Table Of Content

Distributedconvexoptimizationviacontinuous-time ⋆ coordinationalgorithmswithdiscrete-timecommunication SolmazS. Kiaa JorgeCortésb SoniaMart´ınezb 4 1 aDepartment of Mechanical and Aerospace Engineering, University ofCalifornia,Irvine 0 2 bDepartment ofMechanical andAerospace Engineering, University of California,San Diego g u A Abstract 2 2 This paper proposes a novel class of distributed continuous-time coordination algorithms to solve network optimization problemswhosecostfunctionisasumoflocalcostfunctionsassociatedtotheindividualagents.Weestablishtheexponential ] convergence of the proposed algorithm under (i) strongly connected and weight-balanced digraph topologies when the local C costs are strongly convex with globally Lipschitz gradients, and (ii) connected graph topologies when the local costs are O strongly convex with locally Lipschitz gradients. When the local cost functions are convex and the global cost function is strictly convex,we establish asymptotic convergence underconnected graph topologies. Wealso characterize thealgorithm’s . h correctness under time-varying interaction topologies and study its privacy preservation properties. Motivated by practical t considerations, we analyze thealgorithm implementation with discrete-time communication. We provide an upperbound on a thestepsizethatguaranteesexponentialconvergenceoverconnectedgraphsforimplementationswithperiodiccommunication. m Building on this result, we design a provably-correct centralized event-triggered communication scheme that is free of Zeno [ behavior.Finally,wedevelopadistributed,asynchronousevent-triggeredcommunicationschemethatisalsofreeofZenowith asymptoticconvergenceguarantees. Severalsimulations illustrate ourresults. 3 v 2 Keywords: cooperative control, distributedconvexoptimization, weight-balanced digraphs, event-triggered control. 3 4 4 . 1 Introduction Literature review: In distributed convex optimiza- 1 0 An important class of distributed convex optimization, most coordination algorithms are time-varying, 4 consensus-baseddynamics[Boydetal.,2010,Duchietal., tion problems consists of the (un-)constrained net- 1 2012, Johanssonetal., 2009, Nedić andOzdaglar, work optimization of a sum of convex functions, each : 2009, ZhuandMart´ınez, 2012] implemented in dis- v one representing a local cost known to an individual crete time. Recent work [GharesifardandCortés, 2014, i agent. Examples include distributed parameter estima- X LuandTang,2012,WangandElia,2011,Zanellaetal., tion[Rametal.,2010,WanandLemmon,2009],statis- 2011]hasintroducedcontinuous-timedynamicalsolvers r tical learning [Boydetal., 2010], and optimal resource a whose convergence properties can be analyzed via allocation over networks [MadanandLall, 2006]. To classical stability analysis. This has the added advan- find the networkoptimizers, we propose a coordination tage of facilitating the characterization of properties modelwhereeachagentrunsapurelylocalcontinuous- such as speed of convergence, disturbance rejection, time evolution dynamics and communicates at discrete and robustness to uncertainty. WangandElia [2011] instantswithitsneighbors.Wearemotivatedbythede- establish asymptotic convergence under connected sire of combining the conceptual ease of the analysis of graphs and GharesifardandCortés [2014] extend the continuous-timedynamicsandthepracticalconstraints design and analysis to strongly connected, weight- imposed by real-time implementations. Our design is balanced digraphs. The continuous-time algorithms basedonanovelcontinuous-timedistributedalgorithm in [LuandTang, 2012, Zanellaetal., 2011] require whose stability can be analyzedthrough standard Lya- twice-differentiable, strictly convex local cost func- punovfunctions. tions to make use of the inverse of their Hessian and need a careful initialization to guarantee asymptotic ⋆ convergence under undirected connected graphs. The Corresponding author:S.S.Kia novelclassofcontinuous-timealgorithmsproposedhere Emailaddresses: solmaz@uci.edu (Solmaz S.Kia), cortes@ucsd.edu (Jorge Cortés), soniamd@ucsd.edu (upon which our implementations with discrete-time communicationarebuilt)do notsufferfromthe limita- (SoniaMart´ınez). Preprintsubmitted toAutomatica 25August 2014 tions discussed above and have, under some regularity 2 Preliminaries assumptions, exponential convergence guarantees. Our Inthissection,weintroduceournotationandsomebasic work here also touches, albeit slightly, on the concept concepts from convex functions and graph theory. Let of privacy preservation, see e.g., [Weeraddanaetal., RandZ denote,respectively,thesetofrealandnon- 2013,Yanetal.,2013]forrecentworksinthecontextof ≥0 negative integer numbers. We use () to represent the distributed multi-agent optimization. Regarding event- ℜ · realpart of a complex number. The transpose of a ma- triggered control of networked systems, recent years trix A is A⊤.We let 1 (resp. 0 ) denote the vectorof haveseenanincreasingbodyofworkthatseekstotrade n n nones(resp.n zeros),anddenotebyI the n niden- computation and decision making for less communica- n tion, sensing or actuator effort, see e.g. [Heemels etal., titymatrix.WeletΠn =In− n11n1⊤n.Whenc×learfrom 2012, MazoandTabuada, 2011, Wang andLemmon, the context, we do not specify the matrix dimensions. 2011].Closesttotheproblemconsideredhereareworks ForA Rn×m andB Rp×q,weletA Bdenotetheir ∈ ∈ ⊗ that study event-triggered communication laws for av- Kronecker product. For u Rd, u = √u⊤u denotes ∈ k k erage consensus, see e.g., [Dimarogonasetal., 2012, the standard Euclidean norm. For vectors u , ,u , 1 m ··· Garciaetal., 2013, NowzariandCortés, 2014]. The u = (u , ,u ) is the aggregated vector. In a net- 1 m ··· strategies proposed in [WanandLemmon, 2009] save workedsystem,wedistinguishthelocalvariablesofeach communication effort in discrete-time implementations agentbya superscript,e.g.,xi isthe state ofagenti.If by using local triggering events but are not guaranteed pi Rd is a variable of agent i, the aggregated pi’s of to avoid Zeno behavior, i.e., an infinite number of trig- the∈network of N agents is p = (p1, ,pN) (Rd)N. gered events in a finite period of time. Our goal is to Forconvenience,we use r RN andR··· RN×(∈N−1), combine the best of both approaches by synthesizing ∈ ∈ 1 provably-correct continuous-time distributed dynam- r= 1 , r⊤R=0, R⊤R=I , RR⊤=Π . (1) N N−1 N ical systems which only require communication with √N neighbors at discrete instants of time. We are particularly interested in the opportunistic determination of A differentiable function f : Rd R is strictly convex this communicationtimes viaeventtriggeringschemes. overaconvexsetC Rdiff(z x→)⊤( f(z) f(x))> ⊆ − ∇ −∇ Statementof contributions:We proposea novelclassof 0forallx,z C andx=z,anditism-stronglyconvex continuous-time, gradient-based distributed algorithms (m > 0) iff ∈(z x)⊤( 6 f(z) f(x)) m z x 2, for network optimization where the global objective for all x,z C−. A fun∇ction f−:∇Rd R≥d iskLip−schkitz ∈ → functionisthesumoflocalcostfunctions,oneperagent. withconstantM >0,orsimplyM-Lipschitz,overaset We prove that these algorithms converge exponentially C Rd iff f(x) f(y) M x y , for x,y C. ⊂ k − k ≤ k − k ∈ under strongly connected and weight-balanced agent Foraconvexfunctionf withM-Lipschitzgradient,one interactions when the local cost functions are strongly has f(z) f(x) 2 M(z x)⊤( f(z) f(x)) convexandtheirgradientsaregloballyLipschitz.Under forakll∇x,z −C∇ Rd.k ≤ − ∇ −∇ connected, undirected graphs, we establish exponential ∈ ⊂ We briefly review basic concepts from algebraic graph convergence when the local gradients are just locally theoryfollowing[Bulloetal.,2009].Adigraph, isapair Lipschitz, and asymptotic convergence when the local = ( , ), where = 1,...,N is the node set and cost functions are simply convex and the global cost G V E V { } is the edge set.An edgefromi to j,denoted function is strictly convex. We also study convergence E ⊆V ×V by (i,j), means that agent j can send information to under networks with time-varying topologies and char- agenti.Foranedge(i,j) , iis calledanin-neighbor acterizethetopologicalrequirementsonthecommunica- ∈E ofj andj iscalledanout-neighbor ofi.Agraphisundi- tiongraph,algorithmparameters,andinitialconditions rected if (i,j) anytime (j,i) . A directed path necessaryforanagenttoreconstructthelocalgradients ∈ E ∈ E is a sequence of nodes connected by edges. A digraph of local cost functions of other agents. Our technical is strongly connected if for every pair of nodes there is approachbuildsontheidentificationofstrictLyapunov a directed path connecting them. A weighted digraph is functions. The availability of these functions enable a triplet = ( , ,A), where ( , ) is a digraph and our ensuing design of provably-correctcontinuous-time A RN×GN is aVwEeighted adjaceVncEy matrix such that implementations with discrete-time communication. In a ∈> 0 if (i,j) and a = 0, otherwise. A weighted particular, for networks with connected graph topolo- ij ij digraphisundir∈ecEted ifa =a foralli,j .Werefer gies,weobtainanupperboundonthesuitablestepsizes ij ji ∈V to a stronglyconnected andundirected graphas a con- that guarantee exponential convergence under periodic nectedgraph.Theweightedin-andout-degrees ofanode communication.Buildingonthisresult,wedesignacen- i are, respectively, di = ΣN a and di = ΣN a . tralized, synchronous event-triggered communication in j=1 ji out j=1 ij scheme with an exponential convergence guarantees A digraph is weight-balanced if at each node i , ∈ V andZeno-freebehavior.Finally,we developa Zeno-free the weighted out-degree and weighted in-degree coin- asynchronous event-triggered communication scheme cide (although they might be different across different whose execution only requires agents to interchange nodes). Any undirected graph is weight-balanced. The informationwiththeirneighborsandestablishitsexpo- (out-)Laplacian matrix isL=Dout A,whereDout = nential convergence to a neighborhood of the network Diag(d1 , ,dN ) RN×N. Note−that L1 = 0. A optimizer.Severalsimulationsillustrateourresults. digraphouits·w··eighotu-tbal∈anced iff 1TL = 0 iff SNym(L) = N 2 (L+L)/2 is positive semi-definite. Based on the struc- in[GharesifardandCortés, 2014,Wang andElia,2011] tureofL,atleastoneoftheeigenvaluesofLiszeroand require the communication of the corresponding vari- therestofthemhavenonnegativerealparts.Wedenote ables in both x and v. The synthesis of algorithm (4) the eigenvalues of L by λ ,...,λ , where λ = 0 and is inspired by the following feedback control consid- 1 N 1 (λ ) (λ ),fori<j,andtheeigenvaluesofSym(L) erations. In (4b), each agent follows a local gradient i j ℜ ≤ℜ by λˆ ,...,λˆ . For a strongly connected and weight- descentwhiletryingtoagreewithitsneighborsontheir 1 N balanceddigraph, zero is a simple eigenvalue of both L estimate of the final value. However, as the local gra- and Sym(L). In this case, we order the eigenvalues of dients are not the same, this dynamics by itself would Sym(L) asλˆ =0<λˆ λˆ λˆ andwehave never converge. Therefore, to correct this error, each 1 2 ≤ 3 ≤···≤ N agentusesanintegralfeedbacktermvi whoseevolution 0<λˆ I R⊤Sym(L)R λˆ I. (2) isdrivenby the agentdisagreementaccordingto(4a). 2 N ≤ ≤ Ouranalysisofthe algorithmconvergenceis structured Notice that for connected graphs λˆ = λ for i 1, ,N . For convenience, we definie L = Li I an∈d intwoparts,dependingonthedirectedcharacterofthe {Π=··Π· }I todealwithvariablesofdimensio⊗ndd N. interactions. Section 4.1 deals with strongly connected N ⊗ d ∈ andweight-balanceddigraphsandSection4.2dealswith connected graphs. In each case, we identify conditions 3 ProblemDefinition on the agent cost functions that guarantee asymptotic Consider a network of N agents interacting over a convergence.Giventhechallengesposedbydirectedin- strongly connected and weight-balanced digraph . formation flows, it is not surprising that we can estab- G Eachagenti hasadifferentiablelocalcostfunction lish stronger results under less restrictive assumptions fi : Rd R.∈TVhe global cost function of the network, forthe caseofundirectedtopologies. whichwe→assumetobestrictlyconvex,isf =ΣN fi(x). i=1 4.1 StronglyConnected,Weight-Balanced Digraphs Our aim is to design a distributed algorithm such that eachagentsolves Here, we study the convergence of the distributed op- min f(x) timization algorithm (3) over strongly connected and x∈Rd weight-balanced digraph topologies. We first consider using only its own local data and exchanged informa- thecasewheretheinteractiontopologyisfixed,andthen tionwithitsneighbors.Weassumetheaboveoptimiza- discussthetime-varyinginteractiontopologies.Thefol- tionproblemis feasible (which,together withthe strict lowingresultidentifiesconditionsonthelocalcostfunc- convexityoff,impliestheuniquenessoftheglobalopti- tions fi N andthe parameterβ toguaranteethe ex- mizer,whichwedenotex⋆ Rd).Wearealsointerested { }i=1 ponential convergence of (3) to the solution of the dis- ∈ in characterizing the privacy preservation properties of tributedoptimizationproblem. thealgorithmicsolutiontothisdistributedoptimization Theorem 1 (Convergence of (3) over strongly con- problem.Specifically,weaimtoidentifyconditionsguar- nectedandweight-balanceddigraphs): Let beastrongly anteeing that no information about the local costfunc- G connected and weight-balanced digraph. Assume each tion of an agent is revealed to, or can be reconstructed fi, i , is mi-strongly convex, differentiable, and by,any otheragentinthe network. its gra∈dieVnt is Mi-Lipschitz on Rd. Given α > 0, 4 Distributed Continuous-Time Algorithm for m = min m1,...,mN and M = max M1,...,MN , { } { } Convex Optimization letβ,φ>0satisfy φ+1>4M and In this section, we provide a novelcontinuous-time dis- γ=α2(φ+1)m+9βλˆ φα 4α2(Mm+(φ+1)2)>0, (5) 2 − tributed coordination algorithm to solve the problem statedinSection3andanalyzeindetailitsconvergence Then, for each i , starting from xi(0),vi(0) Rd prov˙pie=rtiαesβ.ForNj=i∈1aVija(xnid−wxitjh),α,β>0,consider (3a) lxweiis(tsht)thP→anNi=x1⋆veix(p0o)n=e∈n0tVdia,lltyhefaasltgoarsitthm→(∞3)woivtehraGram∈teakneos 7 1 x˙i = αXfi(xi) β N a (xi xj) vi, (3b) min , γ /(2λ¯F). (6) − ∇ − j=1 ij − − {16 9 } X Here,λ¯F is themaximumeigenvalue of Innetworkvariablesx,v (Rd)N,thealgorithmreadsas 1α(φ+1)I 0 0 v˙ =αβLx∈, (4a) F= 19 0 d α(φ+1)I I . (7) x˙ =−α∇f˜(x)−βLx−v. (4b) 2 0 I(N−1()Nd−1)d α1(IN(N−−11)d)d   Here, f˜: (Rd)N R is defined by f˜(x) = ΣN fi(xi). → i=1 PROOF. For weight-balanced digraphs, we have This algorithm is distributed because each agent only 1⊤L=0.Thus,leftmultiplying(4a)by1⊤ I resultsin needs to receive information from its out-neighbors N N⊗ d about their corresponding variables in x. In con- N v˙i=0 N vi(t)= N vi(0)=0, t 0. (8) trast, the continuous-time coordination algorithms i=1 ⇒ i=1 i=1 ∀ ≥ X X X 3 Next,weobtaintheequilibriumpointof(3),(x¯,v¯),from Next, we show that under φ+1 > 4M and (5), V˙ is negative definite. Invoking the assumptions on the local cost functions in the statement, and using the M- 0=αβLx¯, (9a) Lipschitzness of fãnd the m-strongly convexity of f˜ 0= α f˜(x¯) βLx¯ v¯. (9b) alongwith R⊤ ∇I =1 and z = y ,we have − ∇ − − k ⊗ dk k k k k (R⊤ I )h 2 h 2 My⊤h, (15a) Given(9a),x¯belongstothenull-spaceofL.Forstrongly k ⊗ d k ≤k k ≤ connecteddigraphsthenull-spaceofLisspannedby1N. y⊤h m y 2 =m z 2. (15b) Thus,x¯ =1 θ,whereθ Rd.Leftmultiplying (9b) ≥ k k k k N ⊗ ∈ by1⊤ I andusing(8),weobtain0= N fi(x¯i). Giventhese relationsandinvoking (2),wehave ThenN,t⊗hedoptimalitycondition f(x⋆)=0i=a1lo∇ngwith ∇f(θ)= Ni=1∇fi(θ)imply ∇ Pd V˙ ≤−α2((φ+1)9−4M)mz⊤z−176w⊤2:Nw2:N P x¯i =x⋆, i . φαβλˆ z⊤ z + 4α2(1+φ)2z⊤ z ∈V − 2 2:N 2:N 9 2:N 2:N 3 2α 2α(φ+1) Substituting this valuein(9b),weobtain w + (R⊤ I )h+ z ) 2. −k4 2:N 3 ⊗ d 3 2:N k v¯i = α fi(x⋆), i . (10) − ∇ ∈V Since z⊤z = z⊤z + z⊤ z , it follows that V˙ < 1 1 2:N 2:N Tostudythestabilityof (3),wetransfertheequilibrium min 7 ,1γ p 2 < 0, where γ is a shorthand nota- − {16 9 }k k pointtotheoriginandthenapplyachangeofvariables tion for the expression in (5). Thus, z 0 as t , equivalently xi x⋆, for all i , is e→xponentia→l w∞ith u=v−v¯, y=x−x¯, (11a) ratenolessthan→(6)(cf.[Khalil,∈20V02,Theorem4.10]).✷ u=([r R] I )w, y=([r R] I )z, (11b) ⊗ d ⊗ d InTheorem1,therequirement N vi(0)=0 istriv- whereweused(1).Wepartitionthenewvariablesasfol- i=1 d lows:w=(w ,w )andz=(z ,z ),wherew ,z ially satisfiedby eachagentwith vi(0)=0d. This is an 1 2:N 1 2:N 1 1 P Rd.Inthese new variables,the algorithm(3)readsas∈ advantagewithrespecttothecontinuous-timecoordina- tionalgorithmsproposedin[LuandTang,2012],which w˙1 =0d, requiresthenontrivialinitialization N fi(xi(0))= i=1∇ w˙2:N =αβ(R⊤LR⊗Id)z2:N, 0tida,liaznatdioinn [oZnanaelsltaaetetaclo.,m2m01u1n]i,cawtheidPchamreoqnugirenseitghhebionris- z˙ = α(r⊤ I )h, (12) 1 − ⊗ d andis subjectto communicationerror. z˙2:N =−α(R⊤⊗Id)h−β(R⊤LR⊗Id)z2:N −w2:N, Remark2 (Designparametersin(3)):Weprovidehere severalobservationsontheroleofthedesignparameters where αandβ.First,notetherealwaysexistα,βsatisfying(5), e.g., any β > 4(φ+1)2α/(9φλˆ ). The determination of h= f˜(y+x¯) f˜(x¯). (13) 2 ∇ −∇ theseparameterscanbeperformedbyindividualagents iftheyknowanupperboundonM,alowerboundonm, Note that the first equation in (12) corresponds to the andhaveknowledgeofλˆ ,eitherthroughadedicatedal- constantofmotion(8).Tostudythestabilityintheother 2 gorithmtocomputeit,seee.g.,Yangetal.[2010],oruse variables,considerthe candidate Lyapunovfunction alowerboundonit,seee.g.Mohar[1991].We haveob- 1 φα servedinsimulationthat(5)is onlysufficientandthat, V(z,w2:N)= α(φ+1)z1⊤z1+ z2:N⊤z2:N (14) in fact, the algorithm (3) converges for any positive α 18 2 1 andβ inournumericalexamples.Althoughnotevident + (αz2:N +w2:N)⊤(αz2:N +w2:N), in(6),onecanexpectthelargerαandβ are,thehigher 2α the rate of convergence of the algorithm (3) is. A coefficient α > 1 can be interpreted as a way of increasing with φ>0 as in the statement. Note that V λ¯F p 2, the strong convexity coefficient of the local cost func- ≤ k k withp=(z,w ).TheLiederivativeofV along(12)is 2:N tions.Acoefficientβ >1canbeinterpretedasameans 1 7 of increasing the graph connectivity. The relationship V˙ =− 9α2(φ+1)y⊤h− 16w⊤2:Nw2:N between these parameters and the rate of convergence of (3) is more evidentfor quadraticlocalcostfunctions −φαβz2:N⊤(R⊤Sym(L)R⊗Id)z2:N fi(x)= 1(x⊤x+x⊤ai+bi), i . In this case,the al- 4 4 2 ∈V + α2 (R⊤ I )h 2+ α2(1+φ)2z⊤ z gorithm (3) is a linear time-invariant system where the 9 k ⊗ d k 9 2:N 2:N eigenvaluesofthesystemmatrixare α,withmultiplic- 3w + 2α(R⊤ I )h+ 2α(φ+1)z ) 2. ityofNd,and βλi,i (λi’sareth−eeigenvaluesofL), −k4 2:N 3 ⊗ d 3 2:N k with multiplic−ity d. Th∈eVrefore, (3) converges regardless 4 of the value of α, β > 0 with an exponential rate equal for all t t⋆, it canuse the time history ofxj(t) to nu- ≤ tomin α,β (λ2) .Whendiscussingdiscrete-timecom- mericallyreconstructx˙j(t⋆).Becauseiisthein-neighbor { ℜ } munication,sometrade-offsariseregardingthechoiceof of j and its out-neighbors, it can use its knowledge of the parameters,aswe explainlaterin Section5. a ,k toreconstruct N a (xj(t) xk(t))forall Remark3 (Semiglobal convergence of (3) under loca•l tjk t⋆.∈AVgenticanreconstrukc=t1vjjk(t)from−(3a)uniquely gradientsthatarelocallyLipschitz):Theconvergencere- as≤it knows vj(0). Then,Pagent i has all the elements sult in Theorem 1 is semiglobal [Khalil, 2002] if the lo- to solve for fj(xj(t⋆)) in (3b). The lack of knowledge cal gradients are only locally Lipschitz or, equivalently, ∇ about any of this information would prevent i from re- Lipschitzoncompactsets.Infact,onecanseefromthe constructingexactlythe localgradientofj. ✷ proofofthe result that, for any compactsetcontaining the initial conditions xi(0) Rd andvi(0)=0 , i , d ∈ ∈V 4.2 ConnectedGraphs onecanfindφ>0andβ >0sufficientlylargesuchthat the compactset is containedin the regionof attraction Here,westudytheconvergenceofthealgorithm(3)over ofthe equilibriumpoint. connectedgraphtopologies.Whiletheresultsofthepre- • Next,westudytheconvergenceof (3)overdynamically vious section are of course valid for these topologies, changing topologies. Since the proof of Theorem 1 re- here,usingthestructuralpropertiesoftheLaplacianma- liesonaLyapunovfunctionwithnodependencyonthe trix of undirected graphs, we establish the convergence system parameters and its derivative is upper bounded of (3)foralargerfamilyoflocalcostfunctions.Indoing byaquadraticnegativedefinitefunction,wecanreadily so,wearealsoabletoanalyticallyestablishconvergence extend the convergence result to dynamically changing foranyα,β >0,asweshownext. networks.The proofdetails areomittedforbrevity. Theorem 6 (Exponential convergence of (3) over con- Proposition 4 (Convergence of (3) over dynamically nected graphs): Let be a connected graph. Assume the changinginteractiontopologies): Let beatime-varying local cost function fGi, i , is mi-strongly convex and digraph which is stronglyconnected aGnd weight-balanced differentiableonRd,and∈itVsgradientislocallyLipschitz. at all times and whose adjacency matrix is uniformly Then, for any α,β > 0 and each i , starting from bounded and piecewise constant. Assume the local cost xi(0),vi(0) Rdwith N vi(0)=0∈,tVhealgorithm (3) function fi, i , is mi-strongly convex, differentiable, over satisfi∈es xi(t) xi=⋆1as t exponentially fast. anditsgradien∈tVisMi-Lipschitz onRd.Givenα>0,let G →P →∞ β,φ>0satisfy φ+1>4M and (5)withλˆ replaced by 2 PROOF. We usethe equivalentrepresentation(12)of (λˆ ) =min λˆ (L ) , where is the index set of all 2 min 2 p the algorithm (3) obtained in the proof of Theorem 1. p∈P{ } P possible realizations of . Then, for each i ,starting Considerthe followingcandidate Lyapunovfunction from xi(0),vi(0) RdGwith N vi(0)=∈0 V, the algo- 1 φα rithm (3)over m∈akes xi(t) i=x1⋆ exponentdiallyfast as V(z,w2:N)= 2α(φ+1)z⊤1z1+ 2 z⊤2:Nz2:N (16) t . G P→ 1 →∞ + (αz2:N +w2:N)⊤(αz2:N +w2:N) Ourfinalresultofthissectioncharacterizesthetopolog- 2α ical requirements on the communication graph and the + 1 (φ+1)w⊤ (R⊤LR)−1 I w , knowledgeaboutthealgorithm’sparametersandinitial 2β 2:N ⊗ d 2:N conditionsthatallowapassiveagent(i.e.,anagentthat (cid:0) (cid:1) does not interfere in the algorithmexecution) to recon- withφ 1definedbelow.Given(2),V ispositivedefinite structthelocalgradientsofotheragentsinthenetwork. and rad≥iallyunbounded and satisfies V λ¯E p 2, with Proposition 5 (Privacy preservation under (3)): Let p=(z,w2:N) andλ¯E>0is the maximum≤eigeknvkalueof G beastronglyconnectedandweightbalanceddigraph.For α,β > 0, consider any execution of the coordination al- α(φ+1)Id 0 0 gorithm (3) over starting from xi(0),vi(0) Rd with E=1 0 α(φ+1)I I . N vi(0) = 0 .GThen, an agent i can re∈construct 2 (N−1)d (N−1)d thei=lo1cal gradiendt of another agent ∈j =V i only if j and  0 I(N−1)d α1I+(φ+β1)(R⊤LR)−1⊗Id aPll its out-neighbors are out-neighbors6 of i, and agent i   knows vj(0) and a , k (here we assume that the The Lie derivativeofV alongthe dynamics(12)is jk ∈ V agent i is aware of the identity of neighbors of agent j V˙ = α2(φ+1)y⊤h φαβz ⊤(R⊤LR I )z andithasmemorytosavethetimehistoryofthedatait − − 2:N ⊗ d 2:N receives from its out-neighbors). w ⊤w αw ⊤(R⊤ I )h. − 2:N 2:N − 2:N ⊗ d PROOF. Consideranarbitrarytimet⋆.Letibeanin- Next, we show that V˙ is upper bounded by a negative neighborofagentj andallofits out-neighbors.The al- definite function.We startbyidentifying acompactset gorithm(3)requireseachagenttocommunicateitscom- whose definition is independent of φ and contains the ponentofxtotheirin-neighbors.Sinceagentihasmem- set = (z,w ) RNd R(N−1)d V(z,w ) 0 2:N 2:N S { ∈ × | ≤ orytosaveinformationitreceivesfromitsout-neighbors V(z(0),w (0)) . For any given initial condition, 2:N } 5 let ρ = 1αz (0)⊤z (0) + (α+1)z (0)⊤z (0) + on Rd, and the global cost function f is strictly con- 0 2 1 1 2 2:N 2:N ( 1 + 1 + 1)w (0)⊤w (0)anddefine ¯ = z vex. Then, for any α,β > 0 and each i , starting sRe2Ntβλdis2|co12mα2αzp⊤1aczt12.+No412t:αeNzt⊤2h:Natz(2z:N2,:Nw≤2:Nρ0)}. O0bsiemrvpelSiet0shazt{th¯i∈0s frriothmmx(i3()0)o,vveir(0G)m∈aRkedxwii(tth)P→Nix=⋆1vasi(t0)→=∞∈0.dV, the algo- ∈S ∈S becauseV(z(0),w (0)) (φ+1)ρ and 2:N 0 ≤ PROOF. We use again the equivalent representa- 1 1 tion (12) of the algorithm (3) obtained in the proof 2α(φ+1)z⊤1z1+ 4α(φ+1)z⊤2:Nz2:N ≤ of Theorem 1. To study the stability of this system, considerthe followingcandidate Lyapunovfunction 1 1 2α(φ+1)z⊤1z1+ 2αφz⊤2:Nz2:N ≤V(z,w2:N). V(z,w )=1z⊤z+ 1 w ⊤ (R⊤LR)−1 I w . 2:N 2 2αβ 2:N ⊗ d 2:N Here, we used φ 1 in the first inequality. Since the (cid:16) (cid:17) ≥ change of variables (11a) and (11b) are linear, the cor- Given(2),V ispositivedefiniteandradiallyunbounded. responding x and y for z ¯ belong to compact sets, TheLie derivativeofV along (12)is givenby 0 ∈ S aswell.Then,theassumptiononthegradientsofthelo- V˙ = αy⊤( f(y+x¯) f(x¯)) βy⊤Ly calcostfunctionsimplies thatthereexistsM0 >0with − ∇T −∇T − h M0 y =M0 z ,forall z ¯0.Consequently, = α N yi⊤( fi(yi+x⋆) fi(x⋆)) βy⊤Ly. k k≤ k k k k ∈S − i=1 ∇ −∇ − k(R⊤⊗Id)hk2≤M02kzk2, ∀(z,w2:N)∈S0. To obtainXthe second summand, we have used z2:N = (R⊤ I )y and RR⊤ = I rr⊤. Since the lo- Thenwecanshow−αw2:N⊤(R⊤⊗Id)h≤ 12w⊤2:Nw2:N+ cal c⊗ostd functions are convNex−, the first summand 12α2M02z⊤z for (z,w2:N) ∈ S0. Using this inequality, of V˙ is non-positive for all y. Because the graph factthatthelocalcostfunctionsaremi-stronglyconvex is connected, the second summand is non-positive (andhence (15b)holds)andinvoking(2)we deduce with its null-space spanned by θ⊗1N, θ ∈ Rd. On this null-space, the first summand becomes V˙ ≤−α12w(φ⊤+w1)m(z+⊤1z11α+2Mz⊤22:(Nzz⊤2z:N)+−z⊤φαβzλ2z)⊤2.:Nz2:N b−eαθze⊤ro wNi=h1e(n∇θfi(=θ +0,xb⋆e)c−au∇sefi(xN⋆)), wfih(ixc)h=can fon(xly) − 2 2:N 2:N 2 0 1 1 2:N 2:N and thPe global cost function is stir=ic1t∇ly convex b∇y as- sumption. Then, the two summPands of V˙ can be zero Letφ+1=21mM02+2m1α2δ0,whereδ0>0issuchthatφ≥1 simultaneously only when yi = 0, for all i , (sinceM0doesnotdependonφ,thischoiceisalwaysfea- which is equivalent to z = 0. Thus, V˙ is neg∈atiVve sible). Then, we have V˙ ≤−21min{1,δ0}kpk2. As such, semi-definite, with V˙(z,w2:N) = 0 happening on the the Lyapunov function (16) satisfies all the conditions set = (z,w ) RNd R(N−1)d z = 0 . 2:N of [Khalil, 2002, Theorem 4.10], for the dynamics (12). NoteSthat{(12) on r∈educes t×o w˙ = |0, z˙ = 0}, 2:N 1 Therefore,the convergenceofz 0 as t , equiva- and z˙ = w S. Therefore, the only trajectory lently xi x⋆,for alli ,is ex→ponentia→l. ∞ ✷ of (122):Nthat r−ema2:iNns in is the equilibrium point → ∈V S (z = 0,z = 0,w = 0). The LaSalle invariance 1 2:N 2:N Remark7 (Boundonexponentialrateofconvergence): principle (cf. [Khalil, 2002, Theorem 4.4 and Corol- One can see from the proof of Theorem 6 that, for a lary 4.2]) now implies that the equilibrium is globally giveninitialcondition,therateofconvergenceisatleast asymptoticallystableor,inotherwords,xi x⋆, i 41(min{1,δ0})/λ¯E > 0. The guaranteed rate of conver- globallyasymptotically. → ∈✷V genceisthereforenotuniform,unlessthelocalgradients are globally Lipschitz. In this case, one recoversthe re- Remark9 (Simplification of (3)): For strictly convex sultinTheorem1but forarbitraryα, β >0. local cost functions, using the LaSalle function of the • NotethatinTheorem6thelocalLipschitznessof fiis proof of Theorem 8, one can show that the asymptotic triviallyheldiffi istwicedifferentiable.TheLya∇punov convergetothe optimizer,startingfromthe initialcon- function (16) identified in the proof of this result plays dition stated in Theorem 8, is also guaranteed for the akeyrolelaterinourstudyofthealgorithmimplemen- followingalgorithmoverconnectedgraphs tation with discrete-time communication in Section 5. N Next,westudytheconvergenceofthealgorithm(3)over v˙i = aij(xi xj), j=1 − connectedgraphswhenthelocalcostfunctionsareonly x˙i =Xfi(xi) vi. convex.Here,thelackofstrongconvexitymakesusrely −∇ − • on a LaSalle function, rather than on a Lyapunov one, toestablishasymptoticconvergencetothe optimizer. 5 Continuous-time Evolution with Discrete- TimeCommunication Theorem 8 (Asymptotic convergence of (3) over connected graphs): Let be a connected graph. Assumethe Here,weinvestigatethedesignofcontinuous-timecoor- G local cost function fi, i , is convex and differentiable dination algorithms with discrete-time communication ∈V 6 to solve the distributed optimization problem of Sec- 1 (αM +1)ζ τ= ln 1+ , tion 3. The implementation of (3) requires continuous- αM +1 αM +1+βλ √1+α2(1+ζ) N time communication among the agents. While this (cid:16) (cid:17) (20) abstraction is useful for analysis, in practical scenarios communication is only available at discrete instants where ζ2 = 2ǫ(1−ǫ)λ2min{δ,1} . Then, if ∆ (0,τ), of time. This observation motivates our study here. αβλ2Nφ+4α2λ2(1+φ)2 ∈ Throughout the section, we deal with communication the algorithm evolution starting from initial conditions topologies described by connected graphs. Our results xi(0),vi(0) Rd with N vi(0) = 0 makes xi(t) build on the discussion of Section 4, particularly the x⋆ exponenti∈ally fast as ti=1 , for alldi with a ra→te identificationofLyapunovfunctions forstability. ofnoless than 1ǫ(minP1,→δ )∞/λ¯E >0. ∈V 4 {2 } Westartbyintroducingsomeusefulconventions.Atany given time t R , let xˆj be the last known state of PROOF. We start by transferring the equilibrium ≥0 agent j t∈ransmitted to its in-neighbors. If ti point to the origin using (11a) and then apply the R den∈oteVs the times at which agent i commu{nicka}te⊂s changeofvariables(11b)to write (18)as ≥0 with its in-neighbors, then one has xî = xi(tik) for t ∈ w˙1 =0d, (21a) [tik,tik+1). Consider the next implementation of the al- w˙2:N =αβ(R⊤LR⊗Id)(z2:N +z˜2:N), (21b) gorithm(3)withdiscrete-time communication, z˙ = α(r⊤ I )h, (21c) N 1 − ⊗ d v˙i =αβ a (xî xˆj), (18a) z˙ = α(R⊤ I )h (21d) j=1 ij − 2:N − ⊗ d x˙i = αXfi(xi) β N a (xî xˆj) vi. (18b) −β(R⊤LR⊗Id)(z2:N +˜z2:N)−w2:N, ij − ∇ − j=1 − − where z˜ (t) = z (t ) z (t), for t [t ,t ), X 2:N 2:N k 2:N k k+1 − ∈ Clearly,the evolutionof (18)depends onthe sequences and h is given by (13). To study the stability of (21b)- of communication times for each agent. Here, we con- (21d),considertheLyapunovfunction(16)withφsatis- sider three scenarios. Section 5.1 studies periodic com- fying(19).ItsLiederivativecanbeboundedby(details municationschemeswhereallagentscommunicatesyn- similartotheproofofTheorem6areomittedforbrevity) chronously at ∆ intervals of time, i.e., ti = t = ∆k k k δ(1 ǫ) δǫ for all i .We provide a characterizationof the peri- V˙ − (z⊤z +z⊤ z ) (z⊤z +z⊤ z ) ods that∈guVarantee the asymptotic convergence of (18) ≤− 2 1 1 2:N 2:N − 2 1 1 2:N 2:N totheoptimizer.Ingeneral,periodicschemesmightre- (1−ǫ)w⊤ w ǫw⊤ w φαβλ (1 ǫ)z⊤ z sult in a wasteful use of the communication resources − 2 2:N 2:N−4 2:N 2:N− 2 − 2:N 2:N becauseoftheneedtoaccountforworst-casesituations 1 1 + φαβλ2 ˜z⊤ z˜ + α2(φ+1)2z˜⊤ ˜z indetermining appropriateperiods.This motivates our 4ǫλ N 2:N 2:N ǫ 2:N 2:N 2 study in Section 5.2 of event-triggered communication 1 1 schemes that tie the communication times to the net- ≤−φαβλ2(1−ǫ)z⊤2:Nz2:N − 2ǫmin{δ,2}p⊤p work state for greater efficiency. We discuss two event- 1 triggeredcommunicationimplementations,acentralized − 2(1−ǫ)min{δ,1} p⊤p−ζ−2z˜⊤2:N˜z2:N , (22) synchronousoneandadistributedasynchronousone.In both cases, we pay special attention to ruling out the (cid:0) (cid:1) where p = (z,w ) and ǫ and ζ are given in the the- 2:N presence of Zeno behavior (the existence of an infinite orem’sstatement.Observethatateachcommunication numberofupdates ina finite intervaloftime). time t , ˜z (t ) = 0, then, it grows until next com- k 2:N k k k 5.1 Periodic Communication municationattimetk+1whenitbecomeszeroagain.Our proofproceedsbyshowingthatift <t +τ,whereτ The following result provides an upper bound on the k+1 k isgivenin(20),thenwe havethe guaranteethat sizeofadmissiblestepsizesfortheexecutionof (18)over ˜z (t) <ζ p(t) , t [t ,t ), (23) connectedgraphswithperiodiccommunicationschemes. 2:N k k+1 k k k k ∈ Theorem 10 (Convergence of (18) with periodic com- (note that, from (22), this guarantee ensures that V˙ is munication): Let beaconnectedgraph.Assumethelo- G negativedefiniteforallt 0).Tothisend,westudythe cal cost function fi, i ,is mi-stronglyconvex, differ- ≥ entiable, and its gradi∈enVt is Mi-Lipschitz on Rd. Given dynamicsofq =k˜z2:Nk/kpkandfindalowerboundon the time that it takes for q to evolve from zero (recall α,β >0,consideranimplementationof (18)withagents ˜z (t )=0)to ζ.Notice that communicating over synchronously every ∆ seconds 2:N k . s0ta<rtǫin<g1atatn0d=δ >0,0i.esGu.,cthikth=attk = ∆k for all i ∈ V. Let q˙ = z˜⊤2:Nz˜2:N k˜z2:Nkp⊤p˙ (1+q)kp˙k (1+q) ˜z p − p 3 ≤ p ≤ × 2:N 1 2 1 k kk k k k k k φ= M + δ 1>0, (19) (αM+βλ √1+α2) p + w +βλ √1+α2 ˜z 2m 2mα2 − N k k k 2:Nk N k 2:Nk p k k whereM andm aregiven in Theorem 1, anddefine (αM +1)(1+q)+βλ 1+α2(1+q)2. N ≤ p 7 Here,weusedinthefirstinequalityd/dt(˜z )= z˙ uation of the triggering condition (23) requires knowl- 2:N 2:N − and z˙ p˙ and in the second one the evolution edge of the global optimizer. Our forthcoming discus- 2:N k k ≤ k k ofp˙ givenin(21b)-(21d), R⊤LR L =λ and sion shows how one can circumvent this problem. We N k k≤k k firstconsiderthedesignofcentralizedtriggersrequiring βR⊤LR global network knowledge and then discuss triggering (cid:13)"−αβR⊤LR#(cid:13)=βλN 1+α2. schemesthatonlyrelyoninter-neighborinteraction. (cid:13)(cid:13) (cid:13)(cid:13) p 5.2.1 Centralized Synchronous Implementation (cid:13) (cid:13) UsingtheCo(cid:13)mparisonLem(cid:13)ma(cf.[Khalil,2002,Lemma Here,wepresentacentralizedevent-triggeredschemeto 3.4]),weconcludethatq(t,q ) ψ(t,ψ ),whereψ(t,ψ ) determine the sequence of synchronous communication 0 0 0 isthesolutionofψ˙ =(αM+1)≤(1+ψ)+βλ √1+α2(1+ times in (18). Our discussionbuilds upon the examina- N ψ)2 satisfyingψ(0,ψ )=ψ .Then, tionoftheLiederivativeoftheLyapunovfunctionused 0 0 in the proof of Theorem 10 and the observations made q(t,0) ψ(t,0) aboveregardingthe lackof knowledgeof the globalop- ≤ (αM +1+βλ √1+α2)(eαMt+t 1) timizer. From (24), we see that an event-triggered law N = − . shouldnot employ p, but ratherrely on ˜z and z , 2:N 2:N βλN√1+α2eαMt+t+αM +1+βλN√1+α2 to be independent of x⋆. With this in mind, the exami- − nationofthe upper bound(22)onV˙ revealsthat,if The time τ for ψ(τ,0)=ζ is given by (20). Then, for {tk+1 − tk}k∈Z≥0 < τ, we have (23), and as a result k˜z2:N(t)k2 =kΠ(x(tk)−x(t))k2 from(22)wehaveV˙ < 1ǫmin δ,1 p⊤p.Thus,z 0, κ z2:N(t) 2 =κ Πx(t) 2, (25) −2 { 2} → ≤ k k k k ast ,whichisequivalenttoxi x⋆ ast ,ex- pone→nti∞allyfastwiththe rategiven→inthe stat→em∞ent.✷ whereκ isshorthandnotationfor ǫδλ +2φαβλ2ǫ2(1 ǫ) Remark11 (Dependenceof thecommunication period κ=2 2 2 − , (26) onthedesignparameters):Thevalueofτ inTheorem10 αβφλ2N +2λ2α2(1+φ)2 depends on the graph topology, the parameters of the localcostfunctions,thedesignparametersαandβ,and (here0<ǫ<1andφis givenby (19)), thenwehave the variables ǫ and δ. One can use this dependency to maximizethevalueofτ.Notethattheargumentofln(.) 1 V˙ φαβλ (1 ǫ)2z ⊤z (1 ǫ)w ⊤w (27) in (20) is a monotonically increasing function of ζ > 0. 2 2:N 2:N 2:N 2:N ≤− − −2 − Therefore,thesmallerthevalueofβ,thelargerthevalue 1 1 of τ. However, the dependency of τ on the rest of the δz1⊤z1 min δ,2φαβλ2(1 ǫ)2,(1 ǫ) p⊤p. − 2 ≤−2 { − − } parameterslistedaboveismorecomplex.Forgivenlocal cost functions, fixed network topology and fixed values Then, we can reproduce the proof of Theorem 10 and of α, β, the maximum value of ζ is when φ+1 is at its concludetheexponentialconvergencetotheoptimalso- minimumandǫλ min 1 ǫ,δ is atits maximum. 2 lution. Accordingly, the sequence of synchronous com- { − } • 5.2 Event-Triggered Communication munication times {tk}k∈Z≥0 ⊂ R≥0 for (18) should be determinedby(25).However,foratrulyimplementable This section studies the design of event-triggered com- law,oneshouldruleoutZenobehavior,i.e.,thesequence municationschemesfortheexecutionof(18).Incontrast of times does not have any finite accumulation point. toperiodicschemes,event-triggeredimplementationstie However,observing(25),onecanseethatZenobehavior the determination of the communication times to the willariseatleastneartheagreementsurfaceΠx=0 . currentnetworkstate,resultinginamoreefficientuseof dN Thenextresultdetails howweaddressthis problem. theresources.TheproofofTheorem10revealsthatthe Theorem 12 (Convergence of (18) with Zeno-free cen- satisfactionof condition(23)guaranteesthe monotonic tralizedevent-triggeredcommunication): Let beacon- evolution of the Lyapunov function, which in turn en- G suresthe correctasymptotic behaviorof the algorithm. nected graph. Assume the local cost function fi, i , Onecouldthereforespecifywhencommunicationshould is mi-strongly convex, differentiable, and its grad∈ieVnt occur by determining the times when this condition is is Mi-Lipschitz on Rd. Consider an implementation not satisfied. There is, however, a serious drawback to of (18)withagentscommunicatingover synchronously this approach: the evaluation of the condition (23) re- at{tk}k∈Z≥0 ⊂R≥0,startingat t0 =0,G quires the knowledge of the globalminimizer x⋆, which t =argmax t [t +τ, ) isofcoursenotavailable.To seethis,note that k+1 { ∈ k ∞ | Π(x(t ) x(t)) 2 κ Πx(t) 2 , (28) ˜z2:N = Π(x(tk) x) , (24a) k k − k ≤ k k } k k k − k p = x x¯ 2+ Π(v v¯) 2, (24b) where τ and κ < 1 are defined in (20) and (26), respec- k k k − k k − k tively. Then, for any given α,β > 0 and each i , where we have upsed (1) and (8) (recall Π=Π I ). the algorithm evolution starting from initial condi∈tioVns From(10),v¯i= α fi(x⋆)fori ,andthustNhe⊗evdal- xi(0),vi(0) Rd with N vi(0) = 0 makes xi(t) − ∇ ∈V ∈ i=1 d → P 8 x⋆ exponentially fast as t with a rate no less than Given α>0,let β,φ>0 satisfy φ+1>4M and 14(min{δ,2φαβλ2(1−ǫ)2,(→1−∞ǫ),12ǫ})/λ¯E >0. γ′=α2(φ+1)m+9βλˆ φα 4α2(Mm+(φ+1)2)>0. (30) PROOF. We first show κ < 1. This is an important 2 2 − propertyguaranteeingthat,ifagentsstartinagreement Then, for each i , the evolution starting from ini- at a point other than the optimizer x⋆, then the con- tial conditions xi(∈0),Vvi(0) Rd with N vi(0) = 0 dition(25)iseventuallyviolated,enforcinginformation ∈ i=1 d updates.Noticethat(a)4ǫ2(1−ǫ)λ22 <λ2N and(b)ǫδ < makes kxi(t)−x⋆k ≤ φ4αηβλλ¯FFkǫk2 as tP→ ∞ exponen- α2(1+φ)2 imply that the numerator in (26) is smaller tially fast with a rate no less than η/λ¯ . (Here, η = F thanitsdenominator,andhenceκ<1.(a)followsfrom min 7 ,1γ′ , and λ and λ¯ are the minimum and noting that the maximum of 4ǫ2(1 ǫ) for ǫ (0,1) is max{im16um9 ei}genvaluesFofFinF(7)).Moreover, theinter- − ∈ 16/27 < 1 and the fact that λ λ . We prove (b) 2 N executiontimes of iare lower boundedby reasoning by contradiction. Assum≤e δ > α2(1+φ)2 or equivalently α(1+φ) √δ < 0. Using (19) and multi- τi = 1 ln 1+ αMiǫi , (31) plyingboth sidesofth−e inequalityby 2mα, weobtain αMi 2 di (αMi+2βdi +1)θ out out (cid:16) (cid:17) α2M2+δ 2αm√δ=(√δ αm)2+α2(M2 m2)<0, whereθ = λ¯F x(0)p x¯ 2+ v(0) v¯ 2+φαβλ¯F ǫ 2. − − − λF k − k k − k 4ηλF k k which, since M m, is a contradiction. Having estab- p lished the consis≥tency of (25), consider now the can- PROOF. Given an initial condition, let [0,T) be the didate Lyapunov function V in (16) and let t be the maximal interval on which there is no accumulation k last time at which a communication among all neigh- pointin{tk}k∈Z¯ =∪Ni=1∪k∈Z¯i⊆Z≥0tik.NotethatT >0, boring agentsoccurred.Fromthe proofof Theorem10, since the number ofagents is finite and, for eachi , we know that the time derivativeof V is negative,V˙ < ǫi >0andx˜i(0)=xî(0) xi(0)=0.Thedynamics(∈18V), −21ǫmin{δ,21}p⊤p as long as t < tk + τ. After this under the event-triggere−d communication scheme (29), time, (27) shows that as long as (25) is satisfied, V˙ is has a unique solution in the time interval [0,T). Next, negative,andexponentialconvergencefollows. ✷ we use Lyapunov analysis to show that the trajectory isboundedduring[0,T).ConsiderthefunctionV given in(14),whoseLie derivativealong (21b)-(21d)is Interestingly,giventhat(25)doesnotusethefullstateof the networkbutinsteadreliesonthe disagreement,one 1 7 V˙ = α2(φ+1)y⊤h w⊤ w can interpret it as an output feedback event-triggered −9 − 16 2:N 2:N controller.Guaranteeingtheexistenceoflowerbounded 3 2α 2α inter-executiontimes forsuchcontrollersis ingenerala −k4w2:N + 3 (R⊤⊗Id)h+ 3 (φ+1)z2:N)k2 difficultproblem,seee.g.,[DonkersandHeemels,2012]. 4 4 Augmenting(25)withtheconditiont t +τ results + α2 (R⊤ I )h 2+ α2(1+φ)2z⊤ z k+1 ≥ k 9 k ⊗ d k 9 2:N 2:N in Zeno-free executions by lower bounding the inter- φαβ φαβ event times by τ. The knowledge of this value also al- z⊤ (R⊤LR I )z + s, lows the designer to compute bounds on the maximum − 2 2:N ⊗ d 2:N 2 energyspent bythe networkoncommunication. wheres= z⊤ (R⊤LR I )z 2z⊤ (R⊤LR I )˜z , 5.2.2 DistributedAsynchronous Implementation − 2:N ⊗ d 2:N− 2:N ⊗ d 2:N and z˜ = ˆz z . Using the assumptions on 2:N 2:N 2:N Wepresentadistributedevent-triggeredschemeforde- the local cost funct−ions and following steps similar to terminingthesequenceofcommunicationtimesin(18). those taken in the proof of Theorem 1 to lower bound At each agent, the execution of the communication y⊤h and upper bound (R⊤ I )h along with using schemedependsonlyonlocalvariablesandthetriggered λ z⊤ z z⊤ (R⊤LRk I )⊗z d ,konecanshowthat statesreceivedfromitsneighbors.Thisnaturallyresults 2 2:N 2:N ≤ 2:N ⊗ d 2:N inasynchronouscommunication.Wealsoshowthatthe φαβ V˙ η p 2+ s, resultingexecutionsarefree fromZenobehavior. ≤− k k 2 Theorem 13 (Convergence of (18) with Zeno-free dis- tributedevent-triggeredcommunication): LetGbeacon- where p = (z,w2:N). Next, we show s ≤ 21kǫk2 for t ∈ nected graph. Assume fi, i , is mi-strongly convex, [0,T). Using RR⊤ = Π , LΠ = Π L = L, z = differentiable, and its gradie∈ntVis Mi-Lipschitz on Rd. (R⊤ I )y = (R⊤ I )Nx, andNz˜ N= (R⊤ I2):Ny˜ = Forǫ RN ,consideranimplementation of (18)where ⊗ d ⊗ d 2:N ⊗ d agent∈i >0 communicates with its neighbors in at (R⊤⊗Id)x˜,wegets=−x⊤Lx−2x⊤Lx˜.Then, times {tik∈}kV∈Z¯i⊆Z≥0 ⊂R≥0,startingat ti0 =0, G s=−(xˆ−x˜)⊤L(xˆ−x˜)−2(xˆ−x˜)⊤Lx˜ ti =argmax t [ti, ) (29) =x˜⊤Lx˜ xˆ⊤Lxˆ. k+1 { ∈ k ∞ | − N 4dioutkxî(t)−xi(t)k2 ≤ j=1aijkxî(t)−xˆj(t)k2+(ǫi)2}. GivenL=Dout AandDout+A 0,we have − ≥ X 9 x˜⊤Lx˜ ≤2x˜⊤(Dout⊗Id)x˜ =2 Ni=1dioutkx˜ik2. kxî−xi(t)k≤ci(eαMi(t−tik)−1)/(αMi), t≥tik. X Therefore,we canwrite (recallx˜i =xî xi) Then, the time it takes kxî−xik to reach ǫi/(2 diout) − is lower bounded by τi > 0 given by (31). To show 1 N N p s= 4di xî xi 2 a xî xˆj 2 , T = , we proceed by contradiction. Suppose that 2Xi=1(cid:0) outk − k −Xj=1 ijk − k (cid:1) aTcc<um∞∞ul.aTtihonenp,otihnetasteqTu.eBnceecaoufseevweenthsa{vteka}kfi∈nZ¯itehansuman- which,with(29),yieldss 1 ǫ 2 for t [0,T).Then, ber of agents, this means that there must be an agent ≤2k k ∈ V˙ η p 2+ φαβ ǫ 2, t [0,T), ait∈T,Vimfoprlywinhgicthha{ttika}gke∈nZt¯iihtraasnsamnitasccinufimnuitlaeltyioonftpenoinint ≤− k k 4 k k ∈ the time interval [T ∆,T) for any ∆ (0,T]. How- − ∈ Recall from the proof of Theorem 1 that λ p 2 ever, this is in contradiction with the fact that inter- V(p) λ¯ p 2. Then, using the ComparisonFkLekmm≤a eventtimesarelowerboundedbyτi >0on[0,T).Hav- ≤ Fk k ing established T = , note that this fact implies that (cf.[Khalil, 2002,Lemma3.4]),wededuce that ∞ under the event-triggeredcommunication law (29), the kp(t)k≤λ1 kV(0)ke−λ¯ηFt+φαβ4λη¯λFkǫk2(1−e−λ¯ηFt) amlgoorer,itfhromm(1(83)2)d,oweesndoedtuecxehitbhiattZ,efonroebaechhaivior.,Founrethhears- ≤λλ¯FFFkp(0)ke−λ¯ηFt+φαβ4λη¯λFFkǫFk2(1−e−λ¯ηFt), (32) ktixail(lyt)f−asxt⋆wki≤thkapr(at)tke≤noφw4αηoβλrλ¯FsFektǫhka2naηs/tλ¯→F.∞∈,Vexponen✷- fort [0,T).Notice thatregardlessofvalue ofT, Regardingthe role of the designparametersand condi- ∈ λ¯ φαβλ¯ tion(30),weomitforspacereasonsobservationssimilar p(t) F p(0) + F ǫ 2, (33) tothe onesmadeinRemark2.The lowerboundonthe k k≤ λ k k 4ηλ k k F F inter-eventtimesallowsthedesignertocomputebounds onthemaximumenergyspentbyeachagentoncommu- for t [0,T). Notice that the right-hand side corre- nication during any given time interval. Since the total ∈ spondstoθ.Thiscanbeseenbynotingthat,from(11b), number of agents is finite and each agent’s inter-event we have z = x x¯ and w = w = v v¯ 2:N times are lowerbounded, it follows that the total num- k k k − k k k k k k − k (recall Ni=1vi(0)=0 resultsinw1 =0 forall t≥0). berofevents inanyfinite time intervalis finite. Ingen- Our final objective is to show that T = . To achieve eral,anexplicit expressionlowerbounding the network this,wePfirtestablisalowerboundonthei∞nter-execution inter-eventtimes is notavailable.Itis alsoworthnotic- times of any agent. To do this, we determine a lower ing that the farther away the agents start from the fi- boundonthetimeittakesi tohave xî xi evolve nalconvergencepoint(largerθ in(31)),the smallerthe ∈V k − k guaranteed lower bound between inter-event times be- from0 toǫi/(2 di ).Using (10)and(18b),wehave out comes.Asbefore,τiin(31)dependsonthegraphtopol- d xî xi =p (xî−xi)⊤x˙i x˙i opgayr,atmheetperasraαmaentedrsβo,fatnhdeltohcealvcaorsitabfulencǫtii.oOn,ntehecadnesuigsne dtk − k − xî xi ≤k k k − k thisdependencytomaximizethevalueofτi inasimilar N = α( fi(xi) fi(x⋆)) β a (xî xˆj) fashion as discussed in Remark 11. For quadratic local ij k− ∇ −∇ − j=1 − costfunctions of the form fi(x) = 1(x⊤x+x⊤ai+bi), (vi+α fi(x⋆) X i the claim of Theorem 13 hol2ds for any α,β > 0. − ∇ k ∈ V N Finally, we point out that the discrete-time communi- αMi xi x⋆ +β a xî xˆj + vi v¯i , ij cationstrategiesintroducedaboveenjoysimilarprivacy ≤ k − k j=1 k − k k − k preservation properties as the ones stated in Proposi- X whichleadsto tion5,butwe omitthe details hereforbrevity. d xî xi αMi xî xi +αMi xî x⋆ + 6 Simulations dtk − k≤ k − k k − k N Here,weillustratetheperformanceofthealgorithm(3) β a ( xî x⋆ + xˆj x⋆ )+ vi v¯i . ij and its implementation with discrete-time communica- j=1 k − k k − k k − k tion (18). We consider a network of 10 agents, with X From (33), we have x(t) x¯ θ and v(t) v¯ stronglyconvexlocalcostfunctions onR givenby θ. This implies vi(tk) v¯−i k ≤θ, xî xk⋆ −θ, kan≤d f1(x)=0.5e−0.5x+0.4e0.3x, f2(x)=(x 4)2, Nj=1aij(kxî −kx⋆k+k−xˆj −kx≤⋆k) ≤k 2d−ioutθ,kfo≤r all i ∈ f3(x)=0.5x2ln(1+x2)+x2, f4(x)=x2−+e0.1x, VP. Therefore,fromthe inequality above, ddtkxî−xik ≤ f5(x)=ln(e−0.1x+e0.3x)+0.1x2, f6(x)=x2/ln(2+x2), αMikxî −xik+ci, where ci = (αMi +2βdiout +1)θ. f7(x)=0.2e−0.2x+0.4e0.4x, f8(x)=x4+2x2+2, UsingtheComparisonLemma(cf.[Khalil,2002,Lemma 3.4])andthe factthat xî xi(ti) =0,we deduce f9(x)=x2/ x2+1+0.1x2, f10(x)=(x+2)2. k − k k p 10