A Class of Distributed Optimization Methods with Event-Triggered Communication

Transcription

1 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton Martn C. Mene Mchae Ubrch Sebastan Abrecht the date of recept and acceptance shoud be nserted ater Abstract We present a cass of methods for dstrbuted optmzaton wth event-trggered communcaton. To ths end, we extend Nesterov s frst order scheme to use event-trggered communcaton n a networked envronment. We then appy ths approach to generaze the proxma center agorthm (PCA) for separabe convex programs by Necoara and Suykens. Our method uses dua decomposton and appes the deveoped event-trggered verson of Nesterov s scheme to update the dua mutpers. The approach s shown to be we suted for sovng the actve optma power fow (DC-OPF) probem n parae wth event-trggered and oca communcaton. Numerca resuts for the IEEE 57 bus and IEEE 118 bus test cases confrm that approxmate soutons can be obtaned wth sgnfcanty ess communcaton whe satsfyng the same accuracy estmates as soutons computed wthout event-trggered communcaton. Keywords dstrbuted optmzaton convex optmzaton non-dfferentabe optmzaton dua decomposton dstrbuted reguarzaton oca communcaton event-trggered communcaton DC-OPF probem IEEE test cases 1 Introducton Motvaton. Dstrbuted optmzaton ganed a ot of attenton n recent years to face the need of fast and effcent soutons for probems arsng n the context of arge-scae networks such as utty maxmzaton probems (NUM) [13, 14, 3], dstrbuted estmaton [17, 18], mut-robot coordnaton [3, 15], and the optma power fow (OPF) probem [1, 6, 7, 9]. In ths paper we focus on the OPF probem, but other appcatons are possbe. The goa of dstrbuted optmzaton s to sove these probems n parae by mutpe agents that jonty mnmze (or maxmze) a separabe objectve functon, usuay subject to Martn Mene Lehrstuh für Mathematsche Optmerung, Technsche Unverstät München, Fakutät für Mathematk Botzmannstr. 3, Garchng b. München, Germany E-ma: mene@ma.tum.de Mchae Ubrch Lehrstuh für Mathematsche Optmerung, Technsche Unverstät München, Fakutät für Mathematk Botzmannstr. 3, Garchng b. München, Germany E-ma: mubrch@ma.tum.de Sebastan Abrecht Lehrstuh für Mathematsche Optmerung, Technsche Unverstät München, Fakutät für Mathematk Botzmannstr. 3, Garchng b. München, Germany E-ma: abrecht@ma.tum.de

2 Martn C. Mene et a. coupng constrants that force them to exchange nformaton, such as ther assgned components of the decson varabe, durng the optmzaton process. For detas we refer to [, 11, 4]. Often t s advantageous f the communcaton of cooperatng agents s mnma wth regards to the frequency and amount of transmtted nformaton. Ths s necessary for exampe when ad hoc wreess communcaton s used where the amount of nformaton that can be sent s mted []. Moreover, t s preferabe to keep the communcaton oca,.e., between agents that are n cose proxmty (often ndcated by a connectng edge n the graph representaton of the network) to avod ong communcaton tmes and data package osses. Wth these ssues n mnd, we consder the proxma center agorthm (PCA) by Necoara and Suykens [10] whch s a dua decomposton scheme for convex programs wth separabe structure and where the objectve functons are not requred to be strcty convex or smooth. The two key ngredents of the agorthm are bascay a smoothng technque proposed by Nesterov n [1] to obtan a smooth dua augmented functon that s separabe n the prma varabes, and the usage of Nesterov s optma frst order scheme for smooth optmzaton [1] to update dua mutpers. Generazng the approaches of [1] and [10], we equp Nesterov s agorthm wth an extenson that aows event-trggered communcaton where the agents exchange nformaton n a non-perodc manner ony when t s crucay requred n order to mantan convergence, and show that the agorthm s mpementabe n parae under sutabe condtons. Furthermore, we prove convergence of the deveoped dstrbuted Nesterov agorthm wth event-trggered communcaton (DNA-EC) for the mnmzaton of a contnuousy dfferentabe convex objectve functon wth Lpschtz contnuous gradent, mantanng the convergence rate of the orgna scheme. Fnay, the appcaton of the DNA-EC aows us to mpement the PCA n an entrey parae manner yedng the dstrbuted proxma center agorthm wth event-trggered communcaton (DPCA-EC). Moreover, we manage here as we to mantan the convergence rate of the PCA whch s of the order O(1/ɛ), where ɛ s the desred accuracy of the objectve functon vaue at the approxmate souton. We appy the DPCA-EC to sove the DC-OPF probem [16], whch comprses the optma actve power generaton dspatch n a power system, and obtan approxmate soutons under oca and event-trggered communcaton, sgnfcanty reducng communcaton between agents. As the noton of event-trggered communcaton fnds ts orgn n the context of networked contro systems [8] t appears ony atey n drect connecton wth dstrbuted optmzaton and therefore a mted amount of terature exsts concernng dstrbuted optmzaton wth event-trggered communcaton: Zhong and Cassandras extend n [5] a partay asynchronous dstrbuted optmzaton framework from [] by event-trggered communcaton. Ther agorthm s appcabe for an unconstraned probem wth contnuousy dfferentabe objectve functon that has a Lpschtz contnuous gradent and foows a genera state update scheme, where the current terate of an agent s state s updated by a scaed update drecton (e.g., a gradent step wth a certan step sze). They show convergence to a statonary pont of the objectve functon. Wan and Lemmon proposed a dstrbuted agorthm n [1] wth event-trggered communcaton based on a barrer approach. Recognzng drawbacks of the agorthm such as -condtonng, they formuate n [0] an augmented Lagrangan functon assocated wth the NUM probem that s mpementabe dstrbutvey wth event-trggered communcaton between users and nks, and where the user rates are updated n a steepest ascent fashon. Assumng the utty functons to be twce dfferentabe, strcty ncreasng and strcty concave, they can show asymptotc convergence of the agorthm. In [] Wan and Lemmon use the same approach to sove the DC-OPF probem n a dstrbuted manner wth eventtrggered communcaton and show asymptotc convergence assumng strcty ncreasng, strcty convex and dfferentabe power generaton cost functons. In contrast to the exstng approaches on the one hand the DNA-EC s appcabe to probems wth convex and contnuousy dfferentabe objectve functon constraned by a bock-separabe set. On the other hand the DPCA-EC s appcabe to probems wth separabe and convex but not necessary contnuousy dfferentabe objectve functon constraned by couped near constrants. Moreover, as a hertage of ther

3 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 3 orgns (cf. [1] and [10]), the compextes of the DNA-EC and the DPCA-EC are superor to the compextes of standard gradent projecton methods and dua gradent schemes, respectvey, whch can be mpemented n parae as we. Fnay, accuracy estmates for both agorthms are gven n ths work. Contents. In secton we revew Nesterov s optma frst order scheme [1] for mnmzng a convex functon wth Lpschtz contnuous gradent over a convex and cosed set. We show for a sutabe settng that a sghty modfed verson of the agorthm s mpementabe n parae wth event-trggered communcaton, yedng our agorthm DNA-EC. We prove ts convergence mantanng the compexty O( L/ɛ) of the orgna scheme, where L s the Lpschtz constant of the gradent and ɛ the desred accuracy of the objectve functon vaue at the approxmate souton. In [1] t s mentoned that ths compexty s superor compared to the compexty of the standard gradent projecton method wth O(1/ɛ). In secton 3 we revew the PCA [10] of Necoara and Suykens and propose a scang technque to mprove the accuracy estmates for the agorthm. Moreover, we modfy the PCA by appyng the deveoped DNA- EC to update dua mutpers n parae and wth event-trggered communcaton, yedng the agorthm DPCA-EC. We show convergence of the DPCA-EC preservng the compexty O(1/ɛ) of the PCA whch s better than the compexty of cassca dua gradent schemes wth O(1/ɛ ) [10]. Moreover, we show how to optmay choose the convexty parameters for a certan choce of strongy convex prox-functons that are used to smoothen the dua functon of the consdered convex probem. Addtonay, we show how to optmay choose the smoothng parameters that are used to scae these prox-functons. Fnay, n secton 4 we descrbe n deta the DC-OPF probem and present numerca resuts of the appcaton of the DPCA-EC to the IEEE 57 bus and IEEE 118 bus test cases [4], showng that the communcaton exchange can be reduced sgnfcanty wthout tradng off accuracy of the approxmate souton. Optma frst order scheme wth event-trggered communcaton.1 Nesterov s optma scheme for smooth mnmzaton In [1] Nesterov provdes a frst order method for optmzaton probems of the form mn f (x) (1) x Q where f : Q R s a convex and contnuousy dfferentabe functon on a cosed and convex set Q R m. Further, t s assumed that the gradent of f s Lpschtz contnuous: f (x) f (y) L x y x,y Q, () where denotes the Eucdean norm. Choosng a prox-functon d(x), whch s a contnuous and strongy convex functon on Q wth convexty parameter σ > 0 (.e., t satsfes d(y) d(x) + d(x) T (y x) + σ x y for a x,y Q), the correspondng center x 0 = argmnd(x) (3) x Q of the set Q s determned as the startng vaue for Nesterov s optmzaton scheme that may be stated as foows:

4 4 Martn C. Mene et a. Agorthm.1.1 For k 0 do 1. Compute f (x k ).. Fnd y k = argmn f (x k ),y x k + L y x k }. y Q L 3. Fnd z k = argmn z Q σ d(z) + α j f (x j ),z x j }. 4. Set x k+1 = τ k z k + (1 τ k )y k. Here α k } k 0 are a pror chosen postve step sze parameters and τ k = α k+1 /A k+1 wth A k = k =0 α. Note, that we omtted some constant terms n the argmn-probems of the orgna agorthm n [1]. We dd so to revea the paraezabe nature that s obvous n Agorthm.1.1 f the prox-functon d(x) s chosen appropratey, e.g., d(x) = (σ/) x and f Q has a bock-separabe structure,.e., f Q = Q 1... Q s wth Q R m and s m = m. The foowng convergence resut hods: Theorem.1. [Lemma 1 of [1]] Let the sequence α k } k 0 satsfy the condton: Then the reaton α 0 (0, 1], α k+1 A k+1, α k > 0, k 0. (4) L A k f (y k ) Ψ k = mn z Q σ d(z) + ( α j f (x j ) + f (x j ),z x j )} hods for k 0 and therefore f (y k ) f (x ) Ld(x )/A k, where x s an optma souton to probem (1). The foowng emma gves a possbe choce for α k } k 0 that provdes the compexty estmate of O ( L/ɛ ), where ɛ s the desred accuracy of the objectve functon vaue at the approxmate souton: Lemma.1.3 [Lemma of [1] For k 0 defne α k = (k + 1)/. Then and condtons (4) are satsfed. τ k = k + 3, A k = (k + 1)(k + ), 4 Remark.1.4 Let us note that the foowng equaty for y k computed n step of Agorthm.1.1 hods: y k = argmn f (x k ),y x k + y x k 1 + f (x k y Q L L ) } = argmn y x k + 1 y Q L f (xk ) }. In other words, the second step of Agorthm.1.1 s a projected gradent step wth step sze 1/L. In a sum, the terate x k+1 n step 4 s a convex combnaton of z k and y k wth growng mpact of the projected gradent step f the sequence z k } k 0 Q s bounded.

5 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 5. Dstrbuted Nesterov Agorthm wth event-trggered communcaton To approach a mut-agent framework that aows a parae mpementaton of Nesterov s optma frst order scheme.1.1, we make the foowng assumptons: Assumptons Besdes beng convex and cosed, the set Q R m s bounded and bock-separabe: Q = Q 1... Q s wth Q R m and m = m.. The a pror chosen prox-functon d(x) s separabe n the same way as Q: d(x) = d (x ) wth x Q for = 1,..., s, where d : R m R s a strongy convex functon wth convexty parameter σ. Assumptons..1 suggests to dvde the decson varabes nto s sub-bocks,.e., x = (x 1,..., x s ) Q R m wth x Q R m. Moreover, we denote n the foowng by agent the agent that s responsbe for updatng sub-bock x. In many cases t can be observed that that sub-bock of the gradent f (x) R m, whch s denoted by f (x) R m, does not depend on the whoe vector x Q,.e., agent needs to communcate ony wth the agents that contro the sub-bocks of x necessary to compute f (x). We descrbe ths reaton by a nformaton dependency graph (IDG) whch defnes the communcaton between the agents that correspond to the nodes of ths undrected graph. A more forma descrpton of ths observaton may be the foowng for = 1,..., s: Assumpton.. f (x) = f (y) x j = y j for a j N IDG () }, and x,y Q. Here N IDG () = j 1,..., j η } denotes the set of ndces of agent s η neghbors n the IDG. The cruca pont s that the maxma degree of the IDG, whch w appear n the convergence resut of the DNA-EC, can be assumed to be ndependent of the network sze for probems arsng n arge scae networks as the structure of the objectve functon s usuay ndependent of the network sze. Moreover, especay n the case of spacousy dstrbuted mut-agent networks, t s advantageous to reduce the communcaton traffc to the mnmum by ntroducng event-trggered communcaton. Here nformaton s transmtted by an agent n teraton k ony f the measured devaton of ts content from earer sent nformaton durng the optmzaton process exceeds a certan threshod k. Ths threshod has to be chosen appropratey to ensure overa convergence. To ntroduce event-trggered communcaton n a dstrbuted verson of Agorthm.1.1, we assume n the foowng wthout restrcton that 0 Q and defne smar to [] and [5] for = 1,..., s and k 0 by x,k = ( x,k 1 ),..., x,k s Q R m, where x,k j x k j 1 k R +, f j N IDG (), x,k j = 0, f j N IDG () }, (5) x,k = x k, the terates contanng the possby outdated versons of the sub-bocks x k j 1,..., x k j η for k 0 that are avaabe to agent who updates sub-bock x k. The threshod k n (5) defnes how far the outdated nformaton x,k j s aowed to devate from the atest nformaton x k j that agent j wth j N IDG () hods. If ths threshod s exceeded, agent j transmts the data to agent n order to ensure convergence of the dstrbuted agorthm. In the foowng we w set 0 = 0 whch s natura as there s no outdated nformaton at the begnnng of the optmzaton process,.e., every agent starts wth the same nformaton. Fnay under the above assumptons the DNA-EC can be stated as foows:

6 6 Martn C. Mene et a. Agorthm..3 (DNA-EC) For = 1,..., s and k 0 do n parae 1. Compute f (x,k ).. Fnd ỹ k = argmn f (x,k ),y x k + L k η y x k 1 + L y x k } y Q. L 3. Fnd z k = argmn z Q σ d (z ) + α j f (x, j ),z x j }. 4. Set x k+1 = τ k z k + (1 τ k)ỹ k. 5. Exchange nformaton: For j = 1,..., s f N IDG ( j) and x j,k x k+1 1 > k+1 then x j,k+1 = x k+1, ese x j,k+1 = x j,k. Note that we added the term L k η y x k 1 n step to be abe to prove convergence of the DNA- EC whch concdes wth Agorthm.1.1 f k = 0 for k 0. Furthermore, note that f (x,k ) has to be computed n the frst step ony f x,k x,k 1 for 1,..., s} and k 1,.e., event-trggered communcaton addtonay yeds a reducton of computatona effort. Aso, n a ot of cases the mnmzaton probems n step and 3 can be soved easy. If for nstance the feasbe set Q s component-wsey separabe and the prox-functon d(z) s chosen accordng to d(z) = (σ/) z, t s straghtforward to determne the cosed form soutons of these probems: Exampe..4 Let Q = Q 1 Q Q n wth compact and convex sets Q R and d(z) = (σ/) z. Then the probems n step of Agorthm..3 can be wrtten as: ỹ k = argmn f (x,k )(y x k ) + L kη y x k + L } y Q R (y x k ). Wth Q = [Q, Q ] and L = L k η, the mnmum ỹ k over Q for = 1,...,m s: ỹ k = max mn y, Q } }, Q where y = x k, f (x,k )+ L f (x,k ) L L + x k, f f (x,k )+ L L 0, L + x k, f f (x,k ) L L 0, ese. Accordngy, the soutons of the probems n step 3 are: z k = max mn z, Q } }, Q for = 1,...,m, k wth z α j f (x, j ) = L. Before we can prove the convergence of the DNA-EC we have to provde three emmas startng wth the frst where we make use of the Lpschtz contnuty assumpton of the gradent f (x) smary to [1]: Lemma..5 For y, x k, x,k Q and k 0 the foowng nequaty hods: f (y) f (x k ) + f (x,k ),y x k + L k η y x k 1 + L y x k.

7 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 7 Proof The Lpschtz contnuty assumpton of f (x) s equvaent to: f (y) f (x) + f (x), y x + L y x x,y Q. Wth the defnton of x,k n (5) and Assumpton.. we have for x k,y Q: y x k f (y) f (x k ) + f (x k ), y x k + L = f (x k ) + f (x,k ),y x k + f (x k ) + f (x,k ),y x k + f (x k ) f (x,k ),y x k + L L (x k j 1,..., x k j η ) (x,k j 1,..., x,k j η ) y x k + L y x k f (x k ) + f (x,k ),y x k + L k η y x k 1 + L y x k, where we use the Lpschtz contnuty of the gradent of f smar to [] to obtan (6). y x k Lemma..6 Appyng Lemma..5 to ỹ k computed n step of the DNA-EC we obtan for k 0: f (ỹ k ) f (x k ) + mn f (x y Q,k ),y x k + L k η y x k 1 + L y x k. Let us defne the constant ρ k = η LC (6) α j j for k 0, (7) where η = max,...,s η s the maxmum number of neghbors of each agent n the IDG and C s the dameter of the cosed, convex and bounded set Q wth respect to the 1-norm: C = max y,x Q x y 1. (8) Obvousy, one has ρ 0 = 0 as we assumed 0 = 0,.e., the agents start wth the same nformaton. For k 0 we set Ψ k = mn z Q ρ k + L σ d(z) + α j f (x j ) + f (x, j ),z x j, (9) z k = argmn z Q ρ k + L σ d(z) + α j f (x j ) + f (x, j ),z x j, (10) where z k concdes wth the terate that has to be computed n step 3 of the DNA-EC. Fnay, we w descrbe the error that occurs n the convergence proof of the DNA-EC due to the usage of event-trggered communcaton wth E k = A k 1 τ k 1 f (x k ) f (x,k ),ỹ k 1 z k 1 for k 1, (11)

8 8 Martn C. Mene et a. and set E 0 = 0. Let the startng pont x 0 be chosen accordng to (3). Wthout oss of generaty t can be assumed that d(x 0 ) = 0 (otherwse choose ˆd(x) = d(x) d(x 0 ) nstead). In the foowng emma we prove a reaton between the sequences of ponts x k } k 0, ỹ k } k 0, and z k } k 0 generated by the DNA-EC, budng the bass of our convergence resut: Lemma..7 Let the sequence α k } k 0 satsfy the condton: and set Then the reaton hods for k 0. α 0 (0, 1], α k+1 A k+1, k 0 (1) x k+1 = τ k z k + (1 τ k )ỹ k. (13) Ψ k A k f (ỹ k ) + E j (14) Proof The proof extends that of Lemma 1 n [1] and uses nducton. For k = 0 we have d(z) d(x 0 ) + d(x }} 0 ) T (z x 0 ) + σ z x } } 0 σ z x 0, =0 0 due to the strongy convexty of d(z). It foows that Ψ 0 = mn ρ 0 + L z Q }} σ d(z) + α 0 f (x0 ) + f (x,0 ),z x 0 =0 L α 0 mn z x 0 + f (x z Q 0 ) + f (x,0 ),z x 0 α 0 α 0 f (ỹ 0 ) = A 0 f (ỹ 0 ) + E }} 0, =0 where the ast nequaty foows wth Lemma..6. Now assume that the reaton Ψ k A k f (ỹ k )+ k E j hods for some k N 0. As the functon h k (z) := ρ k + L σ d(z) + α j f (x j ) + f (x, j ),z x j s strongy convex wth convexty parameter L for k 0, we have Ψ k+1 = mn z Q ρ k + η LCα k+1 k+1 + L k+1 σ d(z) + α j f (x j ) + f (x, j ),z x j mn z Q Ψ k + L z z k + η LCα k+1 k+1 + α k+1 f (xk+1 ) + f (x,k+1 ),z x k+1 mn Ψ k + L z z k + η Lα k+1 z k+1 z k 1 + z Q α k+1 f (xk+1 ) + f (x,k+1 ),z x k+1.

9 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 9 Due to the convexty of f, the defnton of x k+1 n (13), and the nducton hypothess, we have Ψ k + α k+1 f (xk+1 ) + A k f (ỹ k ) + α k+1 f (xk+1 ) + f (x,k+1 ),z x k+1 f (x,k+1 ),z x k+1 A k ( f (x k+1 ) + f (x k+1 ), ỹ k x k+1 ) + α k+1 f (xk+1 ) + = A k f (xk+1 ) + α k+1 f (xk+1 ) + = A k+1 f (x k+1 ) + α k+1 f (x,k+1 ),ỹ k xk+1 f (x,k+1 ),z x k+1 + A k + E j f (x,k+1 ),z z k + A k f (x k+1 ) f (x,k+1 ),τ k (ỹ k zk ) + E j, } } = E k+1 Ψ k+1 A k+1 f (x k+1 ) + mn z Q + where the ast equaty foows wth (13) and the fact that τ k = α k+1 /A k+1. From condton (1) t foows that A 1 k+1 τ k and we obtan: η Lα k+1 z k+1 z k 1 + L For arbtrary z Q defne α k+1 f (x,k+1 ),z z k k+1 + = A k+1 f (x k+1 ) + A k+1 mn z Q τ k f (x,k+1 ),z z k k+1 + A k+1 f (x k+1 ) + A k+1 mn z Q τ k E j f (x,k+1 ),z x k+1 + E j f (x k+1 ) f (x,k+1 ),ỹ k xk+1 + E j η L k+1 τ k z z k 1 + E j z z k + L A k+1 z z k + η L k+1 τ k z z k 1 + L τ k z z k + f (x,k+1 ),z z k k+1 + E j (15) y = τ k z + (1 τ k )ỹ k. As τ k [0,1], we have y Q and wth the defnton of x k+1 n (13) we can wrte: y x k+1 = τ k (z z k ).

10 10 Martn C. Mene et a. It foows that mn z Q η L z k+1τ k z k 1 + L τ k z z k + τk f (x,k+1 ),z z k = mn y τ k Q+(1 τ k )ỹ η k L k+1 y x k L y x k+1 + f (x,k+1 ),y x k+1 mn y Q η L k+1 y x k L y x k+1 + f (x,k+1 ),y x k+1 mn y Q L k+1 η y x k L y x k+1 + f (x,k+1 ),y x k+1 f (ỹ k+1 ) f (x k+1 ). (16) Substtutng (16) n (15) yeds Ψ k+1 A k+1 f (ỹ k+1 ) + k+1 E j. Fnay, we use the resut of Lemma..7 to prove convergence of the DNA-EC: Theorem..8 Let the sequence x k } k 0 and ỹ k } k 0 be generated by the DNA-EC wth α k as n Lemma.1.3 and k = βδ k where δ (0,1) and β R +. Then for k 0 the nequaty f (ỹ k ) f (x ) < σ6η βlcg (δ) + 4Ld(x ) σ(k + 1)(k + ) (17) hods, where x s an optma souton to probem (1), C s defned as n (8), and g(δ) = j = 1/(1 δ) for δ (0,1).

11 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 11 Proof To prove the theorem we have to derve an upper bound for the eft-hand sde Ψ k n nequaty (14) and a ower bound for the error k E j occurrng n the rght-hand sde. We start wth Ψ k = mn z Q ρ k + L σ d(z) + α j f (x j ) + f (x, j ),z x j = mn z Q ρ k + L σ d(z) + ( α j f (x j ) + f (x j ), z x j ) + α j f (x, j ) f (x j ),z x j mn z Q ρ k + L σ d(z) + ( α j f (x j ) + f (x j ), z x j ) + α j L (x, j j 1,..., x, j j η ) (x j j 1,..., x j j η ) z x j (18) ρ k + L σ d(x ) + A k f (x ) + Lη α j x j x j 1 (19) η LC α j j + L σ d(x ) + A k f (x ) = η βlc ( j + 1)δ j + L σ d(x ) + A k f (x ) < η βlcg (δ) + L σ d(x ) + A k f (x ), where we used the Lpschtz contnuty assumpton of the gradent of f to obtan (18) and the fact that f s convex (as t was done n Nesterov s proof of Theorem n [1]) to obtan (19). Smary, we derve a ower bound for the accumuated error k E j : E j A j 1 τ j 1 f (x j ) f (x, j ),ỹ j 1 z j 1 j=1 A j 1 τ j 1 L (x j j 1,..., x j j η ) (x, j j 1,..., x, j j η ) ỹ j 1 z j 1 j=1 η L j=1 j=1 A j 1 τ j 1 j ỹ j 1 z j 1 η LC A j 1 τ j 1 j = η βlc 1 j=1 j=1 j( j + 1) 4 j + δ j > η βlc g (δ). Substtutng these bounds n (14) resuts n: ( 3 f (ỹ k ) f (x ) < 4 η βlcg (δ) + L ) σ d(x ) /((k + 1)(k + )).

12 1 Martn C. Mene et a. 3 Appcaton of the DNA-EC n the Proxma Center Agorthm 3.1 Proxma Center Agorthm In ths secton we brefy descrbe the PCA [10] that s appcabe for partay separabe convex probems of the foowng form: mn Φ (x ) : A x = b A, B x b B, (0) x X (,...,n) where Φ : R m R s a contnuous and convex but not necessary smooth functon on a gven compact and convex set X for = 1,...,n. A denotes a m A m matrx, B denotes a m B m matrx ( = 1,...,n), b A R m A, and b B R m B. For the ease of notaton we denote n the foowng by (µ,λ) the vector (λ T,µ T ) T. As the probem structure suggests, the authors use a dua decomposton approach resutng n a bockseparaby constraned dua probem. To obtan a contnuousy dfferentabe dua objectve functon that can be evauated n parae, scaed prox-functons d X wth convexty parameters σ X > 0 ( = 1,...,n) are added to the Lagrangan of (0) resutng n the foowng augmented Lagrangan L c (x 1,..., x n,µ,λ) = Φ (x ) + A x b A,µ + B x b B,λ + c d X (x ), whch s separabe n the prma varabes x ( = 1,...,n). It foows, that the correspondng augmented dua objectve functon f c (µ,λ) = mn Φ (x ) + A x b A,µ + B x b B,λ + c d X (x ) (1) x X (,...,n) can be evauated n parae. As mentoned n [10], dfferent smoothng parameters c X ( = 1,...,n) nstead of snge parameter c can be consdered as we and we show n subsecton 3.5 how to optmay choose them for certan prox-functons. The augmented dua objectve f c s concave and due to the unqueness of the mnmzers x (µ,λ), that sove the rght-hand sde of equaton (1), contnuousy dfferentabe [10]. Moreover, t can be shown (cf. Theorem 3.4 n [10]) that the gradent ( n ) A f c (µ,λ) = x (µ,λ) b A n B x (µ,λ) b B s Lpschtz contnuous wth the Lpschtz constant L c = A + B cσ X. () Let M Λ be the set of optma dua mutpers, assumed to be nonempty n the foowng. Appyng Nesterov s agorthm.1.1 to the probem arg max f c (µ,λ) = argmn f c (µ,λ), (3) (µ,λ) Q A Q B (µ,λ) Q A Q B where Q A Q B R m A R m B + s a gven cosed and convex set that contans at east one optma dua mutper (µ,λ) M Λ, resuts n the foowng PCA [10]:

13 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 13 Agorthm (PCA) For k 0 do 1. Gven (u,h) k Q A Q B, for = 1,...,n compute (v,t) Q A Q B x k+1 = argmn x X Φ (x ) + u k, A x + h k, B x + cdx (x ) }. ( n. Compute f c ((u,h) k A ) = x k+1 ) b A n B x k+1. b B 3. Fnd (µ,λ) k = argmax fc ((u,h) k ),(µ,λ) (u,h) k L c (µ,λ) Q A Q B 4. Fnd (v,t) k = argmax L c σ d(v,t) + 5. Set (u,h) k+1 = k+3 (v,t)k + k+1 k+3 (µ,λ)k. (µ,λ) (u,h) k }. j + 1 fc ((u,h) j ),(v,t) (u,h) j. Note that we omtted agan some constant terms n the orgna verson of the argmax-probems as done n subsecton.1 to revea the paraezabe character of the PCA Fnay, d(v, t) n step 4 denotes the prox-functon for the set Q A Q B wth convexty parameter σ and center (u,h) 0 = argmn (u,h) QA Q B d(u,h). To cose ths secton we state a convergence resut of the PCA that can be proved usng the foowng Lemma whch gves a ower and an upper bound on the prma gap: Lemma 3.1. For every (µ,λ) M Λ, (µ,λ) Q A Q B, and ˆx X ( = 1,...,n) the foowng nequates hod: n (µ,λ) A ˆx b A [ ] n + Φ B ˆx b ( ˆx ) f Φ ( ˆx ) f 0 (µ,λ), (4) B where f = f 0 ((µ,λ) ) and [ ] + denotes the projecton onto R m B +. Proof The proof s smar to the proof of Lemma 3.3 n [10]. We have f = mn Φ x X (,...,n) (x ) + A x b A,µ + B x b B,λ Φ ( ˆx ) + A ˆx b A,µ + B ˆx b B,λ ( n A Φ ( ˆx ) + ˆx b ( A µ n ), ) B ˆx b B λ [ n A Φ ( ˆx ) + ˆx b ] + ( A µ n, ) B ˆx b B λ n A Φ ( ˆx ) + ˆx b A [ ] n + (µ,λ), B ˆx b B where the ast nequaty foows by the Cauchy-Schwarz nequaty. Due to the compactness of the sets X, postve and fnte constants D X exst whch satsfy: D X max x X d X (x ) for = 1,...,n. (5)

14 14 Martn C. Mene et a. Theorem Assume that Q A Q B = R m A R m B + and d(µ,λ) = (σ/) (µ,λ). Takng c = ɛ/ n D X n (1) and k + 1 = L c /ɛ, where L c = ( n D X /ɛ )( n ( A + B ) /σ X ), then after k teratons of Agorthm an approxmate souton to the probem (0) s gven by ˆx = ( j + 1) (k + 1)(k + ) x j+1 for = 1,...,n (6) and wth (ˆµ, ˆλ) = (µ,λ) k the foowng bounds on the duaty gap are satsfed: ( (µ,λ) (µ,λ) ) + (µ,λ) + ɛ as we as the foowng bound on the constrant voaton: n A ˆx b A ( (µ,λ) [ ] n + ɛ ) + (µ,λ) B ˆx b +. B Φ ( ˆx ) f 0 (ˆµ, ˆλ) ɛ, (7) Proof Appyng Lemma 3.1. the proof s amost dentca to the proof of Theorem 3.7 n [10]. 3. Scang the separabe convex probem Before we present the DPCA-EC n the foowng subsecton, we propose a scang technque for probem (0) to baance the bounds (7) on the duaty gap n Theorem In partcuar, f the norm of the optma dua mutpers (µ,λ) n the eft-hand sde of (7) s very arge, ɛ has to be chosen very sma to ensure a tght ower bound on the duaty gap, whch ncreases the necessary number of teratons k as we have k + 1 = Lc ɛ = ɛ D X A + B To remedy ths drawback we propose the foowng scang approach for probem (0) wth scang factor s > 1: Defne b A (s) = sb A, b B (s) = sb B, X (s) = sx, x (s) = sx ( = 1,...,n), and sove mn x (s) X (s)(,...,n) ( ) 1 Φ s x (s) : σ X A x (s) = b A (s), Ths scang approach resuts n optma dua mutpers of the form: (µ(s),λ(s)) 1 = s (µ,λ),. B x (s) = b B (s). (8) and the number of teratons k necessary to reach the desred accuracy ɛ n Theorem s gven by s k = 4 ɛ D X A + B σ X 1. (9)

15 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 15 For the bounds on the duaty gap and the constrant voaton we obtan: 1 (µ,λ) s 1 (µ,λ) 1 + s s (µ,λ) + ɛ Φ ( ˆx ) f ɛ, (30) and n A s ˆx b A [ ] n + ɛ B ˆx b 1 (µ,λ) 1 + B s s (µ,λ) +. (31) It s cear, that ncreasng the scang factor s > 1 rases the Lpschtz constant L c and by mpcaton the number of teratons k n (9) n the same order of magntude as decreasng ɛ does, but wth respect to the ower bound on the duaty gap n (30) and the bound on the constrant voaton n (31), t s favourabe to ncrease s nstead of decreasng ɛ. 3.3 Dstrbuted Proxma Center Agorthm wth event-trggered communcaton We w show now that the compexty O(1/ɛ) of the PCA can be mantaned appyng the DNA-EC deveoped n subsecton. to update the dua mutpers n parae whch yeds the DPCA-EC. To do so we agan use Assumpton.. whch requres a sutabe structure of the matrces A and B n (0) to avod a compete IDG. We omt detas here and antcpate that a sutabe structure of the constrants s gven n network fow probems such as the DC-OPF probem consdered n secton 4. To ensure a parae mpementaton of step 1 n Agorthm after ntroducng event-trggered communcaton wth (u,h),k ( = 1,..., s) defned accordng to (5), we further assume that the agents use the same outdated nformaton of common neghbors n the IDG to compute ther sub-bocks of f c and that agent uses the outdated verson of hs sub-bock of the dua mutpers n the computaton of f c ((u,h),k ): Assumpton Remark 3.3. If (N IDG () }) N IDG ( j) then (u,h),k = (u,h) j,k for k 0 and, j, = 1,..., s. () Note that ths assumpton s not restrctve at a as the convergence resut for the DNA-EC n Theorem..8 st hods substtutng η by η + 1 n (17), where η s the maxma degree of the IDG. () Assumpton aows the defnton of a "goba" vector (ū, h) k that contans the possby outdated nformaton avaabe to the agents: (ū, h) k = (u,h),k for k 0 and = 1,..., s. (3) Note that wth Assumpton.. t foows that f c ((ū, h) k ) = f c ((u,h),k ). To state the DPCA-EC we denote n the foowng by agent x the agent that updates x k n teraton k and has avaabe a the possby outdated sub-bocks of (ū, h) k contaned n the vector (u,h) x,k necessary to compute x k+1 for k 0 and = 1,...,n. Moreover, denotng by agent the agent that updates sub-bock ( = 1,..., s) of the dua mutpers, the DPCA-EC s:

16 16 Martn C. Mene et a. Agorthm (DPCA-EC) For k 0 do For = 1,...,n gven (u,h) x,k, agent x 1. computes x k+1 = argmn x X Φ (x ) + u x,k, A x + h x,k, B x + cdx (x ) }. For = 1,..., s gven the vectors x k+1 necessary for the computaton of f c ((u,h),k ), agent ( n. computes f c ((u,h),k A ) = x k+1 ) b A n B x k+1, b B 3. fnds 4. fnds (ṽ, t) k = argmax (v,t) (Q A Q B ) ( µ, λ) k = argmax f c ((u,h),k ),(µ,λ) (u,h) k (µ,λ) (Q A Q B ) L c k (η + 1) (µ,λ) (u,h) k 1 L ( ) c (µ,λ) (u,h) k }, L c σ d ((v,t) ) + 5. sets (u,t) k+1 = k + 3 (ṽ, t) k + k + 1 k + 3 ( µ, λ) k, 6. and exchanges nformaton: f (u,h),k (u,h) k+1 > k+1 then 1 (u,h) x,k+1 = (u,h) k+1, ese (u,h) x,k+1 = (u,h),k. Remark j + 1 fc ((u,h), j ),(v,t) (u,t) j, () Note that the communcaton takes pace ony between the prma and the dua agents,,.e., the graph representng the communcaton topoogy s dfferent from the IDG n subsecton.. We show n the next secton how ntutve choces of agent x agent 1,...,agent s } ( = 1,...,n) for sovng the DC- OPF probem ensures oca communcaton n the cassca sense,.e., the communcaton network has the same topoogy as the power network. Moreover, ths choce can be apped n genera for sovng network-fow probems wth the DPCA-EC. () Obvousy, x k+1 has to be computed ony f (u,h) x,k (u,h) x,k 1,.e., the event-trggered dua communcaton (the exchange of dua terates) n step 6 of the DPCA-EC nduces a not coser specfed event-trggered prma communcaton n step 1. To prepare the convergence proof of the DPCA-EC under the necessary assumpton that Q A Q B s cosed and bounded (cf. Theorem..8), we provde the foowng emma usng the quantty P( ) defned by P( ) = (η + 1)βL c Cg (δ), (33) where C = max (µ,λ),(χ,ξ) QA Q B (µ,λ) (χ,ξ) 1 s the dameter of Q A Q B, β R + s the coeffcent n the threshod k = βδ k wth δ (0,1), and g(δ) = δ j = 1/(1 δ) ( cf. Theorem..8).

17 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 17 Lemma The foowng nequaty hods for ( µ, λ) k n the DPCA-EC and (ū, h) k defned accordng to (3) for k 0: (k + 1)(k + ) 4 f c (( µ, λ) k ) max (µ,λ) Q A Q B L c σ d(µ,λ) + j + 1 ( fc ((ū, h) j )+ fc ((ū, h) j ),(µ,λ) (ū, h) j )} P( ). (34) Proof For the choce α k = (k + 1)/ for k 0 n Lemma..7 we obtan (k + 1)(k + ) 4 where ρ k s defned by f c (( µ, λ) k ) max (µ,λ) Q A Q B ρ k L c σ d(µ,λ) + fc ((ū, h) j ),(µ,λ) (u,h) j )} ρ k = (η + 1) L c C j=1 j + 1 βδ j, j + 1 ( fc ((u,h) j )+ E j, accordng to Remark It can be easy shown (cf. wth the proof of Theorem..8) that j=1 ρ k P( ) 4 and j=1 E j P( ) 4, whch yeds We have (k + 1)(k + ) 4 f c (( µ, λ) k ) fc ((ū, h) j ),(µ,λ) (u,h) j max (µ,λ) Q A Q B L c σ d(µ,λ) + j + 1 ( fc ((u,h) j )+ fc ((ū, h) j ),(µ,λ) (u,h) j )} P( ). = f c ((u,h) j ),(µ,λ) (u,h) j + f c ((u,h) j ),(µ,λ) (u,h) j L c f c ((u,h), j ) f c ((u,h) j ),(µ,λ) (u,h) j (u,h), j (u,h) j (µ,λ) (u,h) j } } (η +1) j f c ((u,h) j ),(µ,λ) (u,h) j (η + 1) L c (µ,λ) j (u,h) j 1, } } C and accordngy fc ((u,h) j ),(µ,λ) (ū, h) j f c ((ū, h) j ),(µ,λ) (ū, h) j (η + 1) L c C j.

18 18 Martn C. Mene et a. Fnay, wth the concavty of f c we obtan for a (µ,λ) Q A Q B that = j + 1 ( fc ((u,h) j ) + f c ((u,h) j ),(µ,λ) (u,h) j ) P( ) (η + 1) L c C j j + 1 ( fc ((u,h) j ) + f c ((u,h) j ),(µ,λ) (u,h) j ) 3P( ) 4 j + 1 ( fc ((u,h) j ) + f c ((u,h) j ),(µ,λ) (u,h) j + (ū, h) j (ū, h) j ) 3P( ) 4 j + 1 ( fc ((ū, h) j ) + f c ((u,h) j ),(µ,λ) (ū, h) j ) 3P( ) 4 j + 1 ( fc ((ū, h) j ) + f c ((ū, h) j ),(µ,λ) (ū, h) j ) 3P( ) (η + 1) L c C j 4 j + 1 ( fc ((ū, h) j ) + f c ((ū, h) j ),(µ,λ) (ū, h) j ) P( ). Appyng Lemma resuts n an estmate of the duaty gap after k teratons of the DPCA-EC. Theorem After k teratons of the DPCA-EC we obtan an approxmate souton ˆx = ( j + 1) (k + 1)(k + ) x j+1 for = 1,...,n to probem (0) and (ˆµ, ˆλ) = ( µ, λ) k whch satsfes the foowng upper bound on the duaty gap: Φ ( ˆx ) f 0 (ˆµ, ˆλ) c D X max 4L c (k + 1) σ d(µ,λ) + A ˆx b A,µ + (µ,λ) Q A Q B B ˆx b B,λ + 4P( ) (k + 1) Proof Usng nequaty (34) the proof s amost dentca to the proof of Theorem 3.4 n [10]. Wth Theorem we can now prove the convergence of the DPCA-EC mantanng the effcency estmate of O(1/ɛ): Theorem Assume that there exsts R > 0 such that the set Q A Q B has the form Q A Q B = (µ,λ) R m A R m B + : µ max R, λ max R } and contans a (µ,λ) M Λ wth (µ,λ) < R. Denote by D a fnte constant wth D max (µ,λ) QA Q B d(µ,λ). Moreover, et k, g(δ), and C be defned as n Theorem..8. Takng c = ɛ/ ( n =0 D X ) wth ɛ > 0 n (1) and k + 1 = Lc E( ) ɛ,

19 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 19 where L c = ( n =0 D X /ɛ )( n=0 ( A + B ) /σ X ), and E( ) = (D + σ4(η + 1)βCg (δ))/σ. Then after k teratons of the DPCA-EC an approxmate souton to the probem (0) s gven by ˆx = ( j + 1) (k + 1)(k + ) x j+1 for = 1,...,n, and wth (ˆµ, ˆλ) = ( µ, λ) k the foowng bounds on the duaty gap are satsfed: as we as the foowng bound on the constrant voaton: (µ,λ) R (µ,λ) ɛ Φ ( ˆx ) f 0 (ˆµ, ˆλ) ɛ, (35) n A ˆx b A [ ] ɛ n + B ˆx b B R (µ,λ). (36) Proof The proof s smar to the proof of Theorem 3.6 n [10]. If we have a ook at the resut of Theorem Φ ( ˆx ) f 0 (ˆµ, ˆλ) c D X max 4L c (k + 1) σ d(µ,λ)+ (µ,λ) Q A Q B } A ˆx b A,µ + B ˆx b B,λ + 4P( ) (k + 1) (37) the task s to mnmze the rght-hand sde of the nequaty wth respect to c. For the maxmzaton part we obtan wth the defnton of D and Q A Q B = (µ,λ) R m A R m B + : µ max R, λ max R } that max (µ,λ) Q A Q B and for nequaty (37) we obtan 4L c (k + 1) σ d(µ,λ) + 4L cd (k + 1) σ + max µ Q A = 4L cd (k + 1) σ + R A ˆx b A,µ + A ˆx b A,µ + max λ Q B B ˆx b B,λ B ˆx b B,λ + A ˆx b A + R B ˆx b B 1 4L cd (k + 1) σ + R n A ˆx b A [ ] n +, B ˆx b B Φ ( ˆx ) f 0 (ˆµ, ˆλ) c D X + 4L cd (k + 1) σ R n A ˆx b A [ ] n + + 4P( ) B ˆx b B (k + 1). (38) c D X + 4L cd (k + 1) σ + 4P( ) (k + 1). (39) 1

20 0 Martn C. Mene et a. Wth L c = A + B cσ X and P( ) = (η + 1)βL c Cg (δ) we can express the rght-hand sde of (39) as a functon h(c) wth ( 4D h(c) = c D X + L c (k + 1) σ + 8(η + ) 1)βCg (δ) (k + 1) = c D X + 1 A + B 4D + σ8(η + 1)βCg (δ) c (k + 1) σ To get the mnmum of h we have to sove h (c) = c 1, = ± σ X D X 1 A c + B σ X A + B σ X and as c n (1) has to be postve, we choose Fnay, we get c = 1 A k B σ X h(c ) = A k B σ X 4D + σ8(η + 1)βCg (δ) (k + 1) σ 4D + σ8(η + 1)βCg (δ) (k + 1) σ n D X. = 0 4D + σ8(η + 1)βCg (δ) σ n D X. (40) (4D + σ8(η + 1)βCg (δ)) n D X, σ and wth k + 1 = ɛ A + B (4D + σ8(η + 1)βCg (δ)) ( ) n D X, σ σ X we obtan the rght-hand sde of nequaty (35) and the vaue for c = c. Wth nequaty (4) and nequaty (38) we get ( R (µ,λ) ) n A ˆx b A [ n B ˆx b B ] + c D X + 4L cd (k + 1) σ + 4P( ) (k + 1), and the bound on the constrant voaton (36) foows mmedatey by repacng c wth c. Agan appyng nequaty (4) yeds the ower bound on the duaty gap.

21 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton Optma convexty parameters In ths subsecton we show how to optmay determne the convexty parameter σ X ( = 1,...,n) for the foowng choce of prox-functons used for smoothng the dua functon of (0): d X (x ) = σ X x wth x X for = 1,...,n. (41) Ths s mportant as arbtrary chosen convexty parameters can ead to a sgnfcanty arger Lpschtz constant L c whch ncreases the amount of teratons k needed n the PCA and the DPCA-EC necessary to reach the desred accuracy ɛ. We set v = ( A 1 + B 1,..., A n + B n ) T, d X = (max x1 X 1 x 1,...,max xn X n x n ) T, and σ X = (σ X1,...,σ Xn ) T. Then the smoothng parameter c and the Lpschtz constant L c n Theorem can be wrtten n the foowng way: c = ɛ dx T σ X and L c (σ X ) = dt X σ X ɛ v σ X. Note that the scang σ X ζσ X wth ζ > 0 does not change L c (σ X ). We thus can constran the mnmzaton of L c (σ X ) by the normazaton condton d T X σ X = 1 and consder the foowng optmzaton probem: The optmaty condton s Usng the constrant arg mn Lc (σ X ) : dx T σ X = 1 } = argmn σ X >0 (,...,n) v + µd σ X = 0 X σ X > 0 σ X = 1 = d T X σ X = 1 µ σ X >0 (,...,n) v : dx T σ σ X = 1 X. v µd X for = 1,...,n. v d X, the dua mutper µ s gven by µ = v d X, and we obtan the optma convexty parameter σ X = 1 v for = 1,...,n. nj=1 v j d d X X j Note that the same resut s obtaned f c = ɛ/d T X σ X as n Theorem Moreover, the optma convexty parameters can be computed n cosed form and n parae, whch s advantageous n a dstrbuted settng.

22 Martn C. Mene et a. 3.5 Optma smoothng parameters To further reduce the number of teratons of the PCA, we ntroduce mutpe postve smoothng parameters c X ( = 1,...,n) nstead of a snge parameter c n (1) and show how to optmay choose them. Smar to Theorem 3.1 n [10] t can be shown that the Lpschtz constant L c wth c = (c X1,...,c Xn ) T of the gradent of the resutant augmented dua functon s f c (µ,λ) = mn x X (,...,n) Φ (x ) + A x b A,µ + B x b B,λ + c X d X (x ) L c (σ X ) = A + B (4) c X σ X. (43) Takng the smoothng parameters c X ( = 1,...,n) nto account, we can rewrte Theorem as foows: Theorem Let Q A Q B = R m A R m B +, d(µ,λ) = (σ/) (µ,λ), and defne D X = ( D X1,..., D Xn ) T. Take c = ( c X1,...,c Xn ) T n (4) such that c T D X = ɛ and k + 1 = L c /ɛ. Then after k teratons of Agorthm an approxmate souton to the probem (0) s gven by ˆx = ( j + 1) (k + 1)(k + ) x j+1 for = 1,...,n and wth (ˆµ, ˆλ) = (µ,λ) k the foowng bounds on the duaty gap are satsfed: ( (µ,λ) (µ,λ) ) + (µ,λ) + ɛ Φ ( ˆx ) f 0 (ˆµ, ˆλ) ɛ, as we as the foowng bound on the constrant voaton: n A ˆx b A ( (µ,λ) [ ] n + ɛ ) + (µ,λ) B ˆx b +. B Proof Appyng Lemma 3.1. the proof s amost dentca to the proof of Theorem 3.7 n [10]. To answer the arsng queston of how to optmay choose the smoothng parameters c X ( = 1,...,n) that mnmze the Lpschtz constant and satsfy the condton c T D X = ɛ, we consder an optmzaton probem smar to the one n the prevous subsecton: v arg mn : c T D X = ɛ c X σ X, and obtan the souton c X = c X >0 (,...,n) ɛ nj=1 v j D X j σ X j v for = 1,...,n. (44) σ X D X

23 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 3 n the same way. Note that for the choce of prox-functons d X (x ) = ( σ X / ) x and c X accordng to (44) the dua augmented functon f c (µ,λ) n (4) as we as the Lpschtz constant L c of the gradent ( of f c (µ,λ) n (43) do not depend on the convexty parameters σ X > 0 as we have c X σ X = ɛ/ v j v j d X j /) /d X for = 1,...,n, wth d X = (d X1,...,d Xn ) T defned as n the prevous subsecton. 4 Appcaton to the DC optma power fow probem In ths secton we dscuss the resuts of the DPCA-EC apped to sove the DC-OPF probem. The objectve of the DC-OPF probem s to determne the most economca dstrbuton of power generaton to serve gven oads n a power system, restrcted by operatona mts of generaton and transmsson factes as we as Krchhoff s Current Law expressed by the power fow equatons. We show that the probem can be soved n parae wth event-trggered and oca communcaton between the agents,.e., the communcaton topoogy equas the topoogy of the power system. We mode the power system accordng to [5,] by a drected graph G = V, E} wth V = v 1,...,v n } representng the set of buses = 1,...,n, where a buses contan a oad and for smpcty of notaton are drecty connected to a generator. The generazaton of ony p < n buses drecty connected to a generator s straghtforward. The edge e j E V V wth E = m represents the transmsson ne from bus to bus j. Let I be the m n ncdence matrx of the graph G and defne a dagona matrx D R m m wth d = 1/x where x denotes the reactance of the th transmsson ne. Moreover, we defne the varabes of the DC-OPF probem and ther feasbe sets: Let θ Θ be the phase ange of the votage at bus, gven the compact and convex set Θ = [ θ mn for = 1,...,n, and set θ = (θ 1,...,θ n ) T. Let P g P be the generated power at bus, gven the compact and convex set P = [ P g = 1,...,n, and set P g = (P g 1,..., Pg n) T.,θ max ] mn gmax ], P for In the DC mode of a power system the actve power fow between two neghborng buses and j s approxmated by F j = 1 x j (θ θ j ) = 1 x j (θ j θ ) = F j, where x j = x j s the reactance of the transmsson ne connectng bus and bus j. Accordngy, the power fow on every transmsson ne can be expressed wth Aθ after defnng the weghted ncdence matrx A = DI R m n. Gven the oads P d = (P d 1,..., Pd n) T at each bus and the upper and ower ne fow mts F max = (F1 max,..., Fm max ) T and F mn = (F1 mn,..., Fm mn ) T on each transmsson ne, the convex DC-OPF probem that we consder has the form: subject to mn P g P,θ Θ Φ (P g ) (45) Bθ = P g P d (46) F mn Aθ F max (47)

24 4 Martn C. Mene et a. where Φ (x) = a x + a 1 x + a 0 s the quadratc cost of power producton at bus wth non-negatve coeffcents a,a 1,a 0 for = 1,...,n and matrx B R n n s defned as: 1 k N() x B = I T k, f = j, 1 DI = B j = x j, f j N(), 0, ese, where N() = j (v,v j ) E (v j,v ) E } denotes the set of ndces of v s neghbors. Ceary, constrant (47) mts the power fow on each transmsson ne, whereas constrant (46) expresses the power fow equatons. To smoothen the Lagrangan of the DC-OPF probem we choose the optma smoothng parameters c = (c 1,...,c n ) T accordng to (44) n subsecton 3.5 and the prox-functons d (x ) = (σ /) x wth σ > 0 for = 1,...,n, correspondng to the number of prma varabes P g and θ. Note that the convexty parameter σ > 0 can be chosen arbtrary as mentoned n subsecton 3.5. We obtan the foowng augmented dua objectve functon: f c (µ,λ) = = mn P g P, θ Θ m Φ (P g ) + µ ( P g (Bθ) P d mn P g P ) + Φ (P g ) + µ P g + c σ λ ( (Aθ) F max c σ Pg Pg mn θ Θ (λ λ +m )A m L() ( λ+m F mn λ F max } + j N() } + ) + m c +n σ +n ( ) λ +m ( Aθ) + F mn + θ µ j B j θ + c +nσ +n θ + ) µ P d, (48) where L() = v e e E} denotes the set of ndces of nes connected to bus. As mentoned above, the DC-OPF probem can be soved n parae by the DPCA-EC wth oca communcaton for the foowng ntutve dstrbuton of agents: At each bus an agent s nstaed that contros the dua varabe µ correspondng to the power fow baance constrant at bus, as we as the prma varabes P g and θ for = 1,...,n. Moreover, at each transmsson ne two agents, agent +n and agent +n+m, are nstaed that contro the dua mutpers λ and λ +m correspondng to the upper and ower ne fow mts at transmsson ne for = 1,...,m. Ths settng contanng m + n agents ensures that the communcaton takes pace ony ocay,.e., between topoogcay neghbored agents wth respect to the power network as can be seen n the foowng adjusted notaton of the DPCA-EC to sove the DC-OPF probem. We we choose d = (σ/) (µ,λ) accordng to the Assumptons..1 made n subsecton.. Note that for ths choce of prox-functon the convexty parameter σ > 0 can be chosen arbtrary (cf. step 4 of Agorthm 3.3.3):

25 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 5 Agorthm 4.0. For k 0 do n parae: 1. For = 1,...,n, gven ū k j f j N() } and h k, h k +m f bus s connected to transmsson ne,.e., f L(), where ū k and h k are defned accordng to (3), agent soves Φ (P g ) + ūk Pg + c σ }, P g,k+1 θ k+1 = argmn P g P = argmn θ Θ Pg ( h k h k +m )A L() j N() } ū k j B j θ + c +nσ +n f not aready done n a prevous teraton. After that he sends θ k+1 to agent j wth j N(), agent +n, and agent +n+m wth L().. Gven / L c = A + B j (c +n σ +n ) + 1/(c σ ), L() j N() } θ, For = 1,...,n, agent computes k := f c(ū k, h k ) = P g,k+1 µ j N() } B j θ k+1 j P d and soves µ k = argmax µ k (η + 1) L c µ k µ k L c µ R ṽ k = argmax v R L c v + v j + 1 j, } (µ µk ), where G() =, j e = (v,v j ) E } s the set of ndces of buses connected by transmsson ne. For = 1,...,m, agent +n computes the parta dervatve k +n := f c(ū k, h k ) λ = G() A θ k+1 F max and soves λ k = argmax λ k +n (η +n + 1) L c λ k λ k L c λ R + t k = argmax t R + L c t + t j + 1 j +n. } (λ λk ), Agent +n+m computes k +n+m := f c(ū k, h k ) λ +m = G() A θ k+1 + F mn

26 6 Martn C. Mene et a. and soves λ k +m = argmax λ k +n+m (η +n+m + 1) L c λ k λ k L c +m λ R + j + 1 t k +m = argmax t R + L c t + t k +n+m. } (λ λk +m ), +m ac- 3. For = 1,...,n and = 1,...,m the agent, agent +n, and agent +n+m update u k+1, h k+1, and h k+1 cordng to: u k+1 = k + 1 k + 3 µk + k + 3ṽk, h k+1 = k + 1 k + 3 λ k + k + 3 t k, h k+1 +m = k + 1 k + 3 λ k +m + k + 3 t k +m. 4. For = 1,...,n and = 1,...,m agent, agent +n, and agent +n+m exchange nformaton: f ū k u k+1 > k+1 then ū k+1 = u k+1 and sends u k+1 to agent j wth j N(), ese agent sends nothng and sets ū k+1 = ū k as we as agent j wth j N(). f h k hk+1 > k+1 then agent +n h k+1 = h k+1 and sends h k+1 to agent j wth L( j), ese agent +n sends nothng and sets h k+1 = h k as we as agent j wth L( j). Agent +m+n proceeds accordngy wth h k+1 +m and h k+1 +m. Remark A ook at the dervatves k j for j = 1,...,m + n yeds the number of neghbors η j of agent j n the IDG that descrbes the dependence of the jth component j f c (µ,λ) from the components controed by other agents (cf. Assumpton..): 3 N( j) N(), for j = 1,...,n, η j = 3 L( j) N(), for j = n + 1,...,n + m, 3 L( j m) N(), for j = n + m + 1,...,n + m. Ceary, a argmn- and argmax-probems n the above agorthm can be soved anaytcay, smar to as t was shown n Exampe..4, whch s very advantageous wth respect to the computatona compexty of each teraton. Another advantageous feature of the above agorthm s that the power varabes P g, whch can be nterpreted as prvate nformaton [6] that shoud be kept secret f generators beong to dfferent power suppers, doesn t have to be exchanged between the agents. Fnay, for the sake of competeness et us note that n the case of a 0 for = 1,...,n the cost functons Φ (x) are strongy convex wth convexty parameters a. In ths case we ony need to smoothen the Lagrangan of probem (45) wth respect to the phase ange varabes θ resutng n the foowng augmented

27 A Cass of Dstrbuted Optmzaton Methods wth Event-Trggered Communcaton 7 dua functon: f c (µ,λ) = mn Φ (P g P g P ) + µ P g } + mn θ Θ (λ λ +m )A c +n σ +n θ } + m ( λ+m F mn λ F max L() j N() } µ j B j θ + ) µ P d. (49) It s straghtforward to show that the Lpschtz constant of the gradent f c (µ,λ) of (49) s: L c = A + B j /(c +nσ +n ) + 1/(a ). L() j N() } We mpemented the above agorthm as we as the PCA n Matab R01a and compared them wth respect to the communcaton exchange. For the tests we used data from the power systems test case archve of the Unversty of Washngton that can be found n [4]. The frst probem that we dscuss s the IEEE 57 bus test case wth 80 transmsson nes and where 7 of the 57 buses are each connected to a generator (for ustraton we refer to [4]). We chose quadratc cost functons wth a 0 for = 1,...,7 and scaed the probem wth s = 30 accordng to subsecton 3.. We apped the sover IPOPT [19] to determne the optma scaed dua mutpers µ (s),λ (s) = < 1 and the optma vaue f of (45). For comparson we state here the prma gap of the approxmate soutons (51) at the startng pont (µ 0,λ 0 ),.e., at the center of the set Q = R n R m + accordng to (3): ( µ 0,λ 0) = argmnd Q (µ,λ), (50) (µ,λ) Q whch s K$/h. Moreover, the constrant voaton of the approxmate soutons (51) at the startng pont s For the choce ɛ = 0.7 after k = 3386 teratons of the PCA the approxmate soutons ( ˆP g, ˆθ) = ( j + 1) (k + 1)(k + ) (Pg, j+1,θ j+1 ), (51) where ˆP g = 0 f bus s not connected to a generator, shoud satsfy the foowng bounds on the prma gap and the constrant voaton accordng to nequaty (4) and Theorem (combned wth nequates (30) and (31) n subsecton 3.): Φ ( ˆP g ) f 0.7 (5) and ˆP g Bˆθ P [ d Aˆθ F max ] (53) Aˆθ + F mn Ths s verfed by the frst row of the foowng Tabe 1 whch shows the actua prma gap and constrant voaton of the souton obtaned wthout event-trggered communcaton. The rows - 5 of Tabe 1 show the prma gap and the constrant voaton of soutons obtaned wth the DPCA-EC for dfferent choces

28 8 Martn C. Mene et a. of β usng the threshod k = βδ k wth δ = Here δ s chosen such that there hods δ k 0.05 after haf of the requred number of teratons (k 33000/), whch ensures that δ k does not get too cose to zero too eary. In coumn 3 of Tabe 1 we state the amount of the overa communcaton events,.e., the exchange of prma and dua terates between the agents descrbed n Agorthm In coumn 5 we separatey state the amount of the dua communcaton events,.e., the exchange of dua terates, as the framework for event-trggered communcaton was ntroduced nto the DNA-EC deveoped n subsecton. that s used to update dua mutpers. It can be seen that ony for β 10,10 3 } the bounds on the constrant β δ k prma constrant overa dua gap voaton communcaton communcaton e6 (=100%) 16.e6 (=100%) k e6 ( 36 %).0e6 ( 1 %) k e6 ( 50 %) 3.1e6 ( 19 %) k e6 ( 59 %) 4.0e6 ( 4 % ) k e6 ( 64 %) 4.9e6 ( 30 % ) Tabe 1: Resuts for the IEEE 57 bus test case voaton (53) are sghty exceeded whereas for β 10 4,10 5 } the prma gap and the constrant voaton stay wthn the gven accuracy estmates (5) and (53) of the PCA wth the dfference that up to 50% of the overa communcaton and up to 76% of the dua communcaton coud be saved compared to the souton obtaned wthout event-trggered communcaton. Moreover, f nstantaneous communcaton s assumed we measured an overa computaton tme of ess than seconds on a Inte Xeon Pentum X5570 Pro processor wth.6 GHz whch s due to the ow computatona compexty of each teraton. We obtaned smar resuts for another DC-OPF probem that we but wth data from the IEEE 118 bus test case [4] wth 186 transmsson nes and where 34 of the 118 buses are each connected to a generator. Agan, we added quadratc cost functons wth a 0 for = 1,...,34 and scaed the probem wth s = 0 to obtan optma scaed dua mutpers wth µ (s),λ (s) = < 1. For comparson we state here agan the prma gap of the approxmate soutons (51) at the startng pont (µ 0,λ 0 ) whch s K$/h. Moreover, the constrant voaton of the approxmate soutons (51) at the startng pont s For ɛ = 1 after k = 1959 teratons of the PCA the approxmate soutons (51) shoud satsfy the foowng bounds: Φ ( ˆP g ) f 1 and ˆP g Bˆθ P [ d Aˆθ F max ] Aˆθ + F mn Usng a ayout dentca to Tabe 1, Tabe shows the resuts for δ = and β n k = βδ k. Here, the measured overa computaton tme of ess than 8 seconds. 4.1 Concusons We presented the DNA-EC and the DPCA-EC for dstrbuted optmzaton wth event-trggered communcaton. The DNA-EC extends an optma frst order method by Nesterov and s appcabe to probems