Table Of ContentProjectedPrimal-DualGradientFlowofAugmented
LagrangianwithApplicationtoDistributedMaximizationof
(cid:63)
theAlgebraicConnectivityofaNetwork
HanZhanga,JieqiangWeib,PengYic,XiaomingHua,
7
1 aDepartmentofMathematics,KTHRoyalInstituteofTechnology,SE-10044,Stockholm,Sweden
0
2 bDepartmentofAutomaticControl,KTHRoyalInstituteofTechnology,SE-10044,Stockholm,Sweden
n cDepartmentofElectricalandComputerEngineering,UniversityofToronto,Canada
a
J
5
2
Abstract
]
C In this paper, a projected primal-dual gradient flow of augmented Lagrangian is presented to solve convex optimization
O problems that are not necessarily strictly convex. The optimization variables are restricted by a simple convex set as well
as equality constraints. Instead of choosing to project the negative gradient flow of the objective function onto the tangent
.
h coneoftheentirefeasiblesetoftheconvexoptimizationproblem,wechoosetousetheprojectedprimal-dualgradientflowof
t augmentedLagrangiansothatthedynamicsonlyprojectsthenegativegradientflowoftheprimalvariablesontothetangent
a coneofthesimpleconvexset.Hencethecomplexityofcalculatingthetangentconeisdecreased.Weshowthattheprojected
m
dynamical system converges to one of the optimal solutions. We further explain the connection and difference between the
[ continuous-time algorithm and proximal gradient method. Moreover, the problem of distributedly maximizing the algebraic
connectivity of an undirected graph by optimizing the edge weights (communication gains for each channel) of each nodes is
1 considered. The original semi-definite programming (SDP) problem is relaxed into a nonlinear programming (NP) problem
v that will be solved by the aforementioned projected dynamical system. Numerical examples show the convergence of the
5 aforementionedalgorithmtooneoftheoptimalsolutions.
7
4
7 Keywords: ProjectedDynamicalSystems,semi-definiteprogramming,distributedoptimization
0
.
1
0
1 Introduction many powerful mathematical tools to analyze the con-
7
vergenceofthealgorithms.Hencestudyingcontinuous-
1
: Recently,duetothefactofrisinginterestindistributed time algorithms provides us with more insights about
v the theory behind some iterative algorithms as well as
i optimization, there has been a number of advances in
X the field of optimization that involves using continuous moreideasofdesigningthem.
r time dynamical systems to solve smooth convex prob-
a
lems, such as (Yi et al., 2016), (Gharesifard & Cort´es, When solving a convex minimization problem with
2014),(Linetal.,2016),etc,aswellasnonsmoothconvex strong duality, it is well-known that the optimal so-
problems(Qiuetal.,2016).Thoughitseemstobequite lution is the saddle point of the Lagrangian. Hence
differentfromtheclassicaliterativeoptimizationmeth- beingdifferentfromthedynamicsderivedfromKarush-
ods, it has been found that classical methods such as Kuhn-Tucker (KKT) conditions in (Yang et al.) and
steepestdescentandproximalpointmethodarenothing (Liu et al., 2016), it is natural to consider the gradi-
butforwardEulerandbackwardEulerdiscretizationof ent flow of Lagrangians and the primal variable follows
thesamegradientflowdynamics(Parikh&Boyd,2014). the negative gradient flow while the dual variable fol-
On the other hand, continuous-time algorithms enable lows the gradient flow. Gradient flow of Lagrangians is
first studied by Arrow et al. (1959), Kose (1956) and
(cid:63) ThisworkissupportedbyChinaScholarshipCouncil. has been revisited by Feijer & Paganini (2010). (Feijer
Emailaddresses: [email protected](HanZhang), & Paganini, 2010) studies the case of strictly convex
[email protected](JieqiangWei),[email protected] problemsandprovidesmethodologiestotransformnon-
(PengYi),[email protected](XiaomingHu). strictly convex problems to strictly convex problems to
PreprintsubmittedtoAutomatica 27January2017
fit the framework. The convergence is shown by em- projected primal-dual gradient flow of augmented La-
ploying the invariance principle for hybrid automata. grangian.Moreprecisely,thedynamicsonlyprojectsthe
(Cherukurietal.,2016)studiesthesamestrictlyconvex negative gradient flow of the primal variable onto the
problemfromtheangleofprojecteddynamicalsystems tangent cone of the simple convex set. Hence the com-
andisabletoshowtheconvergencebyaLasalle-likein- plexityofcalculatingthetangentconeisdecreased.We
variant principle for Carath´eodory solutions. Instead of showthattheprojecteddynamicalsystemconvergesto
consideringdiscontinuousdynamics,Durr&Ebenbauer oneoftheoptimalsolutions.Wefindthatafterbeingdis-
(2011) propose a smooth vector field for seeking the cretizedbyforwardEulerdiscretization,thecontinuous-
saddle points of strictly convex problems. Wang & Elia time algorithm actually has a strong connection with
(2011) consider a strictly convex problem with equality iterative proximal gradient method but with iteration
constraintsandwithinequalityconstraintsrespectively. also on the dual variables. Second, the problem of dis-
Gradient flow is also used therein, however, it is worth tributedly maximizing the algebraic connectivity of an
noticingthattheirproblemisstillstrictlyconvex.When undirected graph by adjusting the edge weights (com-
they consider the problem with inequality constraints, munication gain for each channel) of each nodes is con-
logarithmic barrier function is used. Though consid- sidered.Itisworthnoticingthattheproblemmotivates
ering nonsmooth problems, Zeng et al. (2016) use the from optimizing a parameter of a graph and the com-
projectedgradientflowofaugmentedLagrangianwhose munication network of the distributed optimization al-
equalityconstraintisthevariableconsensusconstraint, gorithm is actually the physical network itself; one can
and can be viewed as a special case of our problem. not “design” the communication network according to
Instead of using the continuous-time gradient flow, an thestructureoftheproblemorthealgorithm.Wesolve
iterative distributed augmented Lagrangian method is theoriginalproblem,whichisanSDP,byrelaxingitinto
developedin(Chatzipanagiotisetal.,2015). anNPproblem.TheNPproblemisnotstrictlyconvex,
henceweadapttheprojectedgradientflowmethodpro-
Since first introduced by Fiedler (1973), the algebraic posedinthisworktosolvetheaforementionedNPprob-
connectivity of a graph, namely, the second smallest lem.Numericalexamplesshowthattheaforementioned
eigenvalue of the Laplacian matrix has been playing a algorithmconvergestooneoftheoptimalsolutions.
very important role in the networked systems. Alge-
braicconnectivityisboundedbothaboveandbelowby Therestofthispaperisorganizedasfollows.InSection
Cheeger’sconstantandhenceisameasureoftherobust- 2, some preliminaries and notations are presented. In
nessofagraph(Mesbahi&Egerstedt,2010).Italsoin- Section3,weintroducetheproblemformulationaswell
dicates the convergence rate of the variables consensus astheprojectedgradientflowofaugmentedLagrangian.
inmulti-agentsystems(Olfati-Saberetal.,2007).Hence Theconvergenceanalysisoftheprojectedgradientflow
maximizing the algebraic connectivity of a graph is an ispresentedinSection4andtheconnectionandthedif-
importantandfundamentalproblemandhasbeencon- ferencebetweentheaforementionedcontinuous-timeal-
sidered intensively in the past. Ghosh & Boyd (2006) gorithmanditerativeproximalgradientmethodarealso
consider the problem of optimizing the way of adding a described. In Section 5, we apply the projected dynam-
fixed number of edges to the graph to maximizing the ics to distributedly solve the maximization problem of
algebraic connectivity. Since the problem is NP-hard, a the algebraic connectivity of a network. Numerical ex-
greedy heuristic is used to solve the problem. (G¨oring amplesarepresented.ThepaperisconcludedinSection
etal.,2008)considertheproblemofmaximizingtheal- 6.
gebraicconnectivitybyadjustingtheedgeweight.They
explain connections between the problem and the em-
2 PreliminariesandNotations
bedding of nodes in n dimensional space. From the an-
gle of applications, the edge weights of a graph can be
treatedasthegainsofcommunicationchannelsbetween In this section, we introduce the notations we will use
theagentsandwewillconsiderasimilarprobleminthis in the rest of the paper. Some basic knowledge about
paperandsolveitdistributedly. projecteddynamicalsystems,convexanalysisaswellas
SDPwillalsoberevisited.
The main contribution of this paper is two-fold. First,
we construct a projected primal-dual gradient flow of We denote 1 = 11T as an N dimensional all-one ma-
augmented Lagrangian that solves convex optimization trix, where 1 is an N dimensional all-one vector. The
problemswhicharenotnecessarilystrictlyconvex.The elementlocatedontheithrowandandjthcolumnofa
optimization variables are restricted by a simple con- matrixAisdenotedas[A] .MatrixA −A ispositive
ij 1 2
vex set as well as equality constraints. Using the gra- semi-definite, then it will be denoted as A (cid:23) A . We
1 2
dient flow of augmented Lagrangian enables us to han- use(cid:107)·(cid:107)todenote2-normofvectors.|S|denotesthecar-
dle nonstrictly convex problems. In addition, instead dinalityofsetS.Andanynotationwiththesuperscript
of projecting the negative gradient flow of the objec- ∗denotestheoptimalsolutiontothecorrespondingop-
tivefunctionontothetangentconeoftheentirefeasible timization problem. tr(·) denotes the trace of a matrix.
set of the convex optimization problem, we employ the (cid:104)·,·(cid:105) denotes the inner-product in euclidean space and
2
2
(cid:104)A ,A (cid:105) =tr(A A )denotestheinner-productinSn, TakingthegradientoftheLagrangianfunction,together
1 2 M 1 2
whichistheHilbertspaceofn×nsymmetricmatrix. withtheconstraintandthecomplementarityslackcon-
dition,wegetitsKKTconditions
Assume K ⊂ Rn is a closed and convex set, the pro-
m
jection of a point x to the set K is defined as P (x) = (cid:88)
K (cid:104)A ,Φ(cid:105) =−c , x A (cid:22)S,
argmin (cid:107)x−y(cid:107). For x ∈ K, v ∈ Rn, the projection i M i i i
y∈K
ofthevectorv atxwithrespecttoK isdefinedas(see i=1
m
(Nagurney&Zhang,2012),(Brogliatoetal.,2006)) (cid:88)
Φ(cid:23)0, (cid:104) x A −S,Φ(cid:105) =0.
i i M
i=1
P (x+δv)−x
Π (x,v)= lim K =P (v), (1)
K δ→0 δ TK(x)
3 Problem Formulation and Projected Gradi-
where T (x) denotes the tangent cone of K at x. The entFlow
K
interior, the boundary and the closure of K is denoted
asK◦,∂K andcl(K)respectively. In this section, we consider the following optimization
problemdefinedonRn:
InwardnormalsofK atxisdefinedas
minimize f(x)
x∈K (4)
(cid:8) (cid:9)
n(x)= γ|(cid:107)γ(cid:107)=1,(cid:104)γ,x−y(cid:105)2 ≤0,∀y ∈K , subjectto Ax−b=0,
andΠ (x,v)fulfillsthefollowinglemma: where f : Rn (cid:55)→ R and A ∈ Rm×n. K is a simple
K
convex set, namely, it is generated by simple inequality
Lemma1(Nagurney&Zhang(2012)) If x ∈ K◦, constraints such as interval constraints and norm con-
straints. We assume calculating the projection on the
then Π (x,v) = v; if x ∈ ∂K, then Π (x,v) =
K K
v+β(x)n∗(x),wheren∗(x)=argmax (cid:104)v,−n(cid:105)and tangentconeofKiscomputationallycheap.f(x)iscon-
n∈n(x) vex function but not necessarily strictly convex. It is
β(x)=max{0,(cid:104)v,−n∗(x)(cid:105)}.
also assumed that the gradient of f(x) is locally Lips-
chitzcontinuousandtheSlater’sconditionholdsfor(4).
Let F be a vector field such that F : K (cid:55)→ Rn, the
Hencestrongdualityholdsfor(4).
projecteddynamicalsystemisgivenintheform
The Lagrangian L : K ×Rm (cid:55)→ R for the problem (4)
x˙ =Π (x,F(x)). (2) isgivenby
K
Note (1), the right hand side of (2) can be discontin- L(x,v)=f(x)+vT(Ax−b), (5)
uous on the ∂K. Hence given an initial value x ∈ K,
thesystem(2)doesnotnecessarilyhaveaclassic0alsolu- wherev ∈Rm istheLagrangianmultiplierofconstraint
tion. However, if F(x) is Lipschitz continuous, then (2) Ax − b = 0. Since strong duality holds for (4), then
has a unique Carath´eodory solution that continuously (x∗,v∗) is a saddle point of L(x,v) if and only if x∗
dependsontheinitialvalue(Nagurney&Zhang,2012). is an optimal solution to (4) and v∗ is optimal solution
to its dual problem. The augmented Lagrangian L :
A
K×Rm (cid:55)→Rfor(4)isgivenby
NowweintroducesomebasicknowledgeaboutSDP.An
SDPprobleminstandardformcanbeexpressedas ρ
L (x,v)=f(x)+vT(Ax−b)+ (Ax−b)T(Ax−b), (6)
A 2
m
(cid:88)
maximize cixi where ρ > 0 is the damping parameter that suppresses
i=1 theoscillationofx.Withoutlossofgenerality,wechoose
(cid:88)m ρ=1here.
subjectedto x A (cid:22)S,
i i
i=1 We propose to find the saddle point of (6) via the gra-
dientflowdynamicsrestrictedonthesetK,i.e.,
where A ,S ∈ SN. Introducing the Lagrangian multi-
i
plierΦ(cid:23)0,theLagrangianfunctionof(3)canbewrit- x˙ =Π (x,−∇f(x)−ATv−AT(Ax−b))
K
tenas ∂L (x,v)
=Π (x,− A ), (7a)
K ∂x
m m
L(x,Φ)=−(cid:88)cixi+tr(Φ((cid:88)xiAi−S)) (3) v˙ =Ax−b= ∂LA(x,v). (7b)
∂v
i=1 i=1
3
Note that it is assumed that ∇f(x) is locally Lipschitz map such that L V(x) ≤ 0, where L V(x) = ∂Vf(x)
f f ∂x
continuous,thereforethereisauniqueCarath´eodoryso- denotes the Lie derivative along the vector field f(x).
lutionforthedynamics,hencethesystemiswell-defined. Then any solution of x˙ = f(x) starting at S converges
tothelargestinvariantsetincl({x∈S|L V(x)=0}).
f
Remark2 Indeed, one can choose to project the nega- Theorem6 Given an initial value (x(0),v(0)), where
tivegradientflowoftheobjectivefunctionontotheentire x(0)∈K,theprojecteddynamicalsystem (7)asymptot-
feasibleset.However,thisactuallyincreasesthecompu- ically converges to some (x(cid:48),v(cid:48)), where x(cid:48) is one of the
tational complexity since it adds more constraints while optimalsolutionsof (4).
computing the projection. Recall that we assume that
calculating the projection to T (x) is computationally
K
cheap,therefore,primal-dualgradientflowisused. (cid:50) PROOF. Suppose (x∗,v∗) is a saddle point of the La-
grangian (5), namely, x∗ is the optimal solution of (4)
andv∗ istheoptimalsolutionofitsdualproblem.Con-
4 ConvergenceAnalysis
structthefollowingLyapunovfunction
In this section, we will analyse the convergence for (7)
1
and start with the analysis of the equilibrium point of d(t)= ((cid:107)x−x∗(cid:107)2+(cid:107)v−v∗(cid:107)2). (9)
2
(7).
Notethatd(t)iscontinuouslydifferentiableanddenote
Proposition3 (x∗,v∗)istheoptimalsolutionto (4)if
therighthandsideofthedynamics(7)asvectorfieldF.
andonlyifitisanequilibriumof (7).
By the definition of saddle points, it holds for (x∗,v∗)
thatL(x∗,v)≤L(x∗,v∗)≤L(x,v∗).TheLiederiva-
tiveofd(t)alongthevectorfieldF isgivenby
PROOF. Since strong duality holds for (4), the opti-
mality conditions become necessary and sufficient con-
L d(x,v)
ditions. The optimality condition for (4) is given by F
(Ruszczyn´ski,2006) =(x−x∗)TΠK(x,−∇f(x)−ATv−AT(Ax−b))
+(v−v∗)T(Ax−b)
−∇f(x∗)−ATv∗ ∈N (x∗), Ax∗−b=0,
K
ByLemma1,itholdsthat
whichisequivalentto
Π (x,−∇f(x)−ATv−AT(Ax−b))=
−∇f(x∗)−ATv∗+AT(Ax∗−b)∈N (x∗), K
K −∇f(x)−ATv−AT(Ax−b)+β(x)n∗(x),
whereN (x∗)denotesthenormalconeofK atx∗.This
impliesΠK (x,−∇f(x)−ATv−AT(Ax−b))=0,which where β(x) ≥ 0 and (cid:104)n∗(x),x−y(cid:105) ≤ 0, ∀y ∈ K. Since
isequivaleKnttobeinganequilibriumpointof (7). (cid:50) Ax∗−b=0,henceitfollowsthat
L d(x,v)
F
Before we show the asymptotic stability of the equilib- =(x−x∗)T(−∇f(x)−ATv−AT(Ax−b)
riums, we first introduce the definition of omega-limit
+β(x)n∗(x))+(v−v∗)T(Ax−b)
set and a Lasalle-like invariance principle originated
from (Bacciotti & Ceragioli, 2006) and simplified in ≤(x−x∗)T(−∇f(x)−ATv−AT(Ax−b))
(Cherukurietal.,2015). +(v−v∗)T(Ax−b)
∂L(x,v)
Definition4 Assumeγisasolutiontox˙ =f(x),where =−(x−x∗)T −(Ax−b)T(Ax−b)
f :Rn (cid:55)→Rn.Theomega-limitsetisdefinedas ∂x
∂L(x,v)
+(v−v∗)T .
Ω(γ)={y ∈Rn|∃{t }∞ ⊂[0,∞), lim t =∞, ∂v
k k=1 k
k→∞ (8)
lim γ(t )=y} Since L(x,v) is convex with respect to x and concave
k
k→∞ withrespecttov,itfollowsfromthefirstorderproperty
ofconvexandconcavefunction(Boyd&Vandenberghe,
Lemma5 (LasalleinvarianceprincipleforCarath´eodory
2004)that
solutions) Let S ∈ Rn be compact and invariant. As-
sume for each x ∈S, there exists a unique solution for
0
L d(x,v)
x˙ = f(x) starting at x and its omega-limit set is in- F
0
variant.LetV :Rn (cid:55)→Rbeacontinuouslydifferentiable ≤L(x∗,v)−L(x,v∗)−(Ax−b)T(Ax−b).
4
Since L(x∗,v) ≤ L(x,v∗), we have L d(x,v) ≤ ConsidertheindicatorfunctiononK
F
−(Ax − b)T(Ax − b) ≤ 0. Since d(t) is differen-
tiable almost everywhere with respect to t and (cid:26)0, ifx∈K,
ddtd(t) = LFd(x,v), d(t) is non-increasing with respect IK(x)= +∞, otherwise,
totime.Notethatd(t)isradiallyunbounded,hencethe
set Sˆ= {(x,v) ∈ Rn×Rm|d(x,v) ≤ d(x(0),v(0))} is a whose proximal mapping is prox (x) = P (x) =
compactinvariantsetforthesystem(7).Recallthatitis argmin (cid:107)u−x(cid:107)2. tIK K
u∈K
assumedthatf(x)haslocallyLipschitzcontinuousgra-
dient,hencetheuniquenessoftheCarath´eodorysolution Denote ∂L (xk,vk)/∂x = yk, by the definition of the
A
to (7) can be guaranteed. The invariance of the omega- projection (1),wecandiscretizetheprojecteddynamical
limit set can be proved by using the same methodology system (7)usingforwardEulerdiscretization
asLemma4.1in(Khalil&Grizzle,2002)whichisbased
on the continuity and the uniqueness of the solution. (xk+1−xk)/∆t=(P (cid:0)xk−∆tyk)(cid:1)−xk)/∆t,
K
NowbyLemma5,thetrajectoryof (7)convergestothe
largestinvariantsetincl({(x,v)∈Sˆ|LFd(x,v)=0}). andthisgives
(x(cid:48)(t),v(cid:48)(t))∈{x∈Sˆ|LFd(x,v)=0}meansv˙(cid:48) =Ax(cid:48)− xk+1 =PK(xk−∆tyk)=prox∆tIK(xk−∆tyk),
b=0and (11a)
vk+1 =vk+∆t(Axk−b). (11b)
L(x∗,v)=L(x,v∗)⇔f(x∗)=f(x(cid:48))+v∗T(Ax(cid:48)−b)
⇔f(x∗)=f(x(cid:48)). Note that if one choose to project on the tangent cone
(cid:84)
of the entire feasible set Ω = K {x|Ax = b}, then the
Sincex∗ isanoptimalsolutionto(4),x(cid:48) isalsoanopti- continuous-time dynamics will be x˙ = Π (x,−∇f(x)).
Ω
malsolutionto(4).Sincethesetofoptimalsolutionsis OnecandiscretizethedynamicsusingforwardEuleras
closedandv(cid:48)isaconstantin{(x,v)∈Sˆ|L d(x,v)=0},
{(x,v)∈Sˆ|L d(x,v)=0}isalsoclosed.F xk+1 =PΩ(xk−∆t∇f(xk))
F
=prox (xk−∆t∇f(xk)),
∆tIΩ
Denote M as the largest invariant set in {(x,v) ∈
Sˆ|L d(x,v) = 0}. What remains to show is that there whichisexactlyproximalgradientmethod.Duetothefact
F
that the gradient flow only projects on K and the intro-
isnolimitcyclewithinM,namely,ifatrajectorystarts
duction of the dual variable v, the iterative method (11)
in M, it will remain constant for all times. Assume
(xˆ(0),v¯)∈M⊂{(x,v)∈Sˆ|L d(x,v)=0}, where v¯is isdifferentfromclassicalproximalgradientmethod.The
F designof∆tsothat (11)convergestooneoftheoptimal
a constant, then xˆ(0) is also an optimal solution to (4).
solutions of (4) will be studied in the future. Moreover,
Hence there exist some vˆ such that (xˆ(0),vˆ) is another
apart from forward Euler discretization, one can choose
saddlepointof (5).ThentheLyapunovfunctioncanbe
other methods such as Heun Method and Runge-Kutta
constructedsimilarlyas
Methods(Nagurney&Zhang,2012),whichwillresultin
differentiterativeoptimizationmethods. (cid:50)
1
dˆ(t)= ((cid:107)x−xˆ(0)(cid:107)2+(cid:107)v−vˆ(cid:107)2) (10)
2
Example8 Considerthefollowingsimpleproblem:
We have shown that for any arbitrarily chosen saddle
minimize x+3y
point(x∗,v∗),d(t)isnon-increasingglobally;suchstate- x,y∈R
ment also holds for dˆ(t). Furthermore, the trajectory subjectto x≥2,
starts at (xˆ(0),v¯) can not be time-variant, namely, xˆ(t)
x=y,
remains constant for all t > 0. Since dˆ(0) = 1(cid:107)v¯−vˆ(cid:107)2,
2
it contradicts the fact that dˆ(t) is non-increasing if x(t) and apply the projected system dynamics (7) to it. The
istime-variant. evolutionofx(t),y(t)andv(t)isillustratedinFig.1.The
v(0)andy(0)arearbitrarilychosenandx(0)isarbitrarily
Fromtheanalysisabove,thetheoremfollows. (cid:50) chosen among the real numbers which is greater than 2.
(cid:50)
Remark7 Wenowdescribetheconnectionbetweenthe
continuous-timealgorithm (7)andtheproximalgradient Example9 Nowweconsideramorecomplicatedprob-
method(Beck&Teboulle,2010).Asisknown,theproxi- lemin(Chen,2005):
maloperatorofaconvexfunctionh:Rn (cid:55)→Risgivenby
minimize cTx
1 x∈R5+
prox (x)=argmin{h(u)+ (cid:107)u−x(cid:107)2}.
th u 2t subjectto Ax=b,
5
vertex i and j are connected; w ∈ W, (i,j) ∈ E de-
6 ij
notes the edge weight set. To abbreviate the notation,
the edges are labelled with numbers. For instance, an
n 4 edge with label l is denoted as l ∈ E. The set of edges
o
uti (communication channel)(cid:83)adjacent to node i is denoted
ol 2 as E(i). Note that E = i∈VE(i). We only consider
Ev undirected graphs, and hence the weighted Laplacian
e x(t) matrixL issymmetricandcanbeexpressedas
u y(t) w
al 0 v(t)
V (cid:80)
w ifi=jand(i,l)∈E
l il
-2 [Lw]ij = −wij ifi(cid:54)=jand(i,j)∈E
0 4 8 12 16 20 0 otherwise
t(s)
(cid:88)
= w E ,
k k
Fig.1.Theevolutionofx(t),y(t)andv(t)forExample8.
k∈E
where where k is the label of the edges, 0 ≤ w ∈ W, ∀k ∈ E
k
−3 1 1 −1 2 are the edge-weights. If node i and j are connected via
edgek,then
A= 2 0 −1 1 −1,
0 1 2 −1 1 [E ] =[E ] =1, [E ] =[E ] =−1,
k ii k jj k ij k ji
b = [5,6,3]T and c = [2,1,−1,−3,1]T. The optimal
and the other elements of E are zero. If the graph is
value for the problem is cTx∗ = −76 and the evolution k
connected,thentheeigenvaluesofL satisfies:0=λ <
ofcTx(t)isshowninFig.2. (cid:50) w 1
λ ≤ ··· ≤ λ and λ is the algebraic connectivity of
2 N 2
G(V,E,W).
0
(cid:80)
Let L = E be the unweighted Laplacian matrix
k∈E k
-20 ofthegraph.WesupposeLhasonlyonezeroeigenvalue,
namely, the unweighted graph G(V,E) is connected. As
-40 illustratedinFig.3,eachedgeweight(thegainofcom-
x
T municationchannel)isthesumoftheweightcontributed
c -60 X: 186.5
bythetwoendnodesoftheedge.
Y: -76
-80
Moreover,itisalsoassumedthatthereisalimitedcom-
munication resource or budget constraint for each node
-100
0 50 100 150 200 in the network. Without loss of generality, we assume
t(s) (cid:80) w(i) = 1. Note that this differs from the for-
k∈E(i) k
mulation in (G¨oring et al., 2008), while they assume
Fig.2.TheevolutionofobjectivefunctionforExample9. (cid:80) w = 1, if written in the form of our model,
k∈E k
(cid:80) (cid:80) w(i) =1. Then maximizing the algebraic
i∈V k∈E(i) k
connectivity of the edge-weighted Laplacian matrix by
5 Distributed Maximization of the Algebraic
adjustingtheedge-weightscanbeformulatedasthefol-
ConnectivityofaNetwork
lowingSDPproblem:
In this section, we apply the aforementioned algorithm maximize λ
2
tomaximizethealgebraicconnectivityofanetworkina λ2,µ,{wk(i)}
distributed manner. The problem is first formulated as subjectto λ I−µ1(cid:22)(cid:88) (cid:88) w(i)E ,
anSemi-definiteProgramming(SDP)problem,thenby 2 k k
employingthefunction(14),theproblemisrelaxedinto i∈Vk∈E(i) (Pc)
anNonlinearProgramming(NP)problem. (cid:88) w(i) =1,
k
k∈E(i)
5.1 ModelingandProblemEquivalence w(i) ≥0, ∀k ∈E, i∈V.
k
An edge-weighted undirected graph is denoted as
G(V,E,W),whereV ={1,2,··· ,N}denotesthevertex In(P ),thevariableµisusedtoshiftthezeroeigenvalue
c
set; E denotes the edge set and (i,j) ∈ E, i,j ∈ V if oftheLaplacianmatrixL withitseigenvector1.When
w
6
the optimal value is reached, λ∗ would be the smallest (P ):
2 c
eigenvalue of (cid:80) (cid:80) w(i)∗E +µ∗1. Since for any
positive semi-deifi∈nVite mk∈aEtrixk G,kit holds that ξI (cid:22) G, tr(Φ∗)=1, Φ∗ (cid:23)0, ϕ(i)∗ ≥0, tr(1Φ∗)=0, (12a)
k
whereξisthesmallesteigenvalueofG,wegettheabove tr(E Φ∗)−v(i)∗+ϕ(i)∗ =0, w(i)∗ ≥0, (12b)
constraints.Moreover,sinceλ∗isthesmallesteigenvalue k k k
2
of(cid:80)i∈V(cid:80)k∈Ewk(i)∗Ek+µ∗1,theoptimalvalueof (Pc) ϕ(ki)∗wk(i)∗ =0, ∀k ∈E(i), (12c)
can be attained. On the other hand, we can choose λ2 λ∗−(cid:88) (cid:88) w(i)∗tr(E Φ∗)=0, (12d)
smallenoughandµlargeenoughtomakethefirstmatrix 2 k k
i∈Vk∈E(i)
inequalityconstraintstrictlyholds,hencestrongduality
holds for (Pc). In order to solve (Pc) distributedly, we λ∗2I−µ∗1−(cid:88) (cid:88) wk(i)∗Ek (cid:22)0, (12e)
i∈Vk∈E(i)
edge 1 edge 2 (cid:88) w(i)∗ =1, (12f)
k
k∈E(i)
1 2 3
whiletheKKTconditionsof (P )foralli∈V reads
D
tr(Φ(i)∗)=1,Φ(i)∗ (cid:23)0,ϕ(i)∗ ≥0,tr(1Φ(i)∗)=0,
Fig.3.Edgeweightisthesumofweightcontributionbythe k
nodes.
(13a)
considerthefollowingproblem
tr(E Φ(i)∗)−v(i)∗+ϕ(i)∗ =0, w(i)∗ ≥0, (13b)
k k k
maximize (cid:88)λ(i) ϕ(ki)∗wk(i)∗ =0,∀k ∈E(i), (13c)
{µ(i)},{wk(i)},{Z(i)} i∈V 2 λ(2i)∗− (cid:88) wk(i)∗tr(EkΦ(i)∗)
subjectto λ(i)I−µ(i)1− (cid:88) w(i)E k∈E(i)
2 k k (cid:88)
k∈E(i) + tr[(Z(i)−Z(j))Φ(i)]=0, (13d)
(cid:88)
+ (Z(i)−Z(j))(cid:22)0, j∈N(i)
j∈N(i) λ(2i)∗I−µ(i)∗1− (cid:88) wk(i)∗Ek
(cid:88) w(i) =1, k∈E(i)
k (cid:88)
k∈E(i) + (Z(i)∗−Z(j)∗)(cid:22)0, (13e)
w(i) ≥0, ∀k ∈E(i), ∀i∈V, j∈N(i)
k (PD) (cid:88) w(i)∗ =1, (cid:88) Φ(i)∗−Φ(j)∗ =0, (13f)
k
where Z(i),∀i ∈ V are symmetric matrices which can
k∈E(i) j∈N(i)
be written as Z(i) = (cid:80)N(N2+1) z(i)B , where z(i) ∈ R
l=1 l l l where Φ, ϕ and v are the Lagrange multipliers cor-
is the matrix entry and B is the basis matrix for SN. k
l respond to the matrix inequality constraint, inequality
Both are labeled by l. To be more precise, if z(i) is the
l constraints and the equality constraint of (Pc) respec-
entry located on pth row and qth column of Z(i), then tively.Similarly,Φ(i),ϕ(i)andv(i)aretheLagrangemul-
[B ] =[B ] =1andotherentriesofB remainzeros. k
l pq l qp l tipliersof (P ).
D
The idea behind the introduction of Z(i) terms is that
it will result in a consensus condition (13f) in the KKT Sincestrongdualityholdsfor(P ),thentheKKTcon-
D
conditions. ditions(13)becomesnecessaryandsufficientconditions
for optimality. Hence {λ(i)∗,µ(i)∗,{w(i)∗},Z(i)∗}, i ∈ V
2 k
It is worth noticing that the Slater’s conditions holds solves (P ), if and only if there exists Lagrange multi-
D
alsofor(PD)whichimpliesstrongduality. pliers{Φ(i)∗}and{ϕ(i)∗}suchthat(13)holds.
k
Proposition10 If {λ(i)∗,µ(i)∗,{w(i)∗},Z(i)∗}, i ∈ V Meanwhile, (13f) implies Φ(i)∗ = Φ(j)∗ for all i,j ∈ V.
2 k
solves (P ), then λ∗ = (cid:80) λ(i)∗, µ∗ = (cid:80) µ(i)∗ This means that (12a)-(12b) are equivalent to (13a)-
D 2 i∈V 2 i∈V (13b).Further,byadding(13d)and(13e)foreachnode
solves (P ).
c i∈V,thetermstr[(Z(i)−Z(j))Φ(i)]andZ(i)∗−Z(j)∗will
disappear. Denote λ∗ = (cid:80) λ(i)∗, µ∗ = (cid:80) µ(i)∗,
2 i∈V 2 i∈V
wewillget(12).Sincestrongdualityholdsfor(P ),then
c
PROOF. Byfollowingthesameprocedureintroduced KKT conditions is necessary and sufficient conditions
inSection2,itisnothardtogettheKKTconditionsof foroptimality.Hencethestatementfollows. (cid:50)
7
5.2 RelaxingSDPintoNP Remark12 Thoughconvex,f (X)isnotastrictlycon-
ε
vexfunction.Forall0≤α≤1,itholdsthat
(P ) can be solved distributedly by using a similar
D
methodas(Zhang&Hu,2016)whenthegraphisregu- αfε(I)+(1−α)fε(2I)=αεln(Ne1ε)+(1−α)εln(Ne2ε)
lar.However,wewouldliketoconsiderarbitrarygraphs, =εlnN +2−α
notonlyregulargraphs.Inordertoapplytheprojected
gradient flow dynamics to solve (P ), the problem is
D 2−α
relaxedintoanNPfirst. fε(αI+(1−α)2I)=fε((2−α)I)=εln(Ne ε )
=εlnN +2−α
Now we introduce the convex function proposed by Hence αf (I) + (1 − α)f (2I) = f (αI + (1 − α)2I),
ε ε ε
Nesterov (2007) which can be used to approximate f (X)isnotstrictlyconvex. (cid:50)
ε
the largest eigenvalue of a symmetric matrix. Given
X ∈SN,functionf :SN (cid:55)→Randreads
ε 5.3 ProjectedDynamicsandNumericalExamples
N
f (X)=εlntr(eX/ε)=εln[(cid:88)eλi(X)/ε], (14) Now we apply the projected dynmaical system to
ε
solve (P ). To abbreviate the notation, denote x =
i=1 c
(cid:2)x(1)T,··· ,x(N)T(cid:3)T, where x(i) =(cid:2)µ(i),{w(i)},{z(i)}(cid:3)T
k l
anditsderivativewithrespecttoX reads andv =(cid:2)v(1),··· ,v(N)(cid:3)T.
N N
(cid:88) (cid:88) Theprojecteddynamicsisgivenby
∇ f (X)=[ eλi(X)/ε]−1[ eλi(X)/εu uT], (15)
X ε i i
i=1 i=1 ∂L (x,v)
µ˙(i) =(cid:104)∇ f (X(i)),1(cid:105) =− A , (16a)
X(i) ε M ∂µ(i)
where(λ (X),u )areeigen-pairsofX with(cid:107)u (cid:107)=1,∀i.
i i i
It has been proved by Nesterov (2007) that λmax(X)≤ w˙k(i) =ΠR+(cid:0)wk(i),(cid:104)∇X(i)fε(X(i)),Ek(cid:105)M −v(i)
fε(X)≤λmax(X)+εlnN.Hencewhenεissufficiently −( (cid:88) w(i)−1)(cid:1)
small,fε(X)≈λmax(X). l
l∈E(i)
Nnoottheintghebumta−trλi(xi)iinsetqhueallaitrygecsotnesigtreanivnatluieno(fPtDhe)mmaetarnixs =ΠR+(wk(i),−∂L∂Aw((xi),v)), ∀k ∈E(i) (16b)
2 k
−µ(i)1 − (cid:80)k∈Ewk(i)Ek + (cid:80)j∈N(i)(Z(i) − Z(j)) (Boyd z˙l(i) =− (cid:88) (cid:104)∇X(i)fε(X(i))−∇X(j)fε(X(j)),Bl(cid:105)M
& Vandenberghe, 2004). Hence when choosing ε small,
j∈N(i)
SDP(PD)canberelaxedintothefollowingNP ∂L (x,v)
=− A , (16c)
(cid:88) ∂z(i)
minimize f (X(i)) l
{µ(i)},{wk(i)},{Z(i)} i∈V ε v˙(i) = (cid:88) w(i)−1= ∂LA(x,v), (16d)
subjectto (cid:88) w(i) =1 (NP) k∈E(i) k ∂v(i)
k
k∈E(i)
w(i) ≥0, ∀k ∈E(i), whereLA(x,v)=(cid:80)i∈V{fε(X(i))+v(i)((cid:80)k∈E(i)wk(i)−
k
1)+1((cid:80) w(i)−1)2}and∇ f (X(i))isgivenby
2 k∈E(i) k X(i) ε
whereX(i) =−µ(i)1−(cid:80) w(i)E +(cid:80) (Z(i)− (15).
k∈E(i) k k j∈N(i)
Z(j)) to abbreviate the notation. It is clear that the
Corollary13 The system (16) is well-defined and
Slater’s condition also holds for (NP) and hence strong
asymptotically stable with x converging to one of the
dualityholds.Therefore,KKTconditionsbecomesnec-
optimalsolutions.
essaryandsufficientconditionsfor(NP)problem.
Remark11 (NP) explains the reason of the introduc-
tion of µ in (Pc) instead of writing the constraint as PROOF. fε(X) has a Lipschitz(cid:80)continuous gradient
λ (I − 11) (cid:22) (cid:80) (cid:80) w(i)E as in (Ghosh & withrespecttoxgiventhatX = ixiAi,whereallAi
2 N i∈V k∈E(i) k k are symmetric matrices (Nesterov, 2007). By Theorem
Boyd, 2006). A λ2I term is needed for the relaxation. 2.5 in (Nagurney & Zhang, 2012), for any initial value
It is worth noticing that µ∗ does not necessarily equals
µ(i)(0) ∈ R, w(i)(0) ∈ R , and v(i)(0) ∈ R, there ex-
toλ∗/N in (P ).Infact,any(λ∗,µ∗,(cid:8)w(i)∗(cid:9))suchthat k +
2 c 2 k istsauniqueCarath´eodorysolutionwhichcontinuously
µ∗ ≥λ∗2/N isanoptimalsolutionfor (Pc). (cid:50) depends on the initial value. Therefore the system (16)
8
is well-defined and by Theorem 6, the system (16) is
1.2
asymptotically stable with x converging to the optimal X: 18.83
solution. (cid:50) Y: 0.9997
1
Tdyhnoaumghicsit(1se6e)m, sontehahtaswhtoendsoimeuiglaentivnaglutehedepcroomjepcotseid- eulaV thg0.8 www112(((122)))
tion on X(i) to compute ∇X(i)f(X(i)) at each time ieW e0.6 w2(3) XY:: 108.5.83
step. However, since the factors eλi(X(i))/ε decrease gd
E0.4
very rapidly, the gradient numerically only depends on
few largest eigenvalues and correpondant eigenvectors
(Nesterov,2007).Moreprecisely,forsufficientlysmallε, 0.2
0 5 10 15 20
∇ f(X(i)) ≈ 1 (cid:80) u(i)u(i)T, where u(i) is the eigen- t(s)
X(i) η j j j j
vector corresponds to the largest eigenvalue, η is the
Fig.5.Theevolutionoftheedgeweights
multiplicity of the largest eigenvalue of X(i). Extreme
eigenvalues will converge first in numerical methods
such as Lanszos scheme, hence one does not have to do
3 2
the entire eigenvalue decomposition and the numerical
complexityisreduced.
4 1
Example14 Werunasimplenumericalexampleusing
the graph illustrated in Fig. 3 to show that the variables 5 10
do converge to the optimal solutions. We know that the
optimal solution of (P ) is w(1)∗ = w(3)∗ = 1, w(2)∗ =
c 1 2 1 6 9
w(2)∗ = 0.5 and λ∗ = 1.5. Fig. 4 and 5 show that the
2 2
7 8
10
Fig.6.GraphTopology
5
X: 18.69
Y: 1.5
)i(62 0 50
V
i P
0
X: 17.15
-5 -50 optimal Y: 1.061
value:
-100 1.1406
-10
0 5 10 15 20 -150 6(i)
t(s) i 2
6$ 2V
-200 P2
Fig.4.Theevolutionoftheobjectivefunction
-250
objectivefunctionvalueandtheedgeweightsconvergeto 0 5 10 15 20
theoptimalsolution. (cid:50) t(s)
Example15 Now we consider a more complicated Fig.7.Theevolutionoftheobjectivefunction
graphgeneratedbytennodesasillustratedinFig.6The
evolution of the objective function is illustrated in Fig.
and difference between the aforementioned continuous-
7. It shows that the objective function converges to the
optimalvalueasymptotically. (cid:50) timealgorithmandproximalgradientmethodisfurther
explained.Moreover,theproblemofdistributedlymaxi-
mizingthealgebraicconnectivityofanundirectedgraph
6 Conclusion byadjustingtheedgeweights(communicationgainsfor
each channel) of each nodes is considered. The original
In this paper, a projected primal-dual gradient flow of SDP problem is relaxed into an NP problem and then
augmentedLagrangianispresentedtosolveconvexopti- the aforementioned projected dynamical system is ap-
mizationproblemswhicharenotnecessarilystrictlycon- plied to solve the NP. Numerical examples show that
vex.Weshowthattheprojecteddynamicalsystemcon- theprojecteddynamicalsystemsconvergetooneofthe
verges to one of the optimal solutions. The connection optimalsolutions.
9
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