Preconditioning of conjugate gradient iterations in interior point MPC method

Transcription

1 MISUBISHI ELECRIC RESEARCH LABORAORIES Peconditioning of conjugate gadient iteations in inteio point MPC method Malyshev, A; Quiynen, R; Knyazev, A R August 25, 2018 Abstact hee ae seveal efficient diect solves fo stuctued systems of linea equations defining seach diections in pimal-dual inteio point methods applied to constained model pedictive contol poblems We popose eusing matix decompositions of diect solves as peconditiones in Kylov-subspace methods applied to subsequent iteations of the inteio point method, which esults in at least halving its asymptotic computational complexity We also analyze sensitivity of diect solves to the egulaization paametes IFAC Confeence on Nonlinea Model Pedictive Contol his wo may not be copied o epoduced in whole o in pat fo any commecial pupose Pemission to copy in whole o in pat without payment of fee is ganted fo nonpofit educational and eseach puposes povided that all such whole o patial copies include the following: a notice that such copying is by pemission of Mitsubishi Electic Reseach Laboatoies, Inc; an acnowledgment of the authos and individual contibutions to the wo; and all applicable potions of the copyight notice Copying, epoduction, o epublishing fo any othe pupose shall equie a license with payment of fee to Mitsubishi Electic Reseach Laboatoies, Inc All ights eseved Copyight c Mitsubishi Electic Reseach Laboatoies, Inc, Boadway, Cambidge, Massachusetts 02139

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3 Peconditioning of conjugate gadient iteations in inteio point MPC method Alexande Malyshev Rien Quiynen Andew Knyazev Univesity of Begen, Depatment of Mathematics, Postbox 8700, 5020 Begen, Noway ( Mitsubishi Electic Reseach Laboatoies, 201 Boadway, 8th floo, Cambidge, MA 02139, USA ( s: Abstact: hee ae seveal efficient diect solves fo stuctued systems of linea equations defining seach diections in pimal-dual inteio point methods applied to constained model pedictive contol poblems We popose eusing matix decompositions of diect solves as peconditiones in Kylov-subspace methods applied to subsequent iteations of the inteio point method, which esults in at least halving its asymptotic computational complexity We also analyze sensitivity of diect solves to the egulaization paametes Keywods: model pedictive contol, inteio point, peconditioning, conjugate gadient 1 INRODUCION In this pape, peconditioning techniques ae poposed that may be useful fo numeical solution of optimal contol poblems in model pedictive contol (MPC); see (Rawlings et al, 2017) We conside MPC ove a eceding hoizon in the fom of a quadatic pogam (QP), whee the cost function is given by a convex quadatic pefomance index, the equality constaints ae the linea model dynamics, the inequality constaints on the state and contol inputs ae also linea; simila fomulations ae used in (Wang and Boyd, 2010; Shahzad et al, 2012; Fison et al, 2014; Quiynen et al, 2018) Such stuctued QPs appea as subpoblems within a sequential QP fo nonlinea MPC; see, eg, (Diehl et al, 2009) he inequality constaints in the QP poblem ae usually teated by the inteio point (IP) method o by the activeset method; cf (Batlett et al, 2000) he IP method is chaacteized by fast convegence, which is almost independent of the poblem size; see, eg, (Mehota, 1992; Gondzio, 2012, 2013) he active-set solves fo MPC ae highly competitive owing to athe small changes of the active set duing continuation steps, educing aithmetic costs by exploiting low-an updates of a Cholesy factoization when changing the active set as shown in (Gill et al, 1974; Kiches et al, 2011) Matices in systems of linea equations defining seach diections in the Newton method ae spase, consisting of blocs with banded matices Seveal efficient diect methods have been poposed fo solving such systems; see, eg, (Rao et al, 1998; Wang and Boyd, 2010; Domahidi et al, 2012; Fison and Jøgensen, 2013) Let n x, n u, n g, and N denote the dimensions of the state vaiable and the contol input, the numbe of the inequality constaints, and the hoizon length, espectively Aithmetic complexity of bloc stuctued diect solves used in the IP method typically amounts to O(N(n 3 x + n 3 u + n 3 g)) flops Numeical solution of linea systems of the Newton method, eg, using diect solves, is the most timeconsuming online pat of MPC Computational complexity can be educed if saddle-point linea systems of KK conditions ae eplaced by linea systems having symmetic positive definite (SPD) bloc stuctued matices Diect solves fo SPD linea systems ae commonly based on the Cholesy factoization, often implemented to tae advantage of the BLAS3 matix opeations, which is highly efficient on contempoay compute achitectues with advanced memoy hieachies he active-set methods may achieve theoetical complexity O(N(n 2 x+n 2 u+n 2 g)), if based on cost-efficient low-an updates, eg, of the Cholesy factoization, although the low-an updates can only use BLAS2; see (Andeson et al, 1999) Well-nown fast convegence of the IP method, its esilience to inexact implementations, and high pefomance of BLAS3 implementations of the diect solves can mae the IP method competitive with the active-set methods We have consideed the IP method in (Malyshev et al, 2018) using each of the thee diect solves: Riccati ecusion, Schu complement, and augmented Lagangian, with egulaization by adding the identity matix multiplied by a small paamete to some blocs he factoized Riccati ecusion and the augmented Lagangian egulaization have been found the best in tems of aithmetic opeations he spase linea systems of the Newton methods can be solved by iteative methods; see, eg, (Golub and Van Loan, 2013, Ch 11) If an efficient peconditione is available, the theoetical complexity of an iteative method is educed to O(N(n 2 x + n 2 u + n 2 g)), and the peconditioned iteative method outpefoms diect solves fo elatively lage values of n x, n u, n g he est of this pape is oganized as follows Section 2 intoduces the notation and the MPC QP poblem Section 3 discusses eductions to SPD systems We evisit eg-

4 ulaization of the Schu complement in Section 4 and the augmented Lagangian in Section 5 We show theoetically in Section 6 and numeically in Section 7 that the fome is less sensitive to the choice of the egulaization paamete, compaed to the latte Section 7 also illustates behavio of the poposed peconditioning in the conjugate gadient (CG) method, whee Cholesy factoizations computed at odd IP iteations ae eused fo peconditioning at even IP iteations to halve the computational cost Ou main contibutions include (a) euse of the Cholesy factoization fo peconditioning subsequent IP iteations and (b) analysis of sensitivity of the egulaized diect solves fom Sections 4 and 5 to the choice of egulaization paamete values ɛ and τ 2 QUADRAIC PROGRAMMING FOR MODEL PREDICIVE CONROL We conside the following quadatic pogam (QP) with espect to a decision vaiable d R n h, min 2 d Hd + h d subject to F d = f, Gd g, (1) whee n h = (n x + n u )N + n x, n f = n x (N + 1), and the matices H R n h n h, F R n f n h, G R ng n h and the vectos h, f, and g have the fom Q 0 S S 0 R H = 0 0 Q N 1 SN 1, S N 1 R N Q N I A 0 B 0 I F =, A N 1 B N 1 I G x 0 G u G = 0 0 G x N 1 G u N 1 0, G x N h x 0 h u f 0 0 g 0 h = h u, f = f 1, g = g 1 N 1 f N g N d 1 h x N he blocs I in F denote identity matices of ode n x We assume that the matix H is symmetic positive semidefinite We note that the matix H as well as the blocs G x i and Gu i of the matix G ae often diagonal Optimization poblem (1) is solved by the pimal-dual IP method; see, eg, (Wight, 1997; Shahzad et al, 2012) he IP method uses the following Kaush-Kuhn-uce (KK) optimality conditions Hd + F y + G z + h = 0, (2) F d f = 0, (3) Gd g + s = 0, (4) Z1 µs 1 1 = 0, (5) whee 1 = [1 1] R ng, and the matices Z = diag (z) and S = diag (s) ae diagonal he vectos y R n f and z R ng denote the Lagange multiplies he slac vecto s R ng with positive components and the small paamete µ > 0 ae equied by the IP method he KK conditions consist of linea equations (2)-(4) and nonlinea equation (5) he latte is usually substituted by ZS1 µ1 = 0 to impove scaling the esiduals in the Newton method he seach diection at iteation of the Newton method fo solving (2)-(5) is detemined by the system of linea equations H F G 0 d F y H G 0 0 I 0 0 S Z z s = with the esiduals H = Hd + F y + G z + h, F = F d f, G = Gd g + s, F G S, (6) S = Z S 1 σµ 1, whee σ (0, 1) he scala σ in the last fomula is called a centeing paamete he value of µ = (z ) s /n g is elated to a duality gap; eg, (Wight, 1997; Gondzio, 2012) Linea system (6) must be solved online epeatedly fo = 1, 2,, max, whee max is detemined by a stopping citeion, eg, based on the 2-nom of the esidual vecto in the ight-hand side of (6) he matix of linea system (6) becomes inceasingly ill-conditioned when some inequality constaints become active Nevetheless, the Newton IP method pefoms well, if system (6) is solved by diect methods, as discussed and justified in (Wight, 1998) We note that the matix of system (6) depends on the diagonal matices S and Z in the last bloc ow heefoe, a diect solve has to factoize the matix of (6) at each iteation, which is the most time-consuming opeation of the IP method Reducing costs of solving (6) is cucial 3 PRELIMINARY REDUCIONS BY BLOCK GAUSSIAN ELIMINAIONS he slac vaiable s is usually eliminated fom (6), which leads to the system of linea equations H F G F 0 0 d y = H F (7) G 0 W z W with the diagonal matix W = (Z ) 1 S he esidual G is substituted with the esidual W = G (Z ) 1 S he eliminated vaiable s is ecoveed by the fomula s = (Z ) 1 ( S + S z ) H F In (Shahzad et al, 2010a,b, 2012), the bloc is F 0 assumed to be nonsingula and (7) is educed to { W H F 1 } G + [G 0] z = F 0 0 [ M = W H F [G 0] F 0 ] 1 H F

5 he main advantage of such a eduction is that the matix H F 1 G M = [G 0] is computed only once fo all F 0 0 IP iteations he SPD system (W + M) z = M is then solved by peconditioned CG Howeve, cheap peconditioning via the diagonal pat of W + M may be not sufficient to povide fast convegence, while peconditioning using the bloc diagonal o tidiagonal pats of W + M equie expensive computations of the Cholesy factoization at evey iteation In contast, we eliminate the vaiable z fom linea system (7) he esulting 2 2 bloc linea system H F d F 0 y = E F (8) has the esidual E = H + G (W ) 1 W and the bloc H = H + G (W ) 1 G, (9) which is symmetic positive semidefinite and bloc diagonal with the same spasity patten as the matix H he eliminated vaiable z is ecoveed by the fomula z = (W ) 1 ( G d + W ) (10) Systems in the fom (8) ae called saddle point poblems Now that we have educed (6) to (8), futhe eductions of the linea system (8) with espect to the vaiables d o y ae possible In the next two sections, we descibe two popula altenative eductions, Schu complement- and augmented Lagangian-based, both supplemented with egulaization by adding the identity matix multiplied by a small paamete to some blocs 4 INEXAC NEWON MEHOD BY HE SCHUR COMPLEMEN APPROACH he elimination of d is most popula in the pimal-dual IP method; see (Wight, 1997) If the matix H in (9) is singula, it is usually eplaced by a egulaized SPD bloc Ĥ = H + ɛi, whee ɛ is a small positive constant A justification of such a eplacement is due to the fact that the IP algoithm can use an inexact Newton method, ie, the system (6) may be solved only appoximately; see, eg, (Nocedal and Wight, 2006) In the following egulaized analog of system (8), [Ĥ ] F d F 0 y = E F, (11) the fist equation of (11) is equivalent to d + (Ĥ ) 1 F y = (Ĥ ) 1 E and hence [ F (Ĥ ) 1 F ] y = F F (Ĥ ) 1 E (12) Equation (12), often called a nomal equation, has the SPD matix F (Ĥ ) 1 F, which is the Schu complement fo linea system (11) SPD linea system (12) can be solved by peconditioned CG Fo nonlinea MPC, peconditioning is poposed, eg, in (Knyazev et al, 2015; Knyazev and Malyshev, 2015, 2016) fo the Newton method Peconditioning in the IP context is widely discussed; see, eg, the suvey in (Gondzio, 2012) Inteesting elations between peconditiones fo nomal (12) and indefinite (8) equations ae discoveed in (Oliveia and Soensen, 2005) We popose using the Cholesy factoization of the matix F (Ĥ 1 ) 1 F, ie based on the values fom the pevious IP iteation 1, as a peconditione fo CG in iteation he aithmetic complexity of the Cholesy factoization of the matix F (Ĥ 1 ) 1 F is ( 19 N 3 n3 x + 4n 2 xn u + 2n x n 2 u + 1 ) 3 n3 u, whee only the leading tems ae taen into account 5 INEXAC NEWON MEHOD BY AUGMENED LAGRANGIAN REGULARIZAION In this section, we investigate elimination of the vaiable y fom system (8) Since the IP iteations allow the inexact Newton method, we conside the following egulaized analog of system (8), H F d F τi y = E F, (13) whee the positive egulaization paamete τ is sufficiently small Such a egulaization is often used in iteative methods fo numeical solution of saddle point poblems, especially fo peconditioning, eg, (Rees and Geif, 2007; Benzi and Wathen, 2008), and is also connected to the augmented Lagangian method in numeical optimization; see, eg, (Nocedal and Wight, 2006) Linea system (13) can be equivalently tansfomed to the bloc tiangula system [ H + 1 ] [ d τ F ] [ F 0 F τi y = E + 1 ] τ F F (14) he solution of (14) is given by the fomulas d = ( H + 1 τ F F ) 1 ( E + 1 τ F F ), (15) y = 1 τ ( F + F d ) (16) he main computational buden in fomulas (15)-(16) is solving the system with the SPD matix H + 1 τ F F in (15), which depends on Aithmetic complexity of the Cholesy factoization fo the matix H + 1 τ F F equals ( 7 N 3 n3 x + 4n 2 xn u + 2n x n 2 u + 1 ) 3 n3 u By analogy with the pevious section, we popose evaluating (15) by peconditioned CG, whee the peconditione is based on the Cholesy factoization of the system matix H τ F F fom the pevious IP iteation 6 ANALYSIS OF REGULARIZAION IN INEXAC NEWON MEHODS H F Regulaization of the saddle point matices is F 0 a well-nown topic in optimization theoy Fo example, (Vandebei, 1995) investigates LDL factoizations of the following matices, with both diagonal blocs egulaized, H + ɛi F (17) F τi F

6 (Saundes and omlin, 1996) conside the IP iteations, whee the inexact Newton method uses egulaized linea systems with the matices of fom (17) hey give a fomulation of the initial QP poblem, which is equivalent to the egulaized IP iteation, and pesent numeous tests showing high efficiency of the egulaization appoach In this section, we contibute into theoetical analysis of the egulaizations given by equations (11) and (13) Since the matix H is symmetic positive semidefinite, it has its symmetic positive semidefinite squae oot H 1/2 he convex saddle point poblem can be analyzed by the aid of the genealized singula value decomposition (GSVD) of the matix pai H 1/2 and F ; see, eg, (Golub and Van Loan, 2013) In ou case, thee exist othogonal matices U 1 and U 2 and a nonsingula matix V such that H 1/2 C = U 1 V I, F = U 2 [S 0] V, whee the diagonal matices C = c 1 c2 c nf, S = s 1 s2 s nf have nonnegative diagonal elements c i and s i such that c 2 i + s2 i = 1 fo i = 1,, n f We conside below the egulaization H + ɛv V instead of ou actual egulaization Ĥ = H + ɛi fo simplicity of pesentation heoem 1 he esidual matices ( ) ( ) R S = H F H + ɛv V F 1 I F 0 F 0 and ( ) ( ) R L = H F H F 1 I F 0 F τi satisfy the bounds R S ɛ cond(v ) max ( 1, S 1 ), (18) R L τ cond(v ) max ( S(S 2 + τc 2 ) 1, C 2 (S 2 + τc 2 ) 1 ) (19) Poof Bounds (18) and (19) follow fom the identities ( ) 0 ɛs 1 V R S = U2 ɛ ( ɛ I V U2) 0 0 and ( V R L = U2) 0 0 ( ) 1 0 V τs(s 2 + τc 2 ) 1 τc 2 (S 2 + τc 2 ) 1 U2 heoem 1 implies the following asymptotic esults in the situation whee the smallest singula value σ min (S) vanishes he esiduals fo the Schu complement and augmented Lagangian egulaizations ae small if ɛ σ min (S) and τ σ min (S) 2 We thus conclude that the Schu based egulaization has much wide inteval of ɛ, whee the egulaization wos, than the woing inteval of τ in the augmented Lagangian egulaization 7 CASE SUDY: CHAIN OF OSCILLAING MASSES Let us pesent peliminay esults fom numeical expeiments fo the classical linea contol example of a chain of n/2 unit masses, connected by spings of unit stiffness without damping, whee the ends ae attached to fixed walls We analyze computational pefomance and numeical obustness solving one QP at a paticula sampling instant a epesentative case fo pefoming full closedloop MPC simulations with cold initializations 71 Poblem Fomulation and Algoithm Implementation he continuous-time state-space system is discetized using a sample time τ = 05 while eeping the m inputs constant between the sampling instants he discete-time dynamics ae x i+1 = A i x i + B i u i, whee A i = exp ( τa c ), B i = A 1 c (A i I n ) 0 n/2 I m, and 0 n/2 m 2 1 0n/2 I A c = n/ , = n/2 0 n/2 1 2 he contol inputs and states ae subject to the inequality constaints 05 u(i) 05, i = 0,, N 1, 35 x(i) 35, i = 1,, N, whee x( ) denotes the position of the masses he cost matices in the MPC fomulation ae R i = 10 6 I, S i = 0, and Q i = C C = [ I n/2 0 ] [ In/2 0 ] he initial values of the state coespond to f 0 = 35 [ ] Ou numeical expeiments in MALAB ae pefomed on a test poblem of the following size: we conside 6 masses such that the state dimension equals n = 12, the numbe of contol inputs is m = 3 and the contol hoizon length is N = 30 he numbe of inequality constaints at each contol inteval is n + 2m = 18 Ou MALAB code implements the vaiant of the IP method descibed in moe detail by (Shahzad et al, 2012) and uses the same algoithmic constants and stopping citeion such as σ = 01, γ = 10 3, β = 2, ɛ = 10 7 We epot the esults of solving QP fom a cold stat at the initial time of the MPC contolle, whee the initial values fo x, u, y, z, and s ae equal to 1 he IP method fo QP geneally does not benefit much fom wam stating, especially fo S and Z, (Batlett et al, 2000) As a efeence fo futhe compaisons, we initially un the IP method in which the linea system (8) is solved exactly by the bacslash opeato in MALAB Ou implementation etuns a solution to the quadatic pogam afte 20 IP iteations and with the 2-nom of the IP esidual equal to [ H, F, G, S ] 2 =

7 # iteations IP-AL # iteations IP-SC Residual IP-AL Residual IP-SC o # CG iteations IP-AL IP-SC Fig 1 he numbe of IP iteations and the 2-nom of the esidual fo the solution with espect to ɛ o τ Ou poposed implementation of the IP method uses diect solves based on the Schu complement (IP-SC) o the augmented Lagangian (IP-AL) egulaization appoach he IP-SC and IP-AL algoithms use egulaization of the poblem with a small value of the paamete ɛ and τ, espectively Smalle values of ɛ and τ give bette appoximations to the KK matix, but too small values may lead to pohibitively lage condition numbes, fo which the IP method beas down in ou implementation using double pecision aithmetic Choosing the egulaization paamete caefully becomes inceasingly impotant when pefoming computations with a lowe pecision, eg, using single pecision aithmetic 72 Numeical Stability and Paamete Sensitivity Let us investigate the dependence of IP-SC on ɛ and IP- AL on τ Fo this pupose, we choose 100 values of ɛ and τ unifomly distibuted between 10 2 and in the logaithmic scale Figue 1 displays the numbe of IP iteations and the 2-nom of the IP esidual with espect to the value of the paamete ɛ = τ Ou numeical esults fo ɛ o τ between appoximately and 10 3 almost coincide with those fo the bacslash opeato, ie, the numbe of IP iteations equals 20 and the 2-nom of the IP esidual is vey close to 10 7 Both algoithms have simila behavio in Figue 1 fo the ange of values of ɛ and τ he IP-SC method is clealy pefeential fo pactically impotant small values, below 10 11, consistent with ou analysis in Section 6 IP-SC and IP-AL ae damatically diffeent fo the lage values of ɛ and τ within the inteval [10 35, 10 2 ] he algoithm IP-SC continues executing well, while IP-AL has vey lage esiduals and beas down 73 Peconditioned CG within the IP Method We epot ou peliminay esults pefoming expeiments with peconditioned CG inside the IP method he peconditione is applied at even IP iteations and uses the Cholesy factoization fom the pevious odd IP iteation Odd IP iteations ae esolved exactly, which technically o IP iteation numbe Fig 2 he numbe of peconditioned CG iteations in the IP method fo one MPC time step coesponds to one CG iteation he dimension of the state is n x = 240, coesponding to 120 masses, and the numbe of the contol inputs is n u = 3 We incease the dimension of the state 20 times, compaed to the pevious expeiments, in ode to mae the test case epesentative fo using CG, since unning CG on small-size linea systems may esult in atypical convegence behavio Figue 2 shows that the numbe of CG iteations in IP-AL with τ = 10 8 sometimes is gowing with the numbe of IP iteations his behavio can be obseved also less stongly fo the esults of the IP-SC method in Figue 2 We anticipate that tuning of IP-AL and IP-SC paametes may futhe impove the pefomance of the peconditioned CG method in both cases Pactical embedded MPC implementations equie solving each QP within tight timing constaints heefoe, it is common to limit the maximum numbe of IP iteations to a pedetemined fixed value Similaly, one may limit the numbe of CG iteations, eg, up to 10 iteations in Figue 2, at the potential cost of an incease in the oveall numbe of IP iteations Finally, one can change the ule of when to constuct a new peconditione, ie to compute the new stuctued Cholesy factoization Instead of constucting the new peconditione at each odd IP iteation, one could, eg, apply the same peconditione until the numbe of CG iteations passes a theshold, futhe inceasing pefomance 8 CONCLUSION We conside peconditioned CG fo computation of the seach diection in the inexact Newton based IP method to solve each QP in MPC We popose peconditioning by eusing matix factos of diect solves fo the Schu complement of the nomal equations o fo the augmented Lagangian egulaization appoach We can apply peconditioned CG, eg, at evey even iteation of the IP method with the Cholesy factos computed in the pevious odd iteation Such a vaiant could allow computing optimal contol solutions twice as fast compaed to the taditional use of the diect solves at evey IP/Newton iteation

8 We have caied out numeical tests of the two poposed egulaizations fo saddle point linea systems On the one hand, smalle computational complexity of the peconditiones maes the augmented Lagangian egulaization based appoach somewhat pefeential On the othe hand, ou petubation analysis and numeical expeiments demonstate that peconditioning based on the moe computationally expensive Schu complement is geneally moe obust to ound-off eos, which is advantageous fo low-pecision embedded MPC implementations REFERENCES Andeson, E, Bai, Z, Bischof, C, Blacfod, LS, Demmel, J, Dongaa, J, Coz, JD, Geenbaum, A, Hammaling, S, McKenney, A, and Soensen, D (1999) LAPACK Uses Guide SIAM, Philadelphia Batlett, R, Wächte, A, and Biegle, L (2000) Active set vs inteio point stategies fo model pedictive contol In Poc Ameican Contol Conf, Chicago, IL Benzi, M and Wathen, AJ (2008) Some peconditioning techniques fo saddle point poblems In WHA Schildes et al (ed), Model Ode Reduction: heoy, Reseach Aspects and 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