A primal-dual Laplacian gradient flow dynamics for distributed resource allocation problems

Transcription

1 2018 Annual American Conrol Conference (ACC) June 27 29, Wisconsin Cener, Milwaukee, USA A primal-dual Laplacian gradien flow dynamics for disribued resource allocaion problems Dongsheng Ding and Mihailo R. Jovanović Absrac We employ he proximal augmened Lagrangian mehod o solve a class of convex resource allocaion problems over a conneced undireced nework of n agens. The agens are coupled by a linear resource equaliy consrain and heir saes are confined o a nonnegaive orhan. By inroducing he indicaor funcion associaed wih a nonnegaive orhan, we bring he problem ino a composie form wih a nonsmooh objecive and linear equaliy consrains. A primaldual Laplacian gradien flow dynamics based on he proximal augmened Lagrangian is proposed o solve he problem in a disribued way. These dynamics conserve he sum of he agen saes and he corresponding equilibrium poins are he Karush- Kuhn-Tucker poins of he original problem. We combine a Lyapunov-based argumen wih LaSalle s invariance principle o esablish global asympoic sabiliy and use an economic dispach case sudy o demonsrae he effeciveness of he proposed algorihm. I. INTRODUCTION The resource allocaion problem considers how o allocae he resource o individuals by minimizing he cos of he allocaion. I arises in many areas, including communicaion neworks, power grids, and producion managemen in economics [1]. Due o he emergence of he nex-generaion communicaion neworks and he cyber-physical sysems, he resource allocaion problem seeks o allocae he given resource in a disribued way, using local informaion exchange in a nework. Paricular insances of resource allocaion are given by he nework uiliy maximizaion [2] [5] and he economic dispach problems [6] [10]. A class of resource allocaion problems can be formulaed as follows x f(x) x i Ω i, i = 1,..., n where f(x) = n i=1 f i(x i ) is a separable convex objecive funcion, x = [ x 1 x n ] T R n is he opimizaion variable, 1 is he vecor of all-ones, b R is he given resource, and Ω i is a convex se. All x i s are coupled by he resource equaliy consrain and each x i has a se consrain. Disribued resource allocaion problems have araced significan aenion in recen years. In his seup, each node i in a nework has a local objecive funcion f i and a decision variable x i. The nodes can exchange informaion Financial suppor from he Naional Science Foundaion under Award ECCS is graefully acknowledged. D. Ding and M. R. Jovanović are wih he Ming Hsieh Deparmen of Elecrical Engineering, Universiy of Souhern California, Los Angeles, CA s: dongshed@usc.edu, mihailo@usc.edu. (1) wih heir neighbors wih he goal of compuing soluion of he resource allocaion problem in a disribued manner. Many disribued algorihms have been proposed for (1) or have been applied o (1) as a case sudy. Two main classes include he consensus-based algorihms and he primal-dual dynamics. Some recen works in he firs class include [7] [11], where consensus algorihms were incorporaed ino he cenralized gradien descen o enable disribued compuaion. In [8], a robus disribued algorihm was presened for he problem wih box consrains. In [9], wo disribued projeced algorihms were proposed. While [8] and [9] do no require any special iniializaion, algorihms in [7], [10], [11] do need paricular iniializaion. For (1) wihou he se consrain, he exponenial convergence of consensus-based algorihms was esablished in [9] [11]. The second class is based on primal-dual dynamics [12] [15], where he se consrain can be inequaliy consrains. In [12], [13], for a non-smooh augmened Lagrangian, i was shown ha is primal-dual dynamics can converge exponenially for (1) wih eiher he resource equaliy consrain or he se consrain. In [14], [15], global asympoic sabiliy of he projeced primal-dual dynamics was esablished by uilizing local convexiy-concaviy of he saddle funcion. The asympoic convergence of he primal-dual dynamics was furher discussed in [16] [18]. Alhough he primal-dual dynamics are asympoically sable, he se consrain may cause he difficuly in esablishing he exponenial convergence and he resource allocaion consrain can impede he disribued implemenaion. Moivaed by he above resuls, we are ineresed in developing a disribued primal-dual algorihm o solve (1) wih non-negaive consrains. To deal wih such consrains, we uilize he proximal augmened Lagrangian mehod [19] which yields a non-smooh composie opimizaion problem wih linear equaliy consrains. For his class of problems, he gradien flow dynamics associaed wih he proximal Lagrangian are no convenien for disribued implemenaion. Insead, we propose a primal-dual Laplacian gradien flow dynamics o compue he soluion via in-nework opimizaion and prove ha hese dynamics converge o he opima globally and asympoically. Finally, we use he economic dispach problem o illusrae he performance of he proposed algorihm. Our presenaion is organized as follows. In Secion II, we formulae he problem and provide background on he proximal augmened Lagrangian mehod. In Secion III, we propose primal-dual Laplacian gradien flow dynamics for /$ AACC 5316

2 in-nework resource allocaion and combine a Lyapunovbased argumen wih LaSalle s invariance principle o esablish global asympoic sabiliy. In Secion IV, we use compuaional experimens on an economic dispach problem o demonsrae meris and effeciveness of our approach. We conclude he paper and highligh fuure direcions in Secion V. II. PROBLEM FORMULATION AND BACKGROUND In his secion, we formulae he resource allocaion problem as a non-smooh composie opimizaion problem and inroduce he proximal augmened Lagrangian. We consider he resource allocaion problem, x f(x) x 0 where f is a convex objecive funcion, x R n is he opimizaion variable, and b is he given resource. For b > 0, opimizaion problem (2) is feasible. By inroducing an auxiliary variable z := [ z 1 z n ] T, (2) can be cas as a nonsmooh composie opimizaion wih equaliy consrains, x, z f(x) + g(z) x z = 0 where g(z) = n i = 1 g i(z i ) and each g i is an indicaor funcion of he non-negaive orhan, 0, zi 0 g i (z i ) := (4), oherwise. Since he objecive funcion in (3) is non-differeniable, he exising resuls (e.g., [13]) canno be employed direcly o esablish exponenial convergence. The proximal augmened Lagrangian mehod was recenly proposed in [19] o deal wih non-smooh composie opimizaion problems. The proposed approach explois he proximal operaor associaed wih he non-smooh par of he objecive funcion o eliminae he auxiliary variable z from he augmened Lagrangian. The proximal operaor associaed wih he funcion g is he r of he following opimizaion problem prox µg (v) := argmin z g(z) + 1 2µ z v 2. Here, v is a given vecor, µ is a posiive parameer, and he associaed value funcion specifies he Moreau envelope, M µg (v) = g(prox µg (v)) + 1 2µ prox µg(v) v 2. Noably, he Moreau envelope is coninuously differeniable, even when g is no, and is gradien is deermined by M µg (v) = 1 ( v proxµg (v) ). µ For he indicaor funcion of he non-negaive orhan (4), we (2) (3) have prox µg (v i ) = [v i ] + = M µg (v i ) = 1 2µ [v i] 2, M µg (v i ) = 1 µ [v i]. vi, v i 0 0, v i < 0 The Lagrangian associaed wih (3) is given by L(x, z; λ, y) = f(x) + g(z) + λ(1 T x b) + y T (x z) (5) where λ R and y R n are he Lagrange mulipliers. The augmened Lagrangian conains addiional erms ha inroduce quadraic penalies on he violaion of linear consrains, L µ (x, z; λ, y) = L(x, z; λ, y) + 1 2µ ((1T x b) 2 + x z 2 ) where µ is a posiive parameer. Wihou loss of generaliy, we have seleced he same penaly on he violaion of boh linear consrains in (3). The compleion of squares can be used o rewrie L µ as L µ (x, z; λ, y) = f(x) + λ(1 T x b) + 1 2µ (1T x b) 2 + g(z) + 1 2µ z (x + µy) 2 µ 2 y 2 (7) and he explici r of (7) wih respec o z is given by z µ(x, y) = prox µg (x + µy). As demonsraed in [19], he proximal augmened Lagrangian is obained by resricing he augmened Lagrangian on he manifold ha resuls from explici minimizaion of L µ over z. This eliminaes he non-smooh erm from L µ and cass i in erms of he Moreau envelope, L µ (x; λ, y) = f(x) + λ(1 T x b) + 1 2µ (1T x b) 2 + M µg (x + µy) µ 2 y 2 (8) The Arrow-Hurwicz-Uzawa gradien flow dynamics based on proximal augmened Lagrangian (8) is given by or, equivalenly, ẋ = L µ (x; λ, y) λ = + L µ (x; λ, y) ẏ = + L µ (x; λ, y) ẋ = ( f(x) + M µg (x + µy) + 1λ + 1 µ 1(1T x b)) λ = 1 T x b ẏ = µ ( M µg (x + µy) y). (9) In [19], global asympoic sabiliy of he gradien flow dynamics for problems of he form (3), bu wihou he resource allocaion consrain, was esablished. The above gradien flow dynamics accouns for his consrain; however, due o he appearance of 1 T x, he updae of x i in (9) requires global informaion which impedes disribued implemenaion. In wha follows, we modify (9) in order o solve he resource allocaion problem (2) over a conneced undireced nework (6) 5317

3 and esablish global asympoic sabiliy of he resuling primal-dual Laplacian gradien flow dynamics. III. PRIMAL-DUAL LAPLACIAN GRADIENT FLOW DYNAMICS In his secion, we propose a modificaion of he Arrow- Hurwicz-Uzawa gradien flow dynamics ha saisfies he resource allocaion consrain for all imes. We show ha he equilibrium poins of he resuling Laplacian gradien flow dynamics correspond o he KKT poins in (3) and prove global asympoic sabiliy. I is easy o show ha he following primal-dual Laplacian gradien flow dynamics ẋ = L ( f(x) + M µg (x + µy)) ẏ = µ ( M µg (x + µy) y) (10) saisfies 1 T x() b = 0 for all 0 if 1 T x(0) b = 0. This is because 1 T ẋ = 0 implies ha 1 T x() is a conserved quaniy of (10). Furhermore, since f(x) and M µg (x+µy) are sums of n individual funcions, primal-dual Laplacian gradien flow dynamics (10) is a disribued algorihm. The se of KKT poins for (3) is given by } Ω f(x) + 1λ + y = 0, x z = 0 = (x, z, λ, y) y g(z), 1 T x b = 0 where g(z) is he sub-gradien se of g a z. On he oher hand, if 1 T x(0) b = 0, he equilibrium poins ( x, ȳ) of (10) are given by } L ( f(x) + M µg (x + µy)) = 0 Ω = (x, y) M µg (x + µy) y = 0, 1 T x b = 0 The following lemma esablishes correspondence beween he ses Ω and Ω. Lemma 1: For any ( x, ȳ) Ω, here exiss z and λ such ha ( x, z, λ, ȳ) Ω. Similarly, for any (x, z, λ, y ) Ω, (x, λ ) Ω. Proof: For any ( x, ȳ) Ω, f( x) + M µg ( x + µȳ) is in he null space of he Laplacian L and M µg ( x+µȳ) = ȳ. Thus, here is a non-zero λ such ha f( x) + ȳ + λ1 = 0. Furhermore, M µg ( x + µȳ) = 1 µ( x + µȳ proxµg ( x + µȳ) ) = ȳ implies x = prox µg ( x + µȳ) = z and, consequenly, ȳ = ( f( x) + λ1) g( x). Finally, since he equaliy consrain 1 T x = b is, by consrucion, saisfied for all imes, by selecing x = z = x, λ = λ, and y = ȳ, we conclude ha (x, z, λ, y ) Ω. Conversely, for any (x, z, λ, y ) Ω, we have L( f(x ) + y ) = 0, 1 T x = b, and y g(x ). Furhemore, since x = z = prox µg (x + µy ), from he definiion of he gradien of he Moreau envelope, we have M µg (x + µy ) = y. By selecing x = x, ȳ = y, we conclude ha ( x, ȳ) Ω. In wha follows, we conduc sabiliy analysis of primaldual Laplacian gradien flow dynamics (10) wih he iniial condiion 1 T x(0) = b. This is done for objecive funcions f in (2) ha are srongly convex and have Lipschiz coninuous gradiens. Assumpion 1: The objecive funcion f in (2) is srongly convex and is gradien is Lipschiz coninuous. A. Transformed primal-dual Laplacian gradien flow dynamics The eigenvalue decomposiion of he graph Laplacian L of a conneced undireced nework is given by L = V ΛV T = [ [ ] [ ] U 1 n 1] Λ 0 0 U T = UΛ U T 1 n 1T where Λ 0 is a diagonal marix of non-zero eigenvalues of L and he marix U R n (n 1) and saisfies he following properies: (i) U T U = I; (ii) UU T = I 1 n 11T ; (iii) 1 T U = 0 and U T 1 = 0. We inroduce an affine coordinae ransformaion x = Uξ + 1θ (11) where ξ R n 1 and θ = 1 n 1T x(0). From he above properies of U, we have ξ = U T x, and (10) can be ransformed ino he following form ξ = Λ 0 U T ( f(x) + M µg (x + µy) ) ẏ = µ ( M µg (x + µy) y) (12) where ξ(0) R n 1 and y(0) R n are arbirary iniial condiions. B. Global asympoic sabiliy We firs sae a lemma abou proximal operaors which is used in he proof of Theorem 3. Lemma 2: Le g : R n R be a proper, lower semiconinuous, convex funcion and le prox µg : R n R n be he corresponding proximal operaor. Then, for any a, b R n, we can wrie prox µg (a) prox µg (b) = D(a b) (13) where D is a symmeric marix saisfying 0 D I. Proof: See [19, Lemma 2]. We nex esablish global asympoic sabiliy of ransformed primal-dual Laplacian gradien flow dynamics (12). Theorem 3: Le Assumpion 1 hold. Then, (12) is globally asympoically sable. Proof: Le ( ξ, ỹ) denoe perurbaions around he equilibrium poins ( ξ, ȳ) of (12). Then, ξ = Λ 0 U T ( f(x) f( x) + 1 µ m ) ỹ = m µỹ (14) 5318

4 where m = µ( M µg (x + µy) M µg ( x + µȳ)), x = Uξ + 1θ, and x = U ξ + 1θ. Derivaive of he Lyapunov funcion candidae V ( ξ, ỹ) = 1 2 ξ T Λ 1 0 ξ ỹ 2 along he soluions of (14) is deermined by V = ξ T Λ 1 0 ξ + ỹ T ỹ = ξ T U T ( f(uξ + 1θ) f(u ξ + 1θ)) 1 µ ξ T U T m + ỹ T m µ ỹ 2 From Lemma 2, we have m = (I D)(U ξ + µỹ), where 0 D I, and V = ξ T U T ( f(uξ + 1θ) f(u ξ + 1θ)) 1 µ ξ T U T (I D)U ξ µỹ T Dỹ. (15) Srong convexiy of f implies V 0. Furhermore, V = 0 is equivalen o U ξ = 0 and Dỹ = 0. We nex show ha he larges invarian se of V = 0 is conained in Ω. Firs, from x = U ξ, we have x = x. Second, from he propery of he marix D in Lemma 2, we have prox µg ( x + µy) = prox µg ( x + µȳ). Third, x = prox µg ( x+µȳ) implies 1 µ ( x+µy prox µg( x+µy)) y = M µg ( x + µy) y = 0. Thus, (x, y) saisfies he second and hird condiions in Ω. Furhermore, since L( f( x) + M µg ( x + µy)) = L( M µg ( x + µy) M µg ( x + µȳ)) = L(I D)ỹ = Lỹ, subsiuion of U ξ = 0, Dỹ = 0, and x = 0 ino (14) implies ha he saionary poin for ỹ is ỹ = c1 where c is a non-zero scalar. Therefore, we have L( f( x) + M µg ( x + µy)) = 0 and he firs condiion in Ω holds for (x, y). By combing he above expressions, we conclude ha (x, y) = ( x, y) is in Ω. Thus, since V is radially unbounded LaSalle s invariance principle implies ha (12) is globally asympoically sable. IV. AN EXAMPLE: ECONOMIC DISPATCH PROBLEM In his secion, we provide an example of he economic dispach problem o illusrae he performance of he proposed primal-dual Laplacian gradien flow dynamics. An IEEE 118-bus benchmark problem [20] has 54 generaors and each generaor has a quadraic cos funcion, f i (x i ) = a i + b i x i + c i x 2 i, where x i is he power injecion a bus i, a i [6.78, 74.33], b i [8.3391, ], and c i [0.0024, ]. The load is b = The communicaion nework has a ring opology wih several addiional edges (1, 11), (11, 21), (21, 31), (31, 41), and (41, 51). To show he robusness o he flucuaions in elecriciy price, a = 2000 we increase he parameers in he objecive funcion by 20%. We use ODE45 in MATLAB o simulae primal-dual Laplacian gradien flow dynamics (10), wih n = 54 and µ = In all simulaions, he relaive and absolue error olerances are se o and 10 15, respecively, he iniial condiion for each generaor is se o x i (0) = 4200/54, and y(0) = 0. The simulaion resuls are shown in Figs. 1 and 2. Figures 1a and 2a demonsrae ha all power injecions x i say in he feasible region and converge o he opimal soluion compued by CVX [21]. The following relaive error x x 2 + y ȳ 2 x(0) x 2 + y(0) ȳ 2 is used o show he exponenial convergence in righ plos in Figs. 1 and 2, where verical axis is shown in he logarihmic scale. In boh cases, logarihmic relaive errors decrease linearly excep for he ime insan a which he objecive funcion has been perurbed. V. CONCLUDING REMARKS In his paper, we employ he proximal augmened Lagrangian mehod o solve a class of convex resource allocaion problems over a conneced undireced nework wih n agens. We show ha he resource allocaion problem can be formulaed as a composie opimizaion problem. To perform in-nework compuaions, we propose a primal-dual Laplacian gradien flow dynamics based on he proximal augmened Lagrangian. We demonsrae ha he equilibrium poins correspond o KKT poins of he original problem. A Lyapunov-based argumen is used o esablish global asympoic sabiliy and demonsrae ha he proposed gradien flow dynamics globally converge o he opimal soluion. Finally, we apply he proposed algorihm o an economic dispach problem o illusraes is effeciveness. Several fuure direcions are of ineres. Firs, as indicaed in our simulaions, he proposed gradien flow dynamics appear o be exponenially converging. The exponenial convergence of he primal-dual Laplacian gradien flow dynamics is an open problem ha requires furher invesigaion. Second, he robusness of he proposed algorihm o various uncerainy sources is worh exploring. Third, o deal wih a broader class of problems, i is of ineres o exend he proposed gradien flow dynamics o more general consrains. ACKNOWLEDGMENTS We would like o hank Sepideh Hassan-Moghaddam for useful discussion. REFERENCES [1] P. Michael, A survey on he coninuous nonlinear resource allocaion problem, Eur. J. Oper. Res., vol. 185, no. 1, pp. 1 46, [2] E. Wei, A. Ozdaglar, and A. Jadbabaie, A disribued Newon mehod for nework uiliy maximizaion, in Proceedings of he 49h IEEE Conference on Decision and Conrol, 2010, pp [3], A disribued Newon mehod for nework uiliy maximizaion I: Algorihm, IEEE Trans. Auom. 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