Subgradient Methods and Consensus Algorithms for Solving Convex Optimization Problems

Transcription

1 Proceedngs of the 47th IEEE Conference on Decson and Contro Cancun, Mexco, Dec. 9-11, 2008 Subgradent Methods and Consensus Agorthms for Sovng Convex Optmzaton Probems Björn Johansson, Tamás Kevczy, Mae Johansson, Kar Henr Johansson Abstract In ths paper we propose a subgradent method for sovng couped optmzaton probems n a dstrbuted way gven restrctons on the communcaton topoogy. The teratve procedure mantans oca varabes at each node and rees on oca subgradent updates n combnaton wth a consensus process. The oca subgradent steps are apped smutaneousy as opposed to the standard sequenta or cycc procedure. We study convergence propertes of the proposed scheme usng resuts from consensus theory and approxmate subgradent methods. The framewor s ustrated on an optma dstrbuted fnte-tme rendezvous probem. I. ITRODUCTIO Large-scae networed systems are becomng ubqutous and ncreasngy compex to manage. If the desred behavor of the networed system can be formuated as an optmzaton probem, then compexty ssues can be mtgated by sovng the resutng optmzaton probem wth dstrbuted agorthms. We w deveop such a dstrbuted agorthm. Severa mportant probems n appcatons such as resource aocaton n computer networs 1, 2, estmaton n sensor networs 3, and the rendezvous probem n mutagent systems 4, can be posed as couped optmzaton probems. In these probems, each node or agent n the networ s assocated wth a component of the objectve functon, whch depends on a networ-wde decson varabe. Couped optmzaton probems can be soved usng a varety of dstrbuted agorthms. A cassca way s to use dua reaxaton,.e., the constrants are enforced usng a prcng scheme. Another method s to teratvey refne an estmate of the optmzer usng ncrementa subgradent methods, where the estmate typcay s passed around n the networ foowng a ogca rng. In ths paper, we combne consensus negotatons wth a subgradent method, whch s smar n favor to the approach proposed n 5, but wth dfferent propertes and anayss. In our approach, the communcaton topoogy s expcty respected, and the agorthm ony requres communcaton between neghborng nodes. In Secton II, we present our probem formuaton as we as notaton and connectons wth the exstng terature. Then, n Secton III, we ntroduce the agorthm and supportng emmas. We then contnue wth a convergence anayss of B. Johansson, M. Johansson and K.H. Johansson are wth the ACCESS Lnnaeus Centre, Schoo of Eectrca Engneerng, Roya Insttute of Technoogy KTH), Stochom, Sweden, {bjorn.johansson,maej,aej}@ee.th.se. Ther wor was supported by VR, SSF, VIOVA, and HYCO. T. Kevczy s wth the Deft Center for Systems and Contro, Deft Unversty of Technoogy, 2628 CD, Deft, The etherands, t.evczy@tudeft.n the agorthm n Secton IV. Furthermore, the propertes of the agorthm s expored usng a numerca exampe n Secton V. Fnay, we concude the paper wth a dscusson n Secton VI. II. PROBLEM FORMULATIO In ths paper we consder the foowng optmzaton probem mnmze x fx) = subject to x X, f x) =1 where f : R M R are convex functons and X s a nonempty, cosed, and convex subset of R M. Let f denote the optma vaue of 1) and et x denote an optmzer of 1). We w assume that n genera f s nondfferentabe. Our nterest n studyng ths cass of optmzaton probems s motvated by optma contro probems for fnte-tme rendezvous of mutpe dynamca agents 4, 6, resource aocaton n computer networs 1, 2, and estmaton n sensor networs 3. A. Premnares We w mae use of the foowng defntons and assumptons n the paper. Defnton 1: A vector g R M s a subgradent of a convex functon f : R M R at a pont x R M f 1) fy) fx) + g y x), y R M. 2) Defnton 2: The set of a subgradents of a convex functon f at x R M s caed the subdfferenta of f at x, and s denoted by fx): fx)= { g R M fy) fx) + g y x), y R M}. 3) Defnton 3: The ǫ-subdfferenta set of a convex functon f at x s the coecton of ǫ-subgradents: ǫ fx) = { g R M fy) fx) + g y x) ǫ, y R M}, 4) wth ǫ 0. We w use to denote the 2-norm of a vector x,.e., x = x x, and the nduced 2-norm of a matrx W,.e., Wx W = sup x 0 x. Assumpton 1 Subgradent Boundedness): There exsts a scaar C for a = 1,..., such that g x) C, g x) f x), x X. 5) /08/$ IEEE 4185

2 47th IEEE CDC, Cancun, Mexco, Dec. 9-11, 2008 B. Subgradent Methods A popuar method for sovng probems of the type 1) s the subgradent method, whch conssts of the foowng teratve procedure x +1 = P X x + α g x ), 6) =1 where g x ) s a subgradent of f at x, α s a postve stepsze, and P X denotes projecton on the set X R M. References 7 and 8 contan exceent ntroductons and overvews of the theory and use of subgradent methods, as we as ther convergence rate propertes. The subgradent method can be mpemented n an ncrementa fashon as proposed n 8. Ths entas changng the varabe x ncrementay through steps, n each teraton usng ony the subgradent correspondng to a snge component functon f. The advantage of ths method from a computatona aspect s that t can be performed n a dstrbuted way by assgnng each component functon to a processng unt or agent, whch performs the oca ncrementa update on the varabe x. Ths means that x needs to be passed around between the agents, whch perform a subgradent update usng ony a snge subgradent correspondng to the agent s component functon. Ths ncrementa subgradent scheme has advantages over the standard one n terms of convergence rate and dstrbuton of computatons. However, n ts usua form the ncrementa updates are performed n a sequenta manner, whch assumes that the varabe x passes through a agents ether n a cycc or randomzed sequence. Impementaton of such a communcaton scheme can sometmes be probematc n practce. A more reastc dstrbuton of the computatons woud aow a much wder cass of nformaton exchange topooges, whch mght even be tme-varyng or suffer from deays. otce that we can thn of the computng agents mentoned above as each havng a copy of the decson varabe x, whch they mantan ocay and update based on the vaue obtaned from the prevous subteraton the precedng agent n the update sequence) and the oca subgradent of the component functon evauated at ths vaue. Under approprate assumptons and usng a propery chosen dmnshng stepsze, the subgradent teratons converge asymptotcay to an optmzer x of the probem. Ths means that eventuay a oca versons of the decson varabe converge to the same vaue. Ths resembes to some extent agreement or consensus probems n mut-agent systems, whch has a ong hstory and has receved renewed nterest n the recent terature 9. C. Consensus agorthms The consensus probem consders condtons under whch usng a certan message-passng protoco, the oca varabes of each agent w converge to the same vaue. However, ths vaue does not generay represent an optmzer of a probem of nterest, and s typcay a functon of the nta vaues hed by the agents and the nformaton exchange pocy. A wde varety of resuts exst reated to the convergence of oca varabes to a common vaue usng varous nformaton exchange procedures among mutpe agents It s thus nterestng to nvestgate whether the convergence propertes of certan consensus agorthms can be combned wth subgradent teratons n order to optmze probems of type 1) usng a wder varety of communcaton topooges than what the standard ncrementa subgradent method aows. For our nvestgatons we w consder the foowng consensus agorthm n dscrete-tme: y +1 = Wy, 7) where the -th eement of vector y, denoted by y, corresponds to agent. Furthermore, we assocate an undrected graph G = V, E) wth the optmzaton probem 1), where V = {1,..., } and E V V. The nterpretaton s that f, j) E, then agent can communcate wth agent j. The nformaton exchange among agents s represented by the matrx W, for whch we mae the foowng assumptons. Assumpton 2 Consensus Matrx Propertes): The weght matrx W R fufs W j = 0, f, j) / E and j, W = W, W1 = 1, ρ W 1 1 ) γ < 1, where ρ ) s the spectra radus and 1 R s the coumn vector wth a eements equa to one. The matrx W can for exampe be chosen as the socaed Perron matrx of the communcaton graph G wth parameter ε. It s defned as W = I εlg), where LG) represents the Lapacan matrx of the communcaton graph. If G wth maxmum degree s strongy connected wth 0 < ε < 1/, then the mt m W exsts and a consensus s asymptotcay reached for a nta states. Furthermore, the consensus vaue w be the average of the nta states: y j = 1 =1 y 0, for a j = 1,...,. For more detas, see, e.g, 9; for other ways of choosng W that fufs Assumpton 2, see, e.g, 11. III. SUBGRADIET ALGORITHM WITH COSESUS ITERATIOS We propose to combne the subgradent teratons of 6) wth a number of consensus teratons 7) n the foowng way: x +1 = P ) X W ϕ j x j α g j x j ), 8) where g j x j ) fj x j ) and W ϕ j denotes the eement of W ϕ n the -th row and j-th coumn. Agents mantan ther oca varabe x and perform the update procedure of 8) n parae. Frst, they exchange nformaton wth neghborng agents for ϕ number of consensus teratons. More specfcay, 8) mpes that each agent runs ϕ number of consensus teratons wth ts neghbors, defned by 7), for each row n the oca vector x. Then, a agents mpement the component subgradent update ocay. The 4186

3 47th IEEE CDC, Cancun, Mexco, Dec. 9-11, 2008 tota number of teratons n the agorthm up to step s therefore ϕ. For a more compact notaton we defne and u = x α g x ) 9) v = W ϕ j u j, = 1,...,. 10) Let us assume for the moment that n each subgradent teraton ϕ, and the consensus updates converge to the average of the nta vaues ths hods f W fufs Assumpton 2). Then, for a = 1,..., and x 0 RM, m ϕ ) W ϕ j u j 0 = 1 x j 0 α 0g j x j 0 ) ). Let us denote the nta state of the projected consensus varabe wth x 1 = P X 1 x j 0 α 0g j x j 0 ) ). In the next teraton, each agent w possess the same vaue x 1, thus the oca subgradents g j w be evauated at the same pont. The update procedure n 8) for 1 w thus be equvaent to x +1 = P X 1 x α g j x ) ) = P X x α 1 g j x )). 11) Ths s the same process as the standard subgradent method of 6) performed on the consensus varabe x wth a stepsze of α /. Convergence anayss of ths scheme can be done foowng the procedure n for exampe 7 or 8. We are nterested however n a more reastc scenaro, where the consensus teratons are performed ony for a fnte number of ϕ steps. We ntend to anayze such a scheme wth the use of propery chosen approxmate subgradents and the approxmaton error of the average consensus process acheved n a fnte number of steps. Fnay, we assume that the stepsze n 8) s constant, α = α, whch maes the agorthm easy mpementabe. The foowng three emmas w be nstrumenta n the convergence anayss of the proposed scheme n Secton IV. We w denote the average vaue of the oca varabes at tme wth x = 1 =1 x and v = 1 =1 v. The foowng nequates serve to characterze the Eucdean dstance of agent varabes from each other and from ther average. Lemma 1: 1) If x x j β for a, j = 1,...,, then x j x = x j 1 =1 x 1 β. 2) If x x β for a = 1,...,, then x x j 2β. Proof: 1) x j x = 1 x j =1 x 1 ) =1, =j x xj 1 β. 2) x xj x x x + j x 2β. Lemma 2: If y +1 = W ϕ y, wth W fufng Assumpton 2, and y y j σ for a, j = 1,...,, then y +1 y +1 j 2γ ϕ σ for a, j = 1,...,. Proof: Let us wrte y as y = ȳ + a wth ȳ = 1 1 y / and =1 a = 0. The resuts of Lemma 1 show that a σ for a. Furthermore, y +1 = W ϕ ȳ + a ) = ȳ + W ϕ a 0 ). ow we have, y +1 ȳ y +1 ȳ = W ϕ a 0 ) = W ϕ 1 1 ) a W ϕ 1 1 ) ϕ a W 1 1 ϕ a γ ϕ a γ ϕ σ, where we used that W 11 = ρ ) W 11, whch hods for symmetrc matrces. Fnay, from the above dscusson and Lemma 1, we have y +1 y +1 j 2γ ϕ σ for a, j = 1,...,. The foowng emma estabshes a ower bound on the number of consensus steps that w ensure that the oca varabes w reman n a ba of radus β of ther average, from one teraton to the next. Lemma 3: Let {x 1,..., x } =0 be generated by 8) under Assumptons 1 and 2. If v v β for a and ϕ ogβ) og4mβ + αc)) ) / ogγ), then v +1 v +1 β for a. Proof: Usng Lemma 1, the coseness condton means that v v j 2β for a, j = 1,...,, whch mpes that v vj 2β for a, j = 1,..., and = 1,..., M. We can now decde the dstance between the terates before the consensus step. u +1 u j +1 12a) = P X v αg P X v ) P Xv j + αgj P X v j ) v v j + 2αC 2β + αc), 12b) where we used the non-expansve property of the projecton on a convex set. Furthermore, we have v+1 vj +1 = W ϕ n u n +1 W ϕ jn u n +1 M W ϕ n u n +1 W ϕ jn u n +1. =1 Consder now one of the terms n the sum above and notce 4187

4 47th IEEE CDC, Cancun, Mexco, Dec. 9-11, 2008 that W ϕ n u n +1 W ϕ jn u n +1 = W ϕ n u n +1 W ϕ jn u n +1 = W ϕ y W ϕ, y j wth y = u u. +1 ) Usng Lemma 2 and 12), whch states that y y j 2β + αc), we obtan W ϕ y W ϕ y j 4γ ϕ β + αc). Combnng the above resuts yeds v+1 vj +1 4γ ϕ Mβ + αc), and v +1 v j +1 β s fufed f ϕ s set to ϕ ogβ) og4mβ + αc)). ogγ) IV. COVERGECE AALYSIS We w prove convergence of the agorthm n two cases: Frsty, convergence s proved when the feasbe set s the space R M,.e., the unconstraned optmzaton probem. Secondy, wth an addtona assumpton on the objectve functons, convergence s proved for a genera convex feasbe set. A. Unconstraned Case In the foowng, we w anayze the unconstraned case and we mae the foowng assumpton. Assumpton 3: The feasbe set of 1) s X = R M. We need the foowng Lemma whch aows us to nterpret the agorthm as an ǫ-subgradent agorthm. Lemma 4: Under Assumptons 1 and 2 and f x x β for a = 1,...,, then g x ) ǫf x ) and =1 g x ) ǫf x ), wth ǫ = 2βC. Proof: Usng the defnton 2) and the bound 5) on the subgradent eads to f x ) f x ) + g x ) x x ) f x ) g x ) x x f x ) Cβ. For any y X, usng the subgradent nequaty eads to f y) f x ) + g x ) y x ) f x ) + g x ) y x ) Cβ f x ) + g x ) y x + x x ) Cβ f x ) + g x ) y x ) 2Cβ. Usng the defnton of an ǫ-subdfferenta 4), ths mpes g x ) 2βCf x ). Summaton of terms yeds ) fy) f x ) + g x ) y x ) 2βC. =1 Based on Defnton 3, ths mpes =1 g x ) 2βC f x ). ow we are ready for the convergence theorem for the unconstraned case. Theorem 1: Under Assumptons 1, 2, and 3, wth the sequence {x 1,..., x } =0 generated ) by 8) wth ϕ ogβ) og4mβ+αc)) / ogγ) and x 0 x 0 β, we have: If f =, then If f >, then m nf fx ) =, = 1,...,. m nf fx ) f + αc 2 /2 + 3Cβ, = 1,...,. Proof: From Lemma 3, we now that x x β for a = 1,.., and a 0, snce x = v. Furthermore, from Lemma 4, we now that =1 g x ) 2βCf x ) for a 0. Hence, from the defntons of x and x n combnaton wth the resuts above, we have x +1 = 1 ) W ϕ j x ) j αgj x j ) = = 1 =1 ) x j αgj x j ) = x + α h x ), wth h x ) 2βC f x ) and h x ) C. Ths s precsey the approxmate subgradent teraton and from 12, Proposton 4.1 we have and m nf f x ) =, f f = m nf f x ) f +αc) 2 /2)+2Cβ, f f >. By notng that fx ) f x ) + Cβ, = 1,...,, 0, we have the desred resut. Remar 1: We can get m nf fx ) to be arbtrary cose to f, by choosng the constants α and β arbtrary sma. ote that the number of requred consensus negotatons ϕ) to reach a fxed β does not depend on. B. Constraned Case To show convergence n the constraned case, we need the foowng addtona assumpton on the functons f. Assumpton 4: There exst ζ > 0 and τ > 0 such that for a x X, g x) f x), and ν R M wth ν τ, the foowng hods: g x) + ν ζ f x). As mentoned before, ths s an addtona assumpton compared to what s needed for the unconstraned case. However, f X s a compact set, e.g., x X x η, then Assumpton 4 s fufed for convex functons and arbtrary 4188

5 47th IEEE CDC, Cancun, Mexco, Dec. 9-11, 2008 sma but fxed) ζ as ong as τ s suffcenty sma. To see ths, consder x, w X, then f w) f x) + g x) w x) = f x) + g x) + ν ν) w x) f x) + g x) + ν) w x) ν x + w ) f x) + g x) + ν) w x) 2τη, whch fufs Assumpton 4 f τ ζ/2η). Another exampe s when fx) = ψx x, then Assumpton 4 s fufed f τ 2 ζψ wthout X necessary beng a compact set). In our further deveopments we need to eep trac of the dfference x +1 P X v, and to ths end, we defne y and z as y = P X v 1 and z = x y. 17) Furthermore, we need the foowng Lemma, whch s smar to Lemma 4. Lemma 5: Under Assumptons 1, 2, and 4 and f v 1 v 1 β for a and ν R M wth ν τ, then g x ) + ν) ǫf y ) and =1 g x ) + ν) ǫ f y ) wth ǫ = β6c + 3τ) + ζ. Proof: The proof dea rees on the terates beng cose to each other. Usng the assumptons and the non-expansve property of projecton on a convex set we can bound the dstance z as foows z = 1 PX v 1 PX v 1 ) 1 =1 v 1 v 1 β. =1 Usng the defnton 2), the bound on the subgradent 5), the bound above, and Lemma 1 we obtan f x ) f y ) + g y ) x y ) f y ) g y ) x y f y ) C x x + z ) f y ) C3β, where we used x y = x x +z and v v β v v j 2β x x j 2β x x 2β. For any y X, usng Assumpton 4 and the prevous arguments, we get f y) f x ) + g x ) + ν) y x ) ζ f y ) + g x ) + ν) y x ) 3Cβ + ζ) = f y ) + g x ) + ν) y y + y x ) 3Cβ + ζ) f y ) + g x ) + ν) y y ) β6c + 3τ) + ζ). Usng the defnton of an ǫ-subdfferenta 4), ths mpes g x ) + ν) β6c+3τ)+ζ)f y ). Summaton of terms yeds fy) fy )+ ) g x ) + ν) y y ) β6c + τ) + ζ). =1 Based on Defnton 3, ths mpes =1 g x ) + ν) β6c+3τ)+ζ) fy ). We are now ready for the convergence theorem, whch s based on the dea of nterpretng 8) as an approxmate subgradent agorthm. Theorem 2: Under Assumptons 1; 2; and 4, wth the sequence {x 1,..., x } =0 generated ) by 8) wth ϕ ogβ) og4mβ + αc)) / ogγ) ; v0 v 0 β; and β/α τ, we have: If f =, then If f >, then m nf fx ) =, = 1,...,. m nf fx ) f + αc + τ) 2 /2+ β9c + 3τ) + ζ), = 1,...,. Proof: From the defnton of y we have, 1 y +1 = P X =1 v 1 = P X =1 x αg x )) = P X y + z α =1 g x ) = P X y α =1 g x ) z α ). From Lemma 3, we now that v v β for a = 1,.., and a 0. Furthermore, from Lemma 5, we now that =1 g x ) z /α) β6c+3τ)+ζ) fy ) for a 1, snce z /α β/α τ by assumpton. In addton, we now that =1 g x ) z /α) C+τ). Hence, y s updated accordng to an approxmate subgradent method and from 12, Proposton 4.1 we have and m nf fy ) =, f f = m nf fy ) f + αc + τ)) 2) /2)+ β6c + 3τ) + ζ), f f >. Fnay, we have fx ) fy ) + β3c f + αc + τ) 2 /2+ β9c + 3τ) + ζ), = 1,...,, 0. Remar 2: The assumpton β ατ n Theorem 2 may seem restrctve, but t can be fufed wth a fxed number of consensus negotatons accordng to Lemma 3. Remar 3: The nta condtons n Theorem 1 and Theorem 2, x 0 x 0 β and v 0 v 0 β, respectvey, can be fufed wth suffcenty many consensus negotatons before startng the agorthm. Another smpe aternatve s to set x 0 = x j 0,, j and v 0 = v j 0,, j, respectvey. 4189

6 47th IEEE CDC, Cancun, Mexco, Dec. 9-11, 2008 V. UMERICAL EXAMPLE The proposed subgradent agorthm n combnaton wth consensus terates as shown n 8) has been mpemented to sove a dstrbuted fnte-tme optma rendezvous probem nvovng three agents movng n a pane wth doube ntegrator dynamcs. A more detaed probem descrpton can be found n 4. Each agent mantans a oca verson of the optmzaton varabe x the rendezvous poston) and performs the oca subgradent update wth a dmnshng stepsze accordng to 8). The consensus updates are performed wth the foowng matrx W : W = ) Fgure 1 ustrates the convergence of the oca optmzaton varabe for one of the agents to the gobay optma rendezvous ocaton. The evouton of the oca varabe for the other agents s very smar. The dfferent curves overad n the graph correspond to dfferent choces of the consensus teraton mt.e., the maxmum number ϕ of consensus steps performed). Ths ustrates how ncreased communcaton n terms of the number of consensus teratons wth neghborng agents eads to better convergence rate. As we ncrease ϕ we expect to approach the convergence rate of the standard subgradent method gven n 6). The man advantage of our proposed scheme s that the subgradent updates are performed n parae usng nformaton from neghborng agents and there s no need for a centra processng unt to whom each agent shoud communcate ts oca subgradent. x) x x ϕ = Subgradentteraton number ϕ = 0 ϕ = 1 ϕ = 2 ϕ = 5 ϕ = 10 Fg. 1. Convergence pots for ncreasng number of consensus teratons ϕ). VI. COCLUSIOS AD FUTURE WORK In ths paper we have descrbed an teratve subgradentbased method for sovng couped optmzaton probems n a dstrbuted way gven restrctons on the communcaton topoogy. In order to aow great fexbty n the nformaton exchange archtecture and dstrbute cacuatons, we combned the oca subgradent updates wth a consensus process. Ths means that computng agents can wor n parae wth each other and use ocazed nformaton exchange. The generaty of resuts n consensus theory promses reatvey easy extensons of the presented method to stuatons where the nterconnecton topoogy mght be tme-varyng, deayed and the oca updates are performed asynchronousy. Research on the speed of convergence n consensus protocos has an mmedate and cear use n our framewor. For anayss purposes, we used resuts from consensus theory and empoyed approxmate subgradent methods to study convergence propertes of the proposed scheme. A connecton s estabshed between the number of consensus steps and the resutng eve of optmaty obtaned by the subgradent updates. We have ustrated the effect of choosng dfferent consensus teraton mts n an optma dstrbuted fnte-tme rendezvous probem. A dfferent verson of 8) s aso concevabe as ) x +1 =P ) X W ϕ j x j α g W ϕ j x j, where n each update frst a consensus s reached on the oca varabes at east approxmatey), and then the correspondng oca subgradent updates are cacuated and apped. We expect that a smar ne of anayss shown n ths paper can be performed for such a varaton of the orgna scenaro. We are currenty exporng dfferent choces of the stepsze, e.g., dmnshng stepszes, and we beeve that other resuts from approxmate subgradent methods can be used. Fnay, we are aso oong nto how the dfferent parameters shoud be tuned,.e., shoud we decrease the stepsze or ncrease the number of negotatons to get coser to optmaty? REFERECES 1 M. Chang, S. Low, A. Caderban, and J. Doye, Layerng as optmzaton decomposton: A mathematca theory of networ archtectures, Proceedngs of the IEEE, vo. 95, no. 1, pp , Jan B. Johansson, P. Sodat, and M. Johansson, Mathematca decomposton technques for dstrbuted cross-ayer optmzaton of data networs, IEEE Journa on Seected Areas n Communcatons, vo. 24, no. 8, pp , Aug M. Rabbat, R. owa, and J. Bucew, Generazed consensus computaton n networed systems wth erasure ns, n IEEE SPAWC, B. Johansson, A. Speranzon, M. Johansson, and K. H. Johansson, On decentrazed negotaton of optma consensus, Automatca, pp , Ju A. edć and A. Ozdagar, On the rate of convergence of dstrbuted asynchronous subgradent methods for mut-agent optmzaton, n IEEE CDC, T. Kevczy and K. H. Johansson, A study on dstrbuted mode predctve consensus, n IFAC Word Congress, K. C. Kwe, Convergence of approxmate and ncrementa subgradent methods for convex optmzaton, SIAM J. Optm., vo. 14, no. 3, pp , D. P. Bertseas, A. edć, and A. E. Ozdagar, Convex Anayss and Optmzaton. Athena Scentfc, R. Ofat-Saber, J. A. Fax, and R. M. Murray, Consensus and cooperaton n networed mut-agent systems, Proceedngs of the IEEE, vo. 95, no. 1, pp , A. Oshevsy and J. Tstss, Convergence rates n dstrbuted consensus and averagng, n IEEE CDC, L. Xao and S. Boyd, Fast near teratons for dstrbuted averagng, Systems & Contro Letters, vo. 53, no. 1, pp , A. edć, Subgradent methods for convex mnmzaton, Ph.D. dssertaton, MIT,