Accelerated gradient methods and dual decomposition in distributed model predictive control

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1 Deft Unversty of Technoogy Deft Center for Systems and Contro Technca report bs Acceerated gradent methods and dua decomposton n dstrbuted mode predctve contro P. Gsesson, M.D. Doan, T. Kevczky, B. De Schutter, and A. Rantzer If you want to cte ths report, pease use the foowng reference nstead: P. Gsesson, M.D. Doan, T. Kevczky, B. De Schutter, and A. Rantzer, Acceerated gradent methods and dua decomposton n dstrbuted mode predctve contro, Automatca, vo. 49, no. 3, pp , Mar Deft Center for Systems and Contro Deft Unversty of Technoogy Mekeweg 2, 2628 CD Deft The Netherands phone: (secretary) fax: URL: Ths report can aso be downoaded vahttp://pub.deschutter.nfo/abs/12_011_bs.htm

2 Acceerated gradent methods and dua decomposton ndstrbutedmodepredctvecontro PontusGsesson a,mnhdangdoan b,tamáskevczky b, BartDeSchutter b,andersrantzer a a Department of Automatc Contro, Lund Unversty, Sweden b Deft Center for Systems and Contro, Deft Unversty of Technoogy, The Netherands Abstract We propose a dstrbuted optmzaton agorthm for mxed L 1/L 2-norm optmzaton based on acceerated gradent methods usng dua decomposton. The agorthm acheves convergence rate O( 1 ), where k s the teraton number, whch sgnfcanty k 2 mproves the convergence rates of exstng duaty-based dstrbuted optmzaton agorthms that acheve O( 1 ). The performance of the deveoped agorthm s evauated on randomy generated optmzaton probems arsng n dstrbuted mode k predctve contro (DMPC). The evauaton shows that, when the probem data s sparse and arge-scae, our agorthm can outperform current state-of-the-art optmzaton software CPLEX and MOSEK. Key words: Dstrbuted contro; Gradent methods; Predctve contro. 1 Introducton Gradent-based optmzaton methods are known for ther smpcty and ow compexty wthn each teraton. A mtaton of cassca gradent-based methods s the sow rate of convergence. It can be shown [3], [20] that for functons wth a Lpschtz-contnuous gradent,.e., smooth functons, cassca gradent-based methods converge at a rate of O( 1 k ), where k s the teraton number. In [16] t was shown that a ower bound on the convergence rate for gradent-based methods s O( 1 k ). Nesterov showed n hs work [17] that an acceerated 2 gradent agorthm can be constructed such that ths ower bound on the convergence rate s acheved when mnmzng unconstraned smooth functons. Ths resut has been extended and generazed n severa pubcatons to hande constraned smooth probems and smooth probems wth an addtona non-smooth term [18], [19], [2] and [24]. Gradent-based methods are sut- Ths paper was not presented at any IFAC meetng. Correspondng author P. Gsesson. Te Fax Ema addresses: pontusg@contro.th.se (Pontus Gsesson), m.d.doan@tudeft.n (Mnh Dang Doan), t.kevczky@tudeft.n (Tamás Kevczky), b.deschutter@tudeft.n (Bart De Schutter), rantzer@contro.th.se (Anders Rantzer). abe for dstrbuted optmzaton when they are used n combnaton wth dua decomposton technques. Dua decomposton s a we-estabshed concept snce around 1960 when Uzawa s agorthm [1] was presented. Smar deas were expoted n arge-scae optmzaton [6]. Over the next decades, methods for decomposton and coordnaton of dynamc systems were deveoped and refned [9], [13], [23] and used n arge-scae appcatons [5]. In [25] a dstrbuted asynchronous method was studed. More recenty dua decomposton has been apped n the dstrbuted mode predctve contro terature n [7], [8], [11] and [15] for probems wth a strongy convex quadratc cost and arbtrary near constrants. The above mentoned methods rey on gradent-based optmzaton, whch suffers from sow convergence properteso( 1 k ).Asothestepszeparameternthegradent scheme must be chosen appropratey to get good performance. Such nformaton has not been provded or has been chosen conservatvey n these pubcatons. In ths work, we mprove on the prevousy presented dstrbuted optmzaton methods by usng an acceerated gradent method to sove the dua probem nstead of a cassca gradent method. We aso extend the cass of probems consdered by aowng an addtona sparse but non-separabe 1-norm penaty. Such 1-norm terms areusedasreguarzatontermoraspenatyforsoftcon- Preprnt submtted to Automatca

3 strants [22]. Further, we provde the optma step sze parameter for the agorthm, whch s cruca for performance. The convergence rate for the dua functon vaue usng the acceerated gradent method s mpcty known from [2], [24]. However, the convergence rate n the dua functon vaue does not ndcate the rate at whch the prma terate approaches the prma optma souton. In ths paper we aso provde convergence rate resuts for the prma varabes. Reated to our work s the method presented n [14] for systems wth a(non-strongy) convex cost. It s based on the smoothng technque presented by Nesterov n [19]. Otherreevantworkspresentedn[12],[21]nwhchoptmzaton probems arsng n mode predctve contro (MPC) are soved n a centrazed fashon usng acceerated gradent methods. These methods are, however, restrcted to hande ony box-constrants on the contro sgnas. To evauate the proposed dstrbuted agorthm, we sove randomy generated arge-scae and sparse optmzaton probems arsng n dstrbuted MPC and compare the executon tmes to state-of-the-art optmzaton software for arge-scae optmzaton, n partcuar CPLEX and MOSEK. We aso evauate the performance oss obtaned when suboptma step engths are used. Thepapersorganzedasfoows.InSecton2,theprobem setup s ntroduced. The dua probem to be soved s ntroduced n Secton 3 and some propertes of the dua functon are presented. The dstrbuted souton agorthm for the dua probem s presented n Secton 4. In Secton 5 a numerca exampe s provded, foowed by concusons drawn n Secton 6. 2 Probem setup In ths paper we present a dstrbuted agorthm for optmzaton probems wth cost functons of the form J(x) = 1 2 xt Hx+g T x+γ Px p 1. (1) Thefudecsonvector,x R n,scomposedofocadecsonvectors,x R n,accordngtox = [x T 1,...,x T M ]T. ThequadratccostmatrxH R n n sassumedseparabe,.e., H = bkdag(h 1,...,H M ) where H R n n. Further, H s assumed postve defnte wth σ(h)i H σ(h)i, where 0 < σ(h) σ(h) <. The near part g R n conssts of oca parts, g = [g T 1,...,g T M ]T where g R n. Further, P R m n s composed of P = [P 1,...,P m ] T, where each P r = [P T r1,...,p T rm ]T R n and P r R n. We do not assume that the matrx P shoud be bock-dagona whch means that the cost functon J s not separabe. However, we assume that the vectors P r have sparse structure. Sparsty refers to the property that for each r 1,...,m} there exst some 1,...,M} such that P r = 0. We aso have p = [p 1,...,p m ] T and γ > 0. Ths gves the foowng equvaent formuaton of (1) M [ ] 1 m J(x) = 2 xt H x +g T M x + P rx T p r. =1 r=1 =1 (2) Mnmzaton of (1) s subject to near equaty and nequaty constrants A 1 x = B 1, A 2 x B 2 wherea 1 R q n anda 2 R (s q) n contana R n as A 1 = [a 1,...,a q ] T and A 2 = [a q+1,...,a s ] T. Further, each a = [a T 1,...,aT M ]T where a R n. Further we have B 1 R q and B 2 R s q where B 1 = [b 1,...,b q ] T and B 2 = [b q+1,...,b s ] T. We assume that the matrces A 1 and A 2 are sparse. By ntroducng the auxary varabes y and the constrant Px p = y we get the foowng optmzaton probem mn x,y 1 2 xt Hx+g T x+γ y 1 s.t. A 1 x = B 1 A 2 x B 2 Px p = y. The objectve of the optmzaton routne s to sove (3) n a dstrbuted fashon usng severa computatona unts, where each computatona unt computes the optma oca varabes, denoted x, ony. Each computatona unt s assgned a number of constrants n (3) for whch t s responsbe. We denote the set of equaty constrantsthatuntsresponsbeforbyl 1,thesetof nequaty constrants by L 2 and the set of constrants orgnatngfromthe1-normbyr.thsdvsonsobvousy not unque but a constrants shoud be assgned toonecomputatonaunt.furtherfor L 1 and L2 werequrethata 0andforr R thatp r 0.Now we are ready to defne two sets of neghbors to computatona unt N = j 1,...,M} L 1 s.t. a j 0 or L 2 s.t. a j 0 or r R s.t. P rj 0 }, M = j 1,...,M} L 1 j s.t. a 0 or L 2 j s.t. a 0 or r R j s.t. P r 0 }. (3) 2

4 Through the ntroducton of these sets, the constrants thatareassgnedtountcanequvaentybewrttenas a T x = b j N a T jx j = b, L 1 (4) a T x b j N a T jx j b, L 2 (5) and the 1-norm term can equvaenty be wrtten as Pr T x p r = P T rjx j p r, r R. (6) j N In the foowng secton, the dua functon to be maxmzed s ntroduced. Frst, we state an assumpton that w be usefu n the contnuaton of the paper. Assumpton 1 We assume that there exsts a vector x such that A 1 x = b 1 and A 2 x < b 2. Further, we assume that a, = 1,...,q and P r,r = 1,...,m are neary ndependent. Remark 2 Assumpton 1 s known as the Mangasaran- Fromovtz constrant quafcaton (MFCQ). In [10] t was shown that MFCQ s equvaent to the set of optma dua varabes beng bounded. For convex probems, MFCQ s equvaent to Sater s constrant quafcaton wth the addtona requrement that the vectors defnng the equaty constrants shoud be neary ndependent. 3 Dua probem In ths secton we ntroduce a dua probem to (3) from whch the prma souton can be obtaned. We show that ths dua probem has the propertes requred to appy acceerated gradent methods. 3.1 Formuaton of the dua probem We ntroduce Lagrange mutpers, λ R q,µ R s q 0, ν R m for the constrants n (3). Under Assumpton 1 t s we known (cf. [4, 5.2.3]) that there s no duaty gap and we get the foowng dua probem 1 sup nf λ,µ 0,ν x,y 2 xt Hx+g T x+γ y 1 +λ T (A 1 x B 1 )+ } +µ T (A 2 x B 2 )+ν T (Px p y). (7) After rearrangng the terms we get sup λ,µ 0,ν nf x [ (A T 1λ+A T 2µ+P T ν +g) T x+ 1 2 xt Hx ] (8) λ T B 1 µ T B 2 ν T [ p+nf γ y 1 ν T y ]}. y The nfmum over y can be soved expcty nf γ y 1 ν T y } = nf y y } (γ [y] [ν] [y] ) = } nf(γ [y] [ν] [y] ) [y] 0 f ν γ = ese where [ ] denotes the -th eement n the vector. The nfmum over y becomes a box-constrant for the dua varabes ν. Ths s a cruca observaton for dstrbuton reasons. Before we expcty sove the mnmzaton over x n (8) the foowng notaton s ntroduced A = [A T 1 A T 2 P T ] T B = [B T 1 B T 2 p T ] T z = [λ T µ T ν T ] T wherea R (s+m) n,b R s+m andz R s+m.weaso ntroduce the set of feasbe dua varabes z R 1,...,q} Z = z R s+m z 0 q +1,...,s} z γ s+1,...,s+m} (9) The mnmzaton over x n (8) can be soved expcty [ nf (A T z +g) T x+ 1 ] x 2 xt Hx = = 1 2 (AT z +g) T H 1 (A T z +g) and we get the foowng dua probem sup 1 } z Z 2 (AT z +g) T H 1 (A T z +g) B T z. (10) We ntroduce the foowng defnton of the negatve dua functon f(z) := 1 2 (AT z +g) T H 1 (A T z +g)+b T z. Snce f conssts of a quadratc term wth postve semdefnte hessan and a near term, f s dfferentabe and has the foowng gradent f(z) = AH 1 (A T z +g)+b. (11) Further, from the mn-max theorem we have that the smaest Lpschtz constant, L, to f s L = AH 1 A T 2. 3

5 4 Dstrbuted optmzaton agorthm In ths secton we show how the acceerated gradent method can be used to dstrbutvey sove (3) by mnmzng the negatve dua functon f. The acceerated proxma gradent method for probem(10) s defned by the foowng teraton as presented n [24, Agorthm 2] and [2, Eq ] v k = z k + k 1 k +2 (zk z k 1 ) (12) z k+1 = P Z (v k 1 ) L f(vk ) (13) where P Z s the Eucdean projecton onto the set Z. Thus, the new terate, z k+1, s the prevous terate pus a step n the negatve gradent drecton projected onto the feasbe set. x k = H 1 ( A T z k ) g [ = H 1 g + [ a λ k + a µ k + j M L 1 j L 2 j r R j P rν k r (19) ]] Thus, each oca prma update, x k, can be computed after communcaton wth neghbors j M. Through (4)-(6) we note that the dua varabe teratons can be updatedaftercommuncatonwthneghbors N.We get the foowng dstrbuted agorthm. Agorthm 1 Dstrbuted acceerated gradent agorthm Intaze λ 0 = λ 1,µ 0 = µ 1,ν 0 = ν 1 and x 0 = x 1 In every node,, the foowng computatons are performed: For k 0 We defne the prma teraton x k = H 1 ( A T z k g). Usngthsdefnton,straghtforwardnsertonofv k nto (11) gves ( f(v k ) = A x k + k 1 ) k +2 (xk x k 1 ) +B (1) Compute x k accordng to (19) and set x k = x k + k 1 k +2 (xk x k 1 ) (2) Send x k to each j M, receve x k j j N from each By defnng x k = x k + k 1 k+2 (xk x k 1 ) and recang the partton z = [λ T µ T ν T ] T and the defnton (9) of the set Z, we fnd that (12)-(13) can be paraezed: x k = H 1 ( A T z k g) (14) x k = x k + k 1 k +2 (xk x k 1 ) (15) λ k+1 µ k+1 ν k+1 r = λ k + k 1 k +2 (λk λ k 1 )+ 1 L (at x k b ) (16) = max 0,µ k + k 1 k +2 (µk µ k 1 )+ + 1 L (at x k b ) } (17) = mn γ,max [ γ,ν k r + k 1 k +2 (νk r ν k 1 r )+ + 1 L (PT r x k p r ) ]}. (18) From these teratons t s not obvous that the agorthm s dstrbuted. By parttonng the constrant matrx as A = [A 1,...,A M ] where each A = [a 1,...,a s,p 1,...,P m ] T R (s+m) n, and notng that H s bock-dagona, the oca prma varabes are updated accordng to (3) Compute λ k+1 Compute µ k+1 Compute ν k+1 (4) Send λ k+1 } L 1,µ k+1 j N, receve λ k+1 } L 1 j,µ k+1 from each j M accordng to (16), (4) for L 1 accordng to (17), (5) for L 2 accordng to (18), (6) for R } L 2, νr k+1 } r R to each } L 2 j and νr k+1 } r Rj The convergence rates for the dua functon f and the prma varabes when runnng Agorthm 1 are stated n the foowng theorem. Theorem 3 Agorthm 1 has the foowng convergence rate propertes: (1) Denote an optmzer of the dua probem (10) as z. The convergence rate s: f(z k ) f(z ) 2L z0 z 2 2 (k +1) 2, k 1 (20) (2) Denote the unque optmzer of the prma probem as x. The rate of convergence for the prma varabe s x k x 2 2 4L z0 z 2 2 σ(h)(k +1) 2, k 1 (21) 4

6 PROOF. Agorthm 1 s a dstrbuted mpementaton of [24, Agorthm 2] and [2, Eq ] apped to mnmzef.theconvergenceratenargument1foowsfrom [24, Proposton 2] and [2, Theorem 4.4]. For argument 2 we get that the necessary and suffcent KKT condtons [4, p. 244] mpes x = H 1 ( A T z g) snce H s nvertbe. Ths eads to x k x 2 2 = H 1 (A T z k A T z ) 2 2 H 1 A T z k A T z 2 H 1 = 1 σ(h) (zk z ) T AH 1 A T (z k z ) = 1 σ(h) ( (z k ) T AH 1 A T z k (z ) T AH 1 A T z 2(AH 1 A T z ) T (z k z )+ ) +2(B +AH 1 g) T (z k z k +z z ) = 2 ( f(z k ) f(z ) σ(h) ) (AH 1 (A T z +g)+b) T (z k z ) = 2 ( f(z k ) f(z ) f(z ) T (z k z ) ) σ(h) 2 σ(h) (f(zk ) f(z )) 4L z0 z 2 2 σ(h)(k +1) 2 where the frst nequaty comes from the mn-max theorem, the equates are agebra wth addton of some zero-terms, the frst nequaty n the fna row s from the frst-order optmaty condton [20, Theorem 2.2.5], and the fna nequaty s due to (20). Ths competes the proof. 5 Numerca exampe In ths secton we evauate the performance of Agorthm 1. We compare the presented agorthm to state-ofthe-art centrazed optmzaton software for arge-scae optmzaton mpemented n C, namey CPLEX and MOSEK. We aso evauate the performance oss when usng suboptma step szes. Our agorthm s mpemented on a snge processor to be abe to compare executon tmes. The comparson s made on 100 random optmzaton probems arsng n dstrbuted MPC. A batch of random stabe controabe dynamca systems wth random structure and random nta condtons are created. The sparsty fracton,.e., the fracton of non-zero eements n the dynamcs matrx and the nput matrx, s chosen to be 0.1. We have random nequaty constrants that are generated to guarantee a feasbe souton and a 1-norm cost where the P-matrx and p-vector are randomy chosen. The quadratc cost matrces are chosen Q = I and R = I. Tabe 1 shows the numerca resuts obtaned runnng MATLAB on a Lnux PC wth a 3 GHz Inte Core 7 processor and 4 GB memory. The optmzaton software used s CPLEX V12.2 and MOSEK that are accessed va the provded MATLAB nterfaces. Tabe 1 Agorthm comparson wth 1-norm cost term and random state and nput constrants. Agorthm 1 s mpemented n MATLAB whe CPLEX and MOSEK are mpemented n C. Ag. vars./constr. to. # ters exec (ms) mean max mean max 1 (L) 4320/ (L 1) 4320/ (L F) 4320/ MOSEK 4320/ CPLEX 4320/ (L) 2160/ (L 1) 2160/ (L F) 2160/ MOSEK 2160/ CPLEX 2160/ The frst coumn specfes the agorthm used where Agorthm 1 s suppemented wth the step sze used. L s the optma step sze L = AH 1 A T 2, L F = AH 1 A T F and L 1 = AH 1 A T 1 AH 1 A T. We compare to the suboptma step szes L 1 and L F snce they can be computed n dstrbuted fashon. The step szes satsfy L L 1 and L L F. The second coumn specfes the number of varabes and constrants n the optmzaton probems. In the thrd coumn we have nformaton about the duaty gap toerance that s used as stoppng condton n the agorthms (f possbe to set). The two fna coumns present the resuts n terms of number of teratons and executon tme. The dfference between the upper and ower haves of the tabe s the sze of the probems that are soved. Tabe 1 reveas that Agorthm 1 performs better than CPLEX and MOSEK on these arge-scae sparse probems despte the fact that CPLEX and MOSEK are m- pementedncandagorthm1smpementednmat- LAB. We aso concude that the choce of step sze n Agorthm 1 s mportant for performance reasons. 6 Concusons We have presented a dstrbuted optmzaton agorthm for strongy convex optmzaton probems wth sparse probem data. The agorthm s based on an acceerated gradent method that s apped to the dua probem. The 5

7 agorthm was apped to arge-scae sparse optmzaton probems orgnatng from a dstrbuted mode predctve contro formuaton. Our agorthm performed better than state-of-the-art optmzaton software for argescae sparse optmzaton, namey CPLEX and MOSEK, on these probems. Acknowedgements The second author woud ke to thank Quoc Tran Dnh for hepfu dscussons on the topc of ths paper. The second, thrd and fourth authors were supported by the European Unon Seventh Framework STREP project Herarchca and dstrbuted mode predctve contro (HD-MPC), contract number INFSO-ICT , and the European Unon Seventh Framework Programme [FP7/ ] under grant agreement no HYCON2 Network of Exceence. The frst and ast authors were supported by the Swedsh Research Counc through the Lnnaeus center LCCC. References [1] K.J. Arrow, L. Hurwcz, and H. Uzawa. Studes n Lnear and Nonnear Programmng. Stanford Unversty Press, [2] Amr Beck and Marc Teboue. A fast teratve shrnkagethreshodng agorthm for near nverse probems. SIAM J. Imagng Scences, 2(1): , October [3] Dmtr P. Bertsekas. Nonnear Programmng. Athena Scentfc, Bemont, MA, 2nd edton, [4] Stephen Boyd and Leven Vandenberghe. Convex Optmzaton. Cambrdge Unversty Press, New York, NY, [5] Perre Carpenter and Guy Cohen. Apped mathematcs n water suppy network management. Automatca, 29(5): , [6] G.B. Danzg and P. Wofe. The decomposton agorthm for near programmng. Econometrca, 4: , [7] M. D. Doan, T. Kevczky, and B. De Schutter. An teratve scheme for dstrbuted mode predctve contro usng Fenche s duaty. Journa of Process Contro, 21(5): , June Speca Issue on Herarchca and Dstrbuted Mode Predctve Contro. [8] M. D. Doan, T. Kevczky, I. Necoara, M. Deh, and B. De Schutter. A dstrbuted verson of Han s method for DMPC usng oca communcatons ony. Contro Engneerng and Apped Informatcs, 11(3):6 15, [9] W. Fndesen. Contro and Coordnaton n Herarchca Systems. Internatona seres on apped systems anayss. Wey, [10] Jacques Gauvn. A necessary and suffcent reguarty condton to have bounded mutpers n nonconvex programmng. Mathematca Programmng, 12: , [11] P. Gsesson and A. Rantzer. Dstrbuted mode predctve contro wth suboptmaty and stabty guarantees. In Proceedngs of the 49th Conference on Decson and Contro, pages , Atanta, GA, December [12] M. Köge and R. Fndesen. A fast gradent method for embedded near predctve contro. In Proceedngs of the 18th IFAC Word Congress, pages , Man, Itay, [13] M. D. Mesarovc, D. Macko, and Y. Takahara. Theory of Herarchca Muteve Systems. Academc Press, New York, [14] I. Necoara and J. Suykens. Appcaton of a smoothng technque to decomposton n convex optmzaton. IEEE Transactons on Automatc Contro, 53(11): , December [15] R.R. Negenborn, B. De Schutter, and J. Heendoorn. Mutagent mode predctve contro for transportaton networks: Sera versus parae schemes. Engneerng Appcatons of Artfca Integence, 21(3): , Apr [16] A Nemrovsky and D Yudn. Informatona Compexty and Effcent Methods for Souton of Convex Extrema Probems. Wey, NewYork, NY, [17] Y Nesterov. A method of sovng a convex programmng probem wth convergence rate O (1/k 2 ). Sovet Mathematcs Dokady, 27(2): , [18] Y Nesterov. On an approach to the constructon of optma methods of mnmzaton of smooth convex functons. Ékonom.. Mat. Metody, 24: , [19] Yu Nesterov. Smooth mnmzaton of non-smooth functons. Math. Program., 103(1): , May [20] Yur Nesterov. Introductory Lectures on Convex Optmzaton: A Basc Course (Apped Optmzaton). Sprnger Netherands, 1st edton, [21] S. Rchter, C.N. Jones, and M. Morar. Rea-tme nput-constraned MPC usng fast gradent methods. In Proceedngs of the 48th Conference on Decson and Contro, pages , Shangha, Chna, December [22] C. Savorgnan, C. Roman, A. Kozma, and M. Deh. Mutpe shootng for dstrbuted systems wth appcatons n hydro eectrcty producton. Journa of Process Contro, 21(5): , [23] Madan G. Sngh and Andre Tt. Systems: Decomposton, Optmsaton, and Contro. Pergamon, [24] Pau Tseng. On acceerated proxma gradent methods for convex-concave optmzaton. Submtted to SIAM J. Optm. Avaabe: ~tseng/papers/apgm.pdf, May [25] J. Tstsks, D. Bertsekas, and M. Athans. Dstrbuted asynchronous determnstc and stochastc gradent optmzaton agorthms. IEEE Transactons on Automatc Contro, 31(9): , September