thismightbedue
In this section we present some experiments assessing the
tothefactthattheMax-MinWMMSEconvergestostationary
effectiveness of the proposed formulation and algorithms.
solutions of an auxiliary nonconvexproblem (obtained lifting
Example#1 :Centralized algorithm. We start comparing
P ) that are not proved to be stationary also for the original
s
Algorithm 1 with three other approaches presented in the
formulationP;ouralgorithminsteadconvergestod-stationary
literature for the resource allocation over IBCs, namely: 1)
solutions of P.
the Max-Min WMMSE [18], which aims at maximizing the
Example#2:Distributed algorithms. We test now the dis-
minimum rate of the system (a special case of P ); 2) the
s tributed algorithms in Sec. II-D and compare them with the
WMMSEalgorithm[35]andthepartiallinearization-basedal-
centralizedimplementation.Specifically,we simulate i) Algo-
gorithm(termedSJBR)[13],whichconsiderthemaximization rithm 1 based on the solution Wˆ (Wν) (termed Centralized
of the system sum-rate; and 3) the partial linearization-based algorithm); ii) the same algorithm as in i) but with Wˆ (Wν)
algorithm [13], applied to maximize the geometric mean of
computed in a distributed way using Algorithm 2 (termed
the rates (the proportional fairness utility function), termed
Distributed, first-order); and iii) the same algorithm as in i)
GSJBR. As benchmark, we also report the results achieved but with Wˆ (Wν) computed solving the dual problem (18)
using the standard nonlinear programming solver in Matlab,
using a second-order method (termed Distributed, second-
specifically the active-set algorithm in ’fmincon’ (among the
order). The simulated scenario as well as the tuning of the
other options in fmincon, the active-set algorithm was the
algorithmsis as in Example#1.In Algorithm 2, the step-size
one that showed better performance); we refer to it as “AS” sequence{βn}hasbeenchosenasβn =βn−1(1−0.8·βn−1);
algorithm. To allow the comparison, we consider a special also the starting point (λ0,Ω0) is set equal to the optimal
case of Ps as in [18]. We simulated a K = 4 cell IBC with solution (λ⋆,Ω⋆) of the previous round. Fig. 3 shows the
I = 3 randomly placed active MTs per cell; the BSs and
k normalized rate evolution achieved by the aforementioned
MTs are equipped with 4 antennas. Channels are Rayleigh
algorithms versus the iteration index n. For the distributed
fading, whose path-loss are generated using the 3GPP(TR
algorithms,the numberof iterationscountsboththeinnerand
36.814) methodology. We assume white zero-mean Gaussian
outer iterations. Note that all the algorithms converge to the
noise at each receiver, with variance σ2, and same power
same stationary point of Problem P, and they are quite fast.
budget P for all the BSs; the signal to noise ratio is then
As expected, exploiting second order information accelerates
SNR = P/σ2. Algorithm 1 is simulated using τ = 1e−7
R the practical convergence but at the cost of extra signaling
and the step-size rule γν = γν−1(1 − 10−3γν−1), with
among the BSs.
γ0 = 1. The same step-size rule is used for SJBR and
GSJBR. The same random feasible initialization is used for
III. MULTIGROUP MULTICASTBEAMFORMING
all the algorithms. All algorithms are terminated when the
A. System model
absolute value of the difference between two consecutive
valuesoftheobjectivefunctionbecomessmallerthan1e−3.In We study the general Max-Min Fair (MMF) beamforming
Fig.2 we plot the minimum rate versus SNR achieved by the problem for multi-group multicasting [22], where different
aforementioned algorithms. All results are averaged over 300 groups of subscribers request different data streams from the
independent channel/topology realizations. The figures show transmitter; see Fig.4.
that our algorithm yields substantially more fair rate alloca- WeassumethatthereareK BSs(onepercell),eachofthem
tions (larger minimum rates) than all the others. As expected, equipped with N transmit antennas and a total of I active
t
we observed that SJBR and WMMSE achieve higher sum- users,whichhaveasinglereceiveantenna;letI ,{1,...,I}.
rates(notreportedin thefigure)whilesacrificingthe fairness: For notational simplicity, we assume w.l.o.g. that each BS
SJBR and WMMSE can shut off some users (the associated serves a single multicast group; let G denote the group of
k
minimum rate is zero). More specifically, for SNR ∼ 15dB users served by the k-th BS, with k ∈ K , {1,··· ,K};
BS
(resp.SNR∼30dB)weobservedthefollowingaveragelosses G ,...,G is a partition of I. We will denote by i user
1 K k
on the sum-rate w.r.t. SJBR (and WMMSE): ∼ 70% (resp. i belonging to group G . The extension of the algorithm to
k
7
the multi-group case (i.e., multiple groups in each cell) is BS
2
straightforward.Lettingwk ∈CNt bethebeamformingvector
for transmission at BS k (to group G ), the Max-Min Fair
k
(MMF) beamforming problem reads
max U(w), min wkHHikkwk BS1 (cid:31)2
w,(wk)k∈KBS ik∈Gk,k∈KBS ℓ6=kwℓHHikℓwℓ+σi2k …
s. t. kwkk22 ≤Pk, ∀k ∈KPBS, BSK …
(21) (cid:31)1
where H is a positive semidefinite (not all zero) matrix
ikℓ
modeling the channel between the ℓ-th BS and user i ∈G ;
k k
swpiethcifihcikaℓlly∈,HCNiktℓb=eihngiktℓhheHikfℓreiqfuinesntcayn-tflanateoquusasCi-SsItaitsicascshuamnnedel, … … (cid:31)K
vector from BS ℓ to user i ; and H = E(h hH )
k ikℓ ikℓ ikℓ
representsthe spatialcorrelationmatrix if only long-termCSI Figure4:Multicell multi-groupmulticast network.
is available (in the latter case, no special structure for H
ikℓ
is assumed); σi2k is the variance of the AWGN at receiver (ii) w⋆ is a d-stationary solution of (21) if and only
ik; and Pk is the power budget of cell k. Note that different if there exist t⋆ and β⋆ such that (t⋆,β⋆,w⋆) is a
qualitiesofserviceamonguserscanbereadilyaccommodated
stationary solution of (22).
bymultiplyingeachSINRin(21)byapredeterminedpositive
factor, which we will tacitly assume to be absorbed in the Proof: See supporting material.
channelmatricesH .WedenotebyW the(convex)feasible Therefore, we can focus w.l.o.g. on (22). We remark that
ikk
set of (21). We remark that, differently from the literature, the equivalence stated in Proposition 9 is a new result in the
one can add further (convex) constraints in W, such as per- literature.
antennapower,nullorinterferenceconstraints;thealgorithmic
framework we are going to introduce is still applicable.
B. Related works
A simplified instance of problem (21), where there exists
The multicast beamformingproblem has been widely stud-
only one cell (and multiple groups), i.e., K = 1, has been
ied in the literature, under different channel models and
proved to be NP-hard [22]. Therefore, in the following, we
settings(single-groupvs.multi-groupandsingle-cellvs.multi-
aim at computing efficiently (d-)stationary solutions of (21),
cell). While the general formulation is nonconvex, special
the definition of which is analogous to the previous case (cf.
instances exhibit ad-hoc structures that allow them to be
(6) for Problem P).
solved efficiently, leveraging equivalent (quasi-)convex refor-
Equivalent reformulation: We start rewriting the nonconvex
mulations;see,e.g.,[36,37,38].Inthecaseofgeneralchannel
and nonsmooth problem (21) in an equivalent smooth (still
vectors,however,the(single-cell)MMFbeamformingproblem
nonconvex) form: introducing the slack variables t ≥ 0, and
β ,((β ) ) , we have [and thus also Problem (22)] was proved to be NP-hard [22].
ik ik∈Gk k∈KBS This has motivated a lot of interest to pursuit approximate
max t
solutionsthatapproachoptimalperformanceatmoderatecom-
t≥0, β, w
plexity.SemiDefiniteRelaxations(SDR)followedbyGaussian
s. t. (a):t·β −wHH w ≤0,
ik k ikk k randomization (SDR-G) have been extensively studied in the
,gik(t,βik,wk) literaturetoobtaingoodsuboptimalsolutions[22,23,24,25],
∀i ∈G , ∀k ∈K , withtheoreticalboundguarantees[39,40].Foralargenumber
| {z k} k BS
ofantennasorusers,however,thequalityoftheapproximation
(b): wℓHHikℓwℓ+σi2k≤βik, obtained by SDR-G methods deteriorates considerably. In
Xℓ6=k fact, SDR-based approaches return feasible points that in
∀i ∈G , ∀k ∈K ,
k k BS generalmaynotbeevenstationaryfortheoriginalnonconvex
(c):β ≤βmax, ∀i ∈G , ∀k ∈K , problem. Moreover, in a multi-cell scenario, SDR-G is not
ik ik k k BS suitable for a distributed implementation across the cells.
(d):kw k2 ≤P , ∀k ∈K ,
k 2 k BS (22) Two schemes based on heuristic convex approximations
where condition (c) is meant to bound each β by βmax , have been recently proposed in [41] and [42] (the latter
max λ (H )· K P +σ2 , so that thike feasibikle set based on earlier work [43]) for the single-cell multiple-group
ℓ∈KBS max ikℓ k=1 k ik MMF beamforming problem. While extensive experiments
of (22), which we will denote by Z, is compact (a condition
P show that these schemes achieve better solutions than SDR-
that is desirable to strengthen the convergence properties of
G approaches, their theoretical convergence and guarantees
our algorithms, see [3, Th. 2]). Problems (21) and (22) are
remain an open question. Finally, we are not aware of any
equivalent in the following sense.
distributed scheme with provable convergence for the multi-
Proposition 9. Given (21) and (22), the following hold: cell MMF beamforming problem.
(i) Every stationary solution (t⋆,β⋆,w⋆) of (22) satis- LeveragingourNOVA framework,we proposenexta novel
fies the MFCQ; centralized algorithm and the first distributed algorithm, both
8
convergingto d-stationary solutions of Problem (21). Numer- Using (29), the NOVA algorithm based on (3) is described
ical results (cf.Sec.III-E) show that our schemes reach better in Algorithm3, whoseconvergenceis established in Theorem
solutionsthanSDR-Gapproacheswithhighprobability,while 10. Note that tν > 0, for all ν ≥ 1 (provided that t0 > 0),
having comparable computational complexity. whichguaranteesthat,if(26)isusedin(29),theng˜ isalways
i
well defined. Also, the algorithm will never converge to a
C. Centralized solution method
degenerate stationary solution of (21) (U = 0, i.e., t∞ = 0),
Problem(22) isnonconvexdue to the nonconvexconstraint
at which some users will not receive any signal.
functionsg (t,β ,w ). Severalvalidconvexificationsofg
ik ik k ik
are possible; two examples are given next.
Algorithm 3: NOVA Algorithm for Problem (21)
Example #1: Note that g is the sum of a bilinear function
and a concave one, nameilky: g (t,β ,w ) = g (t,β )+ Data:z0 ,(t0,β0,w0) ∈ Z,witht0 >0,and(τt,τw,τβ)>
g (w ), with ik ik k ik,1 ik 0. Set ν =0.
ik,2 k (S.1) If zν is a stationary solution of (21): STOP.
gik,1(t,βik),t·βik, and gik,2(wk),−wkHHikkwk. (S.2) Compute zˆ(zν).
(23) (S.3)Setzν+1 =zν+γν(zˆ(zν)−zν)forsomeγν ∈(0,1].
A valid surrogate g˜ik can be then obtained as follows: i) (S.4) ν ←ν+1 and go to step (S.1).
linearize g (w ) around wν, that is,
ik,2 k k
g˜ (w ;wν),−(wν)HH wν Theorem 10. Let {zν = (tν,βν,wν)} be the sequence
ik,2 k k − ∇wkk∗gik,2ik(kwkνk), wk−wkν ≥gik,2(w(2k4)), gtheenesrtaepte-dsizbeysAeqlguoernicthem{γ3ν.}Chsuocohsethaantyγτνt,∈τw(0,,τ1β],>γν0,→an0d,
awnidthii∇) uwpk∗pgeikr,2bo(wunkνd(cid:10))g= H(ti,kβkw)kνaaronudnhda(,tbν,iβ,ν(cid:11))26=Re(0{,a0H)ba}s; faonrdall ννγ≥ν =1),+an∞d.eTvhereyn,o{fziνts}liims ibtopuonidnetsd((tw∞it,hβt∞ν,w>∞0),
ik,1 ik ik is a Pstationary solution of Problem (22), such that t∞ > 0.
g˜ (t,β ;tν,βν ), 1 βiνk t2+ tν β2 ≥g (t,β ). Therefore,w∞ isd-stationaryforProblem(21).Furthermore,
ik,1 ik ik 2 tν βν ik ik,1 ik if the algorithm does not stop after a finite number of steps,
(cid:18) ik (cid:19) (25) none of the w∞ above is a local minimum of U.
Overall, this results in the followingsurrogatefunctionwhich
Proof: See Appendix D.
satisfies [3, Assumptions 2-4]:
g˜ (t,β ,w ;tν,βν ,wν)
ik ik k ik k (26) D. Distributed implementation
,g˜ (t,β ;tν,βν )+g˜ (w ;wν).
ik,1 ik ik ik,2 k k
Algorithm 3 is centralized, because subproblems (29) do
Example #2: Another valid approximation can be readily ob-
not decouple across the BSs. In this section, we develop a
tainedusingadifferentboundforthebilineartermg (t,β )
ik,1 ik distributed solution method for (29), using the surrogate in
in(23).Rewritingg (t,β )asthedifferenceoftwoconvex
ik,1 ik Example#1 [cf.(26)]. We exploit the additive separability in
functions, g (t,β ) = 1((t + β )2 − (t2 + β2 )), the
ik,1 ik 2 ik ik the BSs’ variablesof the objectivefunction and constraintsin
desired convex upper bound of g (t,β ) can be obtained
ik,1 ik (29), as outlined next.
bylinearizingtheconcavepartofg (t,β )around(tν,βν )
ik,1 ik ik Denoting by λ , (λ , (λ ) ) and
while retaining the convex part, which leads to k ik ik∈Gk k∈KBS
η , (η , (η ) ) the multipliers associated
gik,1(t,βik;tν,βiνk), 12 (t+βik)2−(tν)2− βiνk 2 tiontrtohdeuccoinnkgstrσai2nts,ig˜kiki(k(∈≤σG2k0)ka∈nKdB(S)b) in (2,9β), re,spec(tβive)ly, and,
b − tν(cid:16)(t−tν)+βiνk βik −(cid:0)βiνk(cid:1) (cid:17).(27) and βmax , (βmkax ,ik ik(∈βGimkkaxk)∈ikK∈BGSk)k∈kKBS, theik(piakr∈tiGakl)
Lagrangian of (29) can be shown to have the following
(cid:0) (cid:0) (cid:1)(cid:1)
The resulting valid surrogate function is then
structure: L(t, β,w,λ,η;zν) = L (t,λ,η;tν,βν) +
t
g˜ (t,β ,w ;tν,βν ,wν) K L (w ,λ,η;wν) + K L (β ,λ,η;tν,βν),
ik ik k ik k (28) k=1 wk k k=1 βk k
,g (t,β ;tν,βν )+g˜ (w ;wν). where
ik,1 ik ik ik,2 k k P P
The strongly convex inner approximation of (22) [cf.(2)] L (t,λ,η;tν,βν),−t+ τt (t−tν)2+ λTβν t2+ηTσ2;
then reads: gibven a feasible zν ,(tν,βν,wν), t 2 2tν
max t− τt (t−tν)2−τ kw−wνk2− τβ kβ−βνk2 Lwk(wk,λ,η;wν),τwkwk−wkνk22
t≥0, β, w 2 w 2 2 2 − ik∈GkλikwkνHHikkwkν
s. t. g˜ik(t,βik,wk;tν,βiνk,wkν)≤0, ∀ik ∈Gk, ∀k ∈KBS, −Pik∈GkλikhHikkwkν,wk−wkνi
+ η wHH w ;
(b), (c), and (d) of (22); Pℓ6=k iℓ∈Gℓ iℓ k iℓk k
(29) L (β ,λ,η;tν,βν), τPβkβ P−βνk2−ηTβ
where g˜ is the surrogatedefined either in (26) or in (28). In βk k 2 k k 2 k k
theobjeciktivefunctionof(29)weaddedaproximalregulariza- + λik·tν β2 .
tion to make it strongly convex;therefore,problem(29) hasa ik∈Gk 2βiνk ik
P
unique solution, which we denote by ˆz(zν). The above structure of the Lagrangian leads naturally to the
9
following decomposition of the dual function: compute the closed form solutions wˆ (λn,ηn;wν) and
k
D(λ,η;zν)=minL (t,λ,η;tν,βν) βˆk(λn,ηn;tν,βν), the BSs need only information within
t≥0 t their cell. The update of the t−variable [using tˆ(λn;tν,βν)]
+ min L (w ,λ,η;wν) and multipliers λn+1,ηn+1 require some coordination
kX∈KBSkwkk22≤Pk wk k among the BSs: i(cid:0)t can be eith(cid:1)er carried out by a BS header
+ min L (β ,λ,η;tν,βν). or locally by all the BSs if a consensus-like scheme is
kX∈KBS0≤βk≤βmkax βk k (30) ecmomplpouyteedtˆ(λton;toνb,taβiνn) aloncdaltlhye gthraedieinnftosrimna(t3io2n). required to
The unique solutions of the above optimization problems can
be computed in closed form (the proof is omitted because of
E. Numerical Results
space limitations):
Inthissection,wepresentsomenumericalresultsvalidating
tˆ(λ;tν,βν),argminL (t,λ,η;tν,βν)= 1+τt·tν the proposed approach and algorithmic framework.
t≥0 t (cid:20)τt+λTβν/tν(cid:21)+ Example#1:Centralized algorithm. We compare our NOVA
βˆ (λ,η;tν,βν), argmin L (β ,λ,η;tν,βν) algorithm with the renowned SDR-G scheme in [22]. For the
k 0≤β ≤βmax βk k NOVAalgorithm,weconsideredtwoinstances,corresponding
k k
= τβ·βiνk +ηik βimkax (toseetheExtwamopalpespr1oxaimndat2io)n; wsteratwegililesteirnmtrothdeumced“NinOVSeAc1.”IIaIn-Cd
τ +λ ·tν/βν
(cid:20) β ik ik(cid:21)0 !ik∈Gk “NOVA2”, respectively. The setup of our experiment is the
wˆ (λ,η;wν), argmin L (w ,λ,η;wν) following. We simulated a single BS system; the transmitter
k kwkk22≤Pk wk k is equippedwith Nt =8 transmit antennas and serves K =2
=(ξ⋆I+A )−1bν, multicast groups, each with I single-antenna users. Different
k k k
(31) numbers of users per group are considered, namely: I =
where [x]b ,min(b,max(a,x)), A and bν are defined as 12, 24, 30, 50, 100. NOVA algorithms are simulated using
a k k
the step-size rule γν = γν−1 1−10−2γν−1 , with γ0 = 1;
A ,τ I+ η H
k w ℓ6=k iℓ∈Gℓ iℓ iℓk theproximalgainissettoτ =1e−5.Theiterateisterminated
bν , τ I+ λ H wν, when the absolute value of t(cid:0)he difference o(cid:1)f the objective
k w Pi∈GkPik ikk k
function in two consecutive iterations is less than 1e−3. For
and ξk⋆, which is s(cid:0)uch thaPt 0 ≤ ξk⋆ ⊥ k(cid:1)wˆk(λ,η,ξk⋆)k22 − the SDR-G in [22], 300 Gaussian samples are taken during
P ≤0, can be efficiently computed as follows. Denoting by
k the randomization phase, where the principal component of
U D UH the eigendecompositionof A , we have f (ξ ),
k k k k k k the relaxed SDP solution is also included as a candidate; the
kwˆkν(λ,η;wν)k22−Pk = Nj=t1 [U(Hkξkb+νk[bDνkkH]jUj)k2]jj −Pk.There- best value of the resulting objective function is denoted by
fore, ξk⋆ = 0 if fk(0) P< 0; otherwise ξk⋆ is such that tSDR. To be fair, for our schemes, we considered 300 random
f (ξ⋆) = 0, which can be computed using bisection on feasiblestartingpointsandkeptthebestvalueoftheobjective
k k
[0, tr(UHbνbνHU )/P −min[D ] ]. functionatconvergence,denotedbytNOVA.Wethencompared
k k k k k k jj
j the performance of the two algorithms in terms of the ratio
Fqinally, note that the dualfunctionD(λ,η;zν) is differen-
tNOVA/tSDR.Asbenchmark,wealsoreporttheresultsachieved
tiable on RI ×RI ,with gradient given by
+ + using the standard nonlinear programming solver in Matlab,
∇ Dν(λ,η;zν)=g˜ tˆ(λ;tν,βν),βˆ(λ,η;tν,βν), specifically the active-set algorithm in ’fmincon’; we refer
λik ik to it as “AS” algorithm and denote by tAS the best value
wˆ(λ,η;wν);zν ,
(cid:0) of the objective function at convergence (obtained over the
∇ Dν(λ,η;zν)= wˆ (λ,η;wν)HH wˆ (λ,η;wν)
ηik ℓ6=k ℓ (cid:1) ikℓ ℓ same random initializations of the NOVA schemes). In Fig.
+σ2 −βˆ (λ,η;tν,βν).
Pik ik (32) 5 a) we plot the probability that tNOVA/AS/tSDR ≥ α versus
Using (32), the dual problem max D(λ,η;zν) can be α, for different values of I (number of users per group), and
λ,η≥0
solvedinadistributedwaywithconvergenceguaranteesusing, SNR , P/σ2 = 3dB; this probability is estimated taking
e.g., a gradient-based scheme with diminishing step-size; we 300 independent channel realizations. The figures show a
omit further details. Overall, the proposed algorithm consists significantgainoftheproposedNOVAmethods.Forinstance,
in updating zν via zν+1 = zν + γν(zˆ(zν)−zν) [cf.(3)] when I = 30, the minimum achieved SINR of all NOVA
wherein zˆ(zν) is computed in a distributed way solving the methods is about at least three times and at most 5 times the
dual problem (30), e.g., using a first or second order method. one achieved by SDR-G, with probability one. It seems that
The algorithm is thus a double-loop scheme. The inner loop the gap tends to grow in favor of the NOVA methods, as the
deals with the update of the multipliers (λ,η), given zν; let numberofusersincreases.InFig.5b)weplotthedistribution
(λ∞,η∞) be the limit point (within the desired accuracy) of of tNOVA/AS/tSDR. For instance, when I = 30, the minimum
the sequence {(λn,ηn)} generated by the algorithm solving achievedSINRofallNOVAmethodsisonaverageaboutfour
thedualproblem(30).IntheouterlooptheBSsupdatelocally times the one achieved by SDR-G; the variance is about 1.
their t, β s and w s using the closed form solutionsˆz(zν)= In Fig. 6 we plot the average (normalized) distance of the
k k
(tˆ(λ∞;tν,βν),βˆ(λ∞,η∞;tν,βν),wˆ(λ∞,η∞;wν)) objectivevalueachievedatconvergenceoftheaforementioned
[cf.(31)]. The inner and outer updates can be performed algorithms from the upper bound obtained solving the SDP
in a fairly distributed way among the cells. Indeed, to relaxation (denoted by tSDP). More specifically, we plot the
10
1
NOVA1 (proposed)
0.9 NOVA2 (proposed)
average of 1−tapprox/tSDP versus I (the average is taken over 0.8 AS
300 independent channel realizations), where tapprox = tSDR R≥α)0.7
D
faonrdtthaepprSoxD=R-GtASalfgoorritthhemA,Stapaplrgoxor=ithmtNOinVAMfaotrlaobu,rwmithethtSoDdRs,, NOVA/ASS/t000...456 I=100
toNbOjeVcAtiavnedvatAluSedreefianchededabsyinouFrigm.5e.thTohdesfiisgmuruecshhcolwossetrhtaot tthhee Prob(t0.3 I=21 I=50
0.2
SDP bound than the one obtained by SDR-G. For instance, 0.1 I=30
I=12
when I = 30, the solution of our NOVA methods are within 0
0 2 4 6 8 10 12
α
25% the upper bound. (a)
Example#2: Distributed algorithms. The previous example
0.5
shows that the proposed schemes compare favorably with the NOVA1 (proposed)
commercialoff-the-shelfsoftware and outperformSDR-based 0.45 I=12 NOVA2 (proposed)
0.4 AS
scheme (in terms of quality of the solution and convergence 0.35 I=21
speed). However, differently from off-the-shelf softwares, our 0.3 I=30 I=50
F
schemesallowforadistributedimplementationinamulti-cell PD0.25
scenario with convergenceguarantees. We test the distributed 0.2 I=100
0.15
implementation of the algorithms, as described in Sec. III-D,
0.1
andcompareitwiththecentralizedone.Morespecifically,we
0.05
simulate i) Algorithm 3 based on the solution ˆz(zν) (termed 0
0 2 4 6 8 10 12
Centralized algorithm); ii) the same algorithm as in i) but tNOVA/AS/tSDR
with zˆ(zν) computed in a distributed way using the heavy (b)
ballmethod(termedDistributed,first-order);andiii)thesame Figure 5: (a): Prob(tNOVA/AS/tSDR ≥ α) versus α, for I =
12,21,30,50,100;(b):Estimatedp.d.f.oftNOVA/tSDR andtAS/tSDR.
algorithmasini)butwithˆz(zν)computedbysolvingthedual
problem maxλ,η≥0Dν(λ,η;zν) using the damped Netwon 1
method (termed Distributed, second-order). The simulated 0.9
0.8
sstagrcynraeosdnntuesaopmmruiihottecaroaso)mnf,Ittpeohnru=einrsaiien3tsexgrapsanKietndergim=lisesee-rna4tventirBtnmiesgSnisnntGh,aaeteeu=afdscoehlw1rlsoeh.mwqeIuniunnilpgttibhp.coeeaWtdshatebwglssoioritoomhlupuutsNpel.a(tvtiEena=adnlcue4haer SDPapproxSDPE[(t−t)/t]h000000......234567 SNDORVA1 (proposed)
NOVA2 (proposed)
of the difference of the objective function in two consecutive 0.1 AS
01 2 21 30 50 100
iterations is less than 1e−2. Fig. 7 shows the evolution of Number of Users per Group
Figure6: AveragenormalizeddistanceoftheachievedminimumSINRfrom
the objective function t of (22) versus the iterations. For the
theSDPupperboundversusI.
distributedalgorithms,thenumberofiterationscountsboththe
innerandouteriterations.Notethatallthealgorithmsconverge 2 2
1.8 1.8
to the same stationary point of Problem (21), and they are 1.6 1.6
quite fast. As expected, exploiting second order information 1.4 1.4
1.2 1.2
acceleratesthepracticalconvergencebutwiththecostofextra nt 1 nt 1
signaling among the BSs. 0.8 0.8
0.6 0.6
0.4 0.4
IV. CONCLUDING REMARKS 0.2 DCiesntrtirbaulitzeedd, fairlgsto−roithrdmer 0.2 DCiesntrtirbaulitzeedd, saelgcoornitdh−morder
00 50 100 150 200 250 00 10 20 30 40 50 60 70
In this two-part paper we introduced and analyzed a new Iteration index n Iteration index n
algorithmic framework for the distributed minimization of Figure 7: Minimum rate vs. iterations. Left plot: Centralized algorithm vs.
Distributed, first order; Right plot: Centralized algorithm vs. Distributed,
nonconvex functions, subject to nonconvex constraints. Part
secondorder.
I [44] developed the general framework and studied its
convergence properties. In this Part II, we customized our
general results to two challenging and timely problems in goes much beyond these two specific formulations and, more
communications, namely: 1) the rate profile maximization in generally, applications in communications. Moreover, even
MIMO IBCs; and 2) the max-min fair multicast multigroup within the network systems considered in 1) and 2), one
beamforming problem in multi-cell systems. Our algorithms can consider alternative objective functions and constraints.
i) were proved to converge to d-stationary solutions of the Two examples of unsolved problems to which our frame-
aforementioned problems; ii) represent the first attempt to work readily applies (with convergence guarantees) are: 1)
designdistributedsolutionmethodsfor1)and2);andiii)were the distributed minimization of the BS’s (weighted) transmit
shownnumericallytoreachbetterlocaloptimalsolutionsthan power over MIMO IBCs, subject to rate constraints; and 2)
ad-hoc schemes proposed in the literature for special cases of the maximization of the (weighted) sum of the multi-cast
the considered formulations. multigroup capacity in multi-cell systems.
We remark that, although we considered in details only Our framework finds applications also in other areas, such
problems 1) and 2) above, the applicability of our framework as signal processing, smart grids, and machine learning.