On the Convergence Time of Dual Subgradient Methods for Strongly Convex Programs

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 63(4), PP. 05, APRIL, 08 On he Convergence Time of Dual Subgradien Mehods for Srongly Convex Programs Hao Yu and Michael J. Neely Universiy of Souhern California Absra This paper sudies he convergence ime of dual gradien mehods for general (possibly non-differeniable) srongly convex programs. For general convex programs, he convergence ime of dual subgradien/gradien mehods wih simple running averages (running averages sared from ieraion 0) is known o be O( ). This paper shows ha he convergence ime for general ɛ srongly convex programs is O( ). This paper also considers ɛ a variaion of he average scheme, called he sliding running averages, and shows ha if he dual funion of he srongly convex program is locally quadraic hen he convergence ime of he dual gradien mehod wih sliding running averages is O(log( )). The convergence ime analysis is furher verified by ɛ numerical experimens. I. INTRODUCTION Consider he following srongly convex program: min f(x) () s.. g k (x) 0, k {,,..., m} () x X (3) where se X R n is closed and convex; funion f(x) is coninuous and srongly convex on X (srong convexiy is defined in Seion II-A); funions g k (x), k {,,..., m} are Lipschiz coninuous and convex on X. Noe ha he funions f(x), g (x),..., g m (x) are no necessarily differeniable. I is assumed hroughou ha problem ()-(3) has an opimal soluion. Srong convexiy of f implies he opimum is unique. Convex program ()-(3) arises ofen in conrol applicaions such as model prediive conrol (MPC) [], decenralized muli-agen conrol [3], and nework flow conrol [4], [5]. More specifically, he model prediive conrol problem is o solve problem ()-(3) where f(x) is a quadraic funion and each g k (x) is a linear funion []. In decenralized muliagen conrol [3], our goal is o develop disribuive algorihms o solve problem ()-(3) where f(x) is he sum uiliy of individual agens and consrains g k (x) 0 capure he communicaion or resource allocaion consrains among individual agens. The nework flow conrol and he ransmission conrol proocol (TCP) in compuer neworks can be inerpreed as he dual subgradien algorihm for solving a problem of he form ()-(3) [4], [5]. In paricular, Seion V-B shows ha he dual subgradien mehod based online flow conrol rapidly converges o opimaliy when uiliies are srongly convex. Hao Yu and Michael J. Neely are wih he Deparmen of Elerical Engineering, Universiy of Souhern California, Los Angeles, CA, USA. This work was presened in par a IEEE Conference on Decision and Conrol (CDC), Osaka, Japan, December, 05 []. This work is suppored in par by he NSF gran CCF A. The ɛ-approximae Soluion Definiion. Le x be he opimal soluion o problem ()- (3). For any ɛ > 0, a poin x ɛ X is said o be an ɛ- approximae soluion if f(x ɛ ) f(x ) + ɛ and g k (x ɛ ) ɛ, k {,..., m}. Definiion. Le x(), {,,...} be he soluion sequence yielded by an ieraive algorihm. The convergence ime (o an ɛ-approximae soluion) is he number of ieraions required o achieve an ɛ-approximae soluion. An algorihm is said o have convergence ime O(h(ɛ)) if {x(), O(h(ɛ))} is a sequence of ɛ-approximae soluions for some funion h(ɛ). Noe if x() saisfies f(x()) f(x ) + and g k(x()), k {,..., m} for all, hen error decays wih ime like O( ) and he convergence ime is O( ɛ ). B. The Dual Subgradien/Gradien Mehod The dual subgradien mehod is a convenional mehod o solve ()-(3) [6], [7]. I is an ieraive algorihm ha, every ieraion, removes he inequaliy consrains () and chooses primal variables o minimize a funion over he se X. This can be decomposed ino parallel smaller problems if he objeive and consrain funions are separable. The dual subgradien mehod can be inerpreed as a subgradien/gradien mehod applied o he Lagrangian dual funion of convex program ()-(3) and allows for many differen sep size rules [7]. This paper focuses on a consan sep size due o is simpliciy for praical implemenaions. Noe ha by Danskin s heorem (Proposiion B.5(a) in [7]), he Lagrangian dual funion of a srongly convex program is differeniable, hus he dual subgradien mehod for srongly convex program ()-(3) is in fa a dual gradien mehod. The consan sep size dual subgradien/gradien mehod solves problem ()-(3) as follows: Algorihm. [The Dual Subgradien/Gradien Mehod] Le c > 0 be a consan sep size. Le λ(0) 0, be a given consan veor. A each ieraion, updae x() and λ( + ) as follows: x() = argmin [f(x) + m k= λ k()g k (x)]. x X λ k ( + ) = max {λ k () + cg k (x()), 0}, k. If here exiss z X such ha g k (z) δ, k {,..., m} for some δ > 0, one can conver an ɛ-approximae poin x ɛ o anoher poin x = θx ɛ + ( θ)z, for θ = δ, which saisfies all desired consrains and ɛ+δ has objeive value wihin O(ɛ) of opimaliy.

2 Raher han using x() from Algorihm as he primal soluions, he following running average schemes are considered: ) Simple Running Averages: Use x() = τ=0 x(τ) as he soluion a each ieraion {,,...}. ) Sliding Running Averages: Use x() = x(0) and x() = { τ= x(τ) x( ) if is even if is odd as he soluion a each ieraion {,,...}. The simple running average sequence x() is also called he ergodic sequence in [8]. The idea of using he running average x() as he soluions, raher han he original primal variables x(), daes back o Shor [9] and has been furher developed in [0] and [8]. The consan sep size dual subgradien algorihm wih simple running averages is also a special case of he drif-plus-penaly algorihm, which was originally developed o solve more general sochasic opimizaion [] and used for deerminisic convex programs in []. (See Seion I.C in [] for more discussions.) This paper proposes a new running average scheme, called sliding running averages. This paper shows ha he sliding running averages can have a beer convergence ime when he dual funion of he convex program saisfies addiional assumpions. C. Relaed Work A lo of lieraure focuses on he convergence ime of dual subgradien mehods o an ɛ-approximae soluion. For general convex programs in he form of ()-(3), where he objeive funionf(x) is convex bu no necessarily srongly convex, he convergence ime of he drif-plus-penaly algorihm is shown o be O( ɛ ) in [], [3]. A similar O( ɛ ) convergence ime of he dual subgradien algorihm wih he averaged primal sequence is shown in [4]. A recen work [5] shows ha he convergence ime of he drif-plus-penaly algorihm is O( ɛ ) if he dual funion is locally polyhedral and he convergence ime is O( ) if he dual funion is locally quadraic. For a ɛ 3/ special class of srongly convex programs in he form of ()- (3), where f(x) is second-order differeniable and srongly convex and g k (x), k {,,..., m} are second-order differeniable and have bounded Jacobians, he convergence ime of he dual subgradien algorihm is shown o be O( ɛ ) in []. Noe ha convex program ()-(3) wih second order differeniable f(x) and g k (x) in general can be solved via inerior poin mehods wih linear convergence ime. However, o achieve fas convergence in praice, he barrier parameers mus be scaled carefully and he compuaion complexiy associaed wih each ieraion is high. In conras, he dual subgradien algorihm is a Lagrangian dual mehod and can yield disribued implemenaions wih low compuaion complexiy when he objeive and consrain funions are separable. This paper considers a class of srongly convex programs ha is more general han hose reaed in []. Besides he srong convexiy of f(x), we only require he consrain Noe ha bounded Jacobians imply Lipschiz coninuiy. Work [] also considers he effe of inaccurae soluions for he primal updaes. The analysis in his paper can also deal wih inaccurae updaes. In his case, here will be an error erm δ on he righ of (6). funions g k (x) o be Lipschiz coninuous. The funions f(x) and g k (x) can even be non-differeniable. Thus, his paper can deal wih non-smooh opimizaion. For example, he l norm x is non-differeniable and ofen appears as par of he objeive or consrain funions in machine learning, compressed sensing and image processing applicaions. This paper shows ha he convergence ime of he dual subgradien mehod wih simple running averages for general srongly convex programs is O( ɛ ) and he convergence ime can be improved o O(log( ɛ )) by using sliding running averages when he dual funion is locally quadraic. A closely relaed recen work is [6] ha considers srongly convex programs wih srongly convex and second order differeniable objeive funions f(x) and conic consrains in he form of Gx + h K, where K is a proper cone. The auhors in [6] show ha a hybrid algorihm using boh dual subgradien and dual fas gradien mehods can have convergence ime O( ); and he dual subgradien mehod ɛ /3 can have convergence ime O(log( ɛ )) if he srongly convex program saisfies an error bound propery. Resuls in he curren paper are developed independenly and consider general nonlinear convex consrain funions; and show ha he dual subgradien/gradien mehod wih a differen averaging scheme has an O(log( ɛ )) convergence ime when he dual funion is locally quadraic. Anoher parallel work [7] considers srongly convex programs wih srongly convex and smooh objeive funions f(x) and general consrain funions g(x) wih bounded Jacobians. The auhors in [7] show ha he dual subgradien/gradien mehod wih simple running averages has O( ɛ ) convergence. This paper and independen parallel works [6], [7] obain similar convergence imes of he dual subgradien/gradien mehod wih differen averaging schemes for srongly convex programs under slighly differen assumpions. However, he proof echnique in his paper is fundamenally differen from ha used in [6] and [7]. Works [6], [7] and oher previous works, e.g., [], follow he classical opimizaion analysis approach based on he descen lemma, while his paper is based on he drif-plus-penaly analysis ha was originally developed for sochasic opimizaion in dynamic queuing sysems [8], []. Using he drif-plus-penaly echnique, we furher propose a new Lagrangian dual ype algorihm wih O( ɛ ) convergence for general convex programs (possibly wihou srong convexiy) in a following work [9]. II. PRELIMINARIES AND BASIC ANALYSIS A. Preliminaries and Assumpions Definiion 3 (Lipschiz Coninuiy). Le X R n be a convex se. Funion h : X R m is said o be Lipschiz coninuous on X wih modulus L if here exiss L > 0 such ha h(y) h(x) L y x for all x, y X. Noe ha in he above definiion can be general norms. However, hroughou his paper, we use o denoe he veor Euclidean norm. Definiion 4 (Srongly Convex Funions). Le X R n be a convex se. Funion h is said o be srongly convex on X

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 63(4), PP. 05, APRIL, 08 wih modulus α if here exiss a consan α > 0 such ha h(x) α x is convex on X. Lemma. [See [0] or Corollary in [9]] Le h(x) be srongly convex on convex se X wih modulus α. If x op is a global minimum, h(x op ) h(y) α y xop, y X. Denoe he sacked veor of muliple funions g (x), g (x),..., g m (x) as g(x) = [ g (x), g (x),..., g m (x) ] T. Assumpion. In convex program ()-(3), funion f(x) is srongly convex on X wih modulus α; and funion g(x) is Lipschiz coninuous on X wih modulus β. Assumpion. There exiss a Lagrange muliplier veor λ = [λ, λ,..., λ m] T 0 such ha q(λ ) = min x X {f(x) : g k(x) 0, k {,,..., m}}, where q(λ) = min {f(x)+ m x X k= λ kg k (x)} is he Lagrangian dual funion of problem ()-(3). B. Properies of he Lyapunov drif Denoe λ() = [ λ (),..., λ m () ] T. Define Lyapunov funion L() = λ() and drif () = L( + ) L(). Lemma. A each ieraion in Algorihm, c () = λt ( + )g(x()) c λ( + ) λ() (4) Proof: The updae equaions λ k ( + ) = max{λ k () + cg k (x()), 0}, k {,,..., m} can be rewrien as λ k ( + ) = λ k () + c g k (x()), k {,,..., m}, (5) { gk (x()), if λ where g k (x()) = k () + cg k (x()) 0 c λ, k(), else k {,,..., m}. Fix k {,,..., m}. Squaring boh sides of (5) and dividing by faor yields: [λ k( + )] = [λ k()] + c [ g k(x())] + cλ k () g k (x()) = [λ k()] + c [ g k(x())] + cλ k ()g k (x()) + cλ k ()[ g k (x()) g k (x())] (a) = [λ k()] + c [ g k(x())] + cλ k ()g k (x()) c g k (x())[ g k (x()) g k (x())] = [λ k()] c [ g k(x())] + c[λ k () + c g k (x())]g k (x()) (b) = [λ k()] [λ k( + ) λ k ()] + cλ k ( + )g k (x()) where (a) follows from λ k ()[ g k (x()) g k (x())] = c g k (x())[ g k (x()) g k (x())], which can be shown by separaely considering cases g k (x()) = g k (x()) and g k (x()) g k (x()); and (b) follows from he fa ha λ k ( + ) = λ k () + c g k (x()). Summing over k {,,..., m} and dividing boh sides by faor c yields he resul. III. CONVERGENCE TIME ANALYSIS This seion analyzes he convergence ime of x() for srongly convex program ()-(3). A. Objeive Value Violaions Lemma 3. Le x be he opimal soluion o ()-(3). Assume c α/β. A each ieraion in Algorihm, we have c () + f(x()) f(x ), 0. (6) Proof: Fix 0. Since f(x) is srongly convex wih modulus α and for all k {,..., m} funions g k (x) are convex and scalars λ k () are non-negaive, he funion f(x)+ m k= λ k()g k (x) is also srongly convex wih modulus α. Noe ha x() = argmin [f(x) + m k= λ k()g k (x)]. By x X Lemma wih x op = x() and y = x, we have f(x()) + m k= λ k()g k (x()) [ f(x ) + m k= λ k()g k (x ) ] α x() x Adding his o equaion (4) yields c () + f(x()) f(x ) + B(), where B() = c λ( + ) λ() α x() x + λ T ()[g(x ) g(x())] + λ T ( + )g(x()) (7) I remains o show ha B() 0. Since x is he opimal soluion o problem ()-(3), we have g k (x ) 0 for all k. Noe ha λ k (+) 0 for all k. Thus, λ T (+)g(x ) 0. Adding he nonnegaive quaniy λ T (+)g(x ) o he righhand-side of (7) gives: B() c λ( + ) λ() α x() x + λ T ()[g(x ) g(x())] + λ T ( + )g(x()) λ T ( + )g(x ) = c λ( + ) λ() α x() x + [λ T () λ T ( + )][g(x ) g(x())] (a) c λ( + ) λ() α x() x + λ() λ( + ) g(x()) g(x ) (b) c λ( + ) λ() α x() x + β λ() λ( + ) x() x = ( λ( + ) λ() cβ x() x ) c (c) 0 (α cβ ) x() x where (a) follows from he Cauchy-Schwarz inequaliy; (b) follows from Assumpion ; and (c) follows from c α β. Theorem (Objeive Value Violaions). Le x X be he opimal soluion o problem ()-(3). If c α β in Algorihm, hen f(x()) f(x ) + λ(0),.

4 Proof: Fix. Summing (6) over τ {0,,..., } yields c τ=0 (τ) + τ=0 f(x(τ)) f(x ). Dividing by faor and rearranging erms yields τ=0 f(x(τ)) f(x ) + L(0) L() = f(x ) + λ(0) λ() Finally, f(x()) B. Consrain Violaions Lemma 4. For any > 0,. τ=0 f(x(τ)) by Jensen s inequaliy. λ k ( ) λ k ( ) + c g k (x(τ)), k {,,..., m} τ= In paricular, λ k () λ k (0) + c τ=0 g k(x(τ)) for all and k {,,..., m}. Proof: Fix k {,,..., m}. Noe ha λ k ( + ) = max{λ k ( ) + cg k (x( )), 0} λ k ( ) + cg k (x( )). By induion, his lemma follows. Lemma 5. Le λ 0 be given in Assumpion. If c α β in Algorihm, hen λ() saisfies λ() λ(0) + λ + λ,. (8) Proof: Le x be he opimal soluion o problem ()- (3). Assumpion implies ha f(x ) = q(λ ) f(x(τ)) + m k= λ k g k(x(τ)), τ {0,,...}, where he inequaliy follows from he definiion of q(λ). Thus, we have f(x ) f(x(τ)) m k= λ k g k(x(τ)), τ {0,,...}. Summing over τ {0,,..., } yields = f(x ) f(x(τ)) c m k= λ k m k= τ=0 [ g k (x(τ)) τ=0 ] (a) τ=0 k= c m λ kg k (x(τ)) m λ k[λ k () λ k (0)] k= λ kλ k () (b) c λ λ() (9) where (a) follows from Lemma 4 and (b) follows from he Cauchy-Schwarz inequaliy. On he oher hand, summing (6) in Lemma 3 over τ {0,,..., } yields f(x ) f(x(τ)) τ=0 Combining (9) and (0) yields L() L(0) c = λ() λ(0) c (0) λ() λ(0) c c λ λ() ( λ() λ ) λ(0) + λ λ() λ(0) + λ + λ Theorem (Consrain Violaions). Le λ 0 be defined in Assumpion. If c α β in Algorihm, hen he consrain funions saisfy for all : λ(0) + λ g k (x()) + λ, k {,..., m} Proof: Fix and k {,,..., m}. Recall ha x() = τ=0 x(τ). Thus, g k(x()) (a) τ=0 g k(x(τ)) (b) λ k () λ k (0) λ k() λ() (c) λ(0) + λ + λ, where (a) follows from he convexiy of g k (x); (b) follows from Lemma 4; and (c) follows from Lemma 5. Theorems - show ha using he simple running average sequence x() ensures he objeive value and consrain error decay like O(/). A lower bound of f(x()) f(x ) O( ) easily follows from srong dualiy and Theorem. See full version [0] for more discussions. IV. EXTENSIONS This seion shows ha he convergence ime of sliding running averages x() is O(log( ɛ )) when he dual funion of problem ()-(3) saisfies addiional assumpions. A. Smooh Dual Funions Definiion 5 (Smooh Funions). Le X R n and funion h(x) be coninuously differeniable on X. Funion h(x) is said o be smooh on X wih modulus L if x h(x) is Lipschiz coninuous on X wih modulus L. Define q(λ) = min x X {f(x)+λt g(x)} as he dual funion of problem ()-(3). Recall ha f(x) is srongly convex wih modulus α by Assumpion. For fixed λ R m +, f(x) + λ T g(x) is srongly convex wih respe o x X wih modulus α. Define x(λ) = argmin x X {f(x) + λ T g(x)}. By Danskin s heorem (Proposiion B.5 in [7]), q(λ) is differeniable wih gradien λ q(λ) = g(x(λ)). Lemma 6 (Smooh Dual Funions). The dual funion q(λ) is smooh on R m + wih modulus γ = β α. Proof: Fix λ, µ R m +. Le x(λ) = argmin x X {f(x) + λ T g(x)} and x(µ) = argmin x X {f(x) + µ T g(x)}. By Lemma, we have f(x(λ)) + λ T g(x(λ)) f(x(µ)) + λ T g(x(µ)) α x(λ) x(µ) and f(x(µ))+µ T g(x(µ)) f(x(λ)) + µ T g(x(λ)) α x(λ) x(µ). Summing he above wo inequaliies and simplifying gives α x(λ) x(µ) [µ λ] T [g(x(λ)) g(x(µ))] (a) µ λ g(x(λ)) g(x(µ)) (b) β µ λ x(λ) x(µ) where (a) follows from he Cauchy-Schwarz inequaliy and (b) follows because g(x) is Lipschiz coninuous. This implies x(λ) x(µ) β λ µ () α

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 63(4), PP. 05, APRIL, 08 Thus, we have q(λ) q(µ) (a) = g(x(λ)) g(x(µ)) (b) β x(λ) x(µ) (c) β α λ µ where (a) follows from λ q(λ) = g(x(λ)); (b) follows from he Lipschiz coninuiy of g(x); and (c) follows from (). Thus, q(λ) is smooh on R m + wih modulus L = β α. Since λ q(λ()) = g(x()), he dynamic of λ() can be inerpreed as he projeed gradien mehod wih sep size c o solve max λ R m + {q(λ)} where q( ) is a smooh funion by Lemma 6. Thus, we have he nex lemma. Lemma 7. Assume problem ()-(3) saisfies Assumpions -. If c α β, hen q(λ ) q(λ()) λ(0) λ,. Proof: Recall ha a projeed gradien algorihm wih sep size c < γ maximizes a concave funion wih smooh modulus γ wih he error decaying like O( ). Thus, his lemma follows. See [0] for he complee proof. B. Problems wih Locally Quadraic Dual Funions In addiion o Assumpions -, his subseion furher requires he nex assumpion. Assumpion 3 (Locally Quadraic Dual Funions). Le λ be a Lagrange muliplier of problem ()-(3) defined in Assumpion. There exiss D q > 0 and L q > 0, where he subscrip q denoes locally quadraic, such ha for all λ {λ R m + : λ λ D q }, he dual funion q(λ) saisfies q(λ ) q(λ) + L q λ λ. Lemma 8. Suppose problem ()-(3) saisfies Assumpions -3. Le q(λ), λ, D q and L q be given in Assumpion 3. ) If λ R m + and q(λ ) q(λ) L q D q, hen λ λ D q. ) The λ defined in Assumpion is unique. Proof: See [0] for he proof. Define T q = λ(0) λ cl q Dq. () Lemma 9. Assume problem ()-(3) saisfies Assumpions -3. If c α β in Algorihm, hen λ() λ D q for all T q, where T q is defined in (). Proof: By Lemma 7 and Lemma 8, if λ(0) λ L q D q, hen λ() λ D q. Noe ha λ(0) λ implies ha λ(0) λ L q D q. cl qd q Lemma 0. Assume problem ()-(3) saisfies Assumpions -3. If c α β in Algorihm, hen ) λ() λ λ(0) λ, T q, where clq T q is defined in (). ) λ() λ ( ) Tq λ(tq +cl q ) λ ( ) Dq ( + cl q ) Tq, T q, where T q is +clq defined in (). Proof: ) By Lemma 7, q(λ ) q(λ()) λ(0) λ,. By Lemma 9 and Assumpion 3, q(λ ) q(λ()) L q λ() λ, T q. Thus, we have L q λ() λ λ(0) λ, T q, which implies ha λ() λ λ(0) λ, T q. clq ) By par (), we know λ() λ D q, T q. The second par essenially follows from Theorem in [], which shows ha he projeed gradien mehod for se consrained smooh convex opimizaion converge geomerically if he objeive funion saisfies a quadraic growh condiion. See [0] for he proof. Corollary. Assume problem ()-(3) saisfies Assumpions -3. If c α β in Algorihm, hen λ() λ() ( ) Dq ( + cl q ) Tq, T q, where T q is defined +clq in (). Proof: λ() λ() λ() λ + λ() λ (a) ( ) Dq ( + cl q ) Tq ( ) Dq + ( + cl q ) Tq + clq + clq (b) ( ) Dq ( + cl q ) Tq, + clq where (a) follows from par () in Lemma 0; and (b) follows from +clq <. Theorem 3. Assume problem ()-(3) saisfies Assumpions -3. Le x be he opimal soluion and λ be defined in Assumpion 3. If c α β in Algorihm, hen ( ) f( x()) f(x ) + η, Tq, where η = +clq D q (+clq)tq +D Tq q(+cl q) ( λ(0) + λ + λ ) c is defined in (). and T q Proof: Fix T q. By Lemma 3, we have c (τ) + f(x(τ)) f(x ) for all τ {0,,...}. Summing over τ {, +,..., } yields c τ= (τ)+ τ= f(x(τ)) f(x ). Dividing by faor yields Thus, we have f(x(τ)) f(x ) + τ= L() L() f( x()) (a) f(x(τ)) (b) f(x L() L() ) + τ= = f(x ) + λ() λ() = f(x ) + λ() λ() + λ() λ() (c) f(x ) + λ() λ() + λ() λ() λ() ( (d) ( ) ) Dq ( + cl q ) Tq f(x +clq ) + 4 ( ) Dq ( + cl q ) Tq λ() +clq + (3)

6 (e) f(x ) + ( ) ( D q ( + cl q ) Tq + clq c + D q( + cl q ) Tq λ() ) (f) f(x ) + ( ) η + clq c where (a) follows from x() = τ= x(τ) and he convexiy of f(x); (b) follows from (3); (c) follows from he Cauchy-Schwarz inequaliy; (d) follows from Corollary ; (e) follows from < ; and (f) follows from (8) and he definiion of η. +clq Theorem 4. Assume problem ()-(3) saisfies Assumpions -3. If c α β in Algorihm, hen g k ( x()) ( ), k {,,..., m}, Tq, +clq D q(+cl q) Tq where T q is defined in (). Proof: Fix T q and k {,,..., m}. Thus, we have g k ( x()) (a) g k (x(τ)) (b) ( λk () λ k () ) τ= λ() λ() (c) D q( + cl q ) Tq ( + clq ) where (a) follows from he convexiy of g k (x); (b) follows from Lemma 4; and (c) follows from Corollary. Under Assumpions -3, Theorems 3 and 4 show ha if c α β, hen x() provides an ɛ-approximae soluion wih convergence ime O(log( ɛ )). C. Discussions ) Praical Implemenaions: Assumpion 3 in general is difficul o verify. However, we noe ha o ensure x() provides he beer O(log( ɛ )) convergence ime, we only require c α β, which is independen of he parameers in Assumpions 3. Namely, in praice, we can blindly apply Algorihm o problem ()-(3) wih no need o verify Assumpion 3. If problem ()-(3) happens o saisfy Assumpion 3, hen x() enjoys he faser convergence ime O(log( ɛ )). If no, hen x() (or x()) a leas has convergence ime O( ɛ ). ) Local Assumpion and Local Geomeric Convergence: Since Assumpion 3 only requires he quadraic propery o be saisfied in a local radius D( q around λ, he error of ( ) ) Algorihm sars o decay like O only afer +clq λ() arrives a he D q local radius afer T q ieraions, where T q is independen of he approximaion requiremen ɛ and hence is order O(). Thus, Algorihm provides an ɛ-approximae soluion wih convergence ime O(log( ɛ )). However, i is possible ha T q is relaively large if D q is small. In fa, T q > 0 because Assumpion 3 only requires he dual funion o have he quadraic propery in a local radius. Thus, he heory developed in his seion can deal wih a large class of problems. On he oher hand, if he dual funion has he quadraic propery globally, i.e., for all λ ( 0, hen T q = 0 and he error of Algorihm decays ( ) like O ),. +clq 3) Locally Srongly Concave Dual Funions: The following assumpion is sronger han Assumpion 3 bu can be easier o verify in cerain cases. Assumpion 4 (Locally Srongly Concave Dual Funions). Le λ be a Lagrange muliplier veor defined in Assumpion. There exiss D c > 0 and L c > 0, where he subscrip c denoes locally srongly concave, such ha he dual funion q(λ) is srongly concave wih modulus L c over {λ R m + : λ λ D c }. In fa, Assumpion 4 implies Assumpion 3. Lemma. If problem ()-(3) saisfies Assumpion 4, hen i also saisfies Assumpion 3 wih D q = D c and L q = Lc. Proof: See [0] for he proof. Since Assumpion 4 implies Assumpion 3, by he resuls from he previous subseion, x() from Algorihm provides an ɛ-approximae soluion wih convergence ime O(log( ɛ )). V. APPLICATIONS A. Problems wih Non-Degenerae Consrain Qualificaions Theorem 5. Consider srongly convex program ()-(3) where f(x) and g k (x), k {,,..., m} are second-order coninuously differeniable. Le x be he unique soluion. ) Le K {,,..., m} be he se of aive consrains, i.e., g k (x ) = 0, k K, and denoe he veor composed by g k (x), k K as g K. If g(x) has a bounded Jacobian and rank( x g K (x ) T ) = K, hen Assumpions -3 hold. ) If g(x) has a bounded Jacobian and rank( x g(x ) T ) = m, hen Assumpions -4 hold. Proof: See [0] for he proof. Corollary. Consider min{f(x) : Ax b}, where f(x) is second-order coninuously differeniable and srongly convex funion; and A is an m n marix. ) Le x be he opimal soluion. Assume Ax b has l rows ha hold wih equaliy, and le A be he l n submarix of A corresponding o hese aive rows. If rank(a ) = l, hen Assumpions -3 hold. ) If rank(a) = m, hen Assumpions -4 hold wih D c =. B. Nework Uiliy Maximizaion (NUM) Consider a nework wih l links and n flow sreams. Le {b, b,..., b l } be he capaciies of each link and {x, x,..., x n } be he raes of each flow sream. Le N (k) {,,..., n}, k l be he se of flow sreams ha use link k. This problem is o maximize he uiliy funion n i= w i log(x i ) wih consans w i > 0, i n, which represens a measure of nework fairness [], subje o he capaciy consrain of each link. This problem is known as

7 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 63(4), PP. 05, APRIL, 08 he nework uiliy maximizaion (NUM) problem and can be formulaed as follows 3 : n min w i log(x i ) (4) i= s.. Ax b (5) x 0 (6) where A = [a,, a n ] is a 0- marix of size m n such ha a ij = if and only if flow x j uses link i and b > 0. Noe ha problem (4)-(6) saisfies Assumpions and. By he resuls from Seion III, x() has convergence ime O( ɛ ). The nex heorem provides sufficien condiions such ha x() has beer convergence ime O(log( ɛ )). Theorem 6. The NUM problem (4)-(6) saisfies: ) Le b max = max i n b i and x max > 0 such ha x max i > b max, i {,..., n}. If we replace consrain (6) wih 0 x x max in problem (4)-(6), hen we obain an equivalen problem. ) Le x be he opimal soluion. Assume Ax b has m rows ha hold wih equaliy, and le A be he m n submarix of A corresponding o hese aive rows. If rank(a ) = m, hen Assumpions -3 hold for he above equivalen problem. 3) If rank(a) = m, hen Assumpions -4 hold for he above equivalen problem. Proof: See [0] for he proof. VI. NUMERICAL RESULTS A. Nework Uiliy Maximizaion Problems Consider he simple NUM problem described in Figure. Le x, x and x 3 be he daa raes of sream, and 3 and le he nework uiliy be minimizing log(x ) log(x ) 3 log(x 3 ). I can be checked ha capaciy consrains oher han x + x + x 3 0, x + x 8, and x + x 3 8 are redundan. By Theorem 6, he NUM problem can be formulaed as follows: where A = min log(x ) log(x ) 3 log(x 3 ) s.. Ax b, 0 x x max 0 0, b = and x max = [,, ] T. The opimal soluion o his NUM problem is x =, x = 3., x 3 = 4.8 and he opimal value is Since he objeive funion is separable, he dual subgradien/gradien mehod can yield a disribued soluion. This is why he dual subgradien/gradien mehod is widely used o solve NUM problems [4]. The objeive funion is srongly convex wih modulus α = on X = {0 x xmax } and g( ) is Lipschiz coninuous wih modulus β 6 on X. Figure verifies he convergence of x() wih c = α β = Wihou loss of opimaliy, we define log(0) = and hence log( ) is defined over R +. Or alernaively, we can replace he non-negaive rae consrains wih x i x min i, i {,,..., n} where x min i, i {,,..., n} are sufficienly small posiive numbers. and λ(0) = 0. Since λ(0) = 0, by Theorem, we have f(x()) f(x ), > 0. To verify he convergence ime of consrain violaions, Figure 3 plos g (x()), g (x()), g 3 (x()) and / wih boh x-axis and y-axis in log 0 scales. As observed in Figure 3, he curves of g (x()) and g 3 (x()) are parallel o he curve of / for large. Noe ha g (x()) 0 is saisfied early because his consrain is loose. Figure 3 shows ha error decays like O( ) and suggess ha he convergence ime is aually Θ( ɛ ) for his NUM problem. Fig.. sream)) capaciy=8) capaciy=8) sream)) capaciy=0) sream)3) A simple NUM problem wih 3 flow sreams capaciy=8) capaciy=8) Noe ha rank(a) = 3. By Theorem 6, his NUM problem saisfies Assumpions -4. Figure 4 verifies Theorem 6 ha he convergence ime of x() is O(log( ɛ )) by showing ha error decays like O( ) wih c = α β = 363. B. Large Scale Quadraic Programs Consider quadraic program min x R N {x T Qx + d T x : Ax b} where Q, A R N N and d, b R N. Q = UΣU H R N N where U is he orhonormal basis for a random N N zero mean and uni variance normal marix and Σ is he diagonal marix wih enries from uniform [, 3]. A is a random N N zero mean and uni variance normal marix. d and b are random veors wih enries from uniform [0, ]. In a PC wih a 4 core.7ghz Inel i7 Cpu and 6GB Memory, we run boh Algorihm and quadprog from Malab, which by defaul is using he inerior poin mehod, over randomly generaed large scale quadraic programs wih N = 400, 600, 800, 000 and 00. For differen problem size N, he running ime is he average over 00 random quadraic programs and is ploed in Figure 5. Fig.. Objeive Values Consrain Values NUM: Convergence of Objeive Values of x() Ieraions: g() = 0 opimal value f(x()) NUM:Convergence of Consrain Values of x() g(x()) = x() + x() + x3()! 0 g3(x()) = x() + x3()! 8 g(x()) = x() + x()! Ieraions: The convergence of x() for a NUM problem.

8 Fig. 3. Fig. 4. Consrain Violaions 0 NUM: Convergence Time of Consrain Violaions of x() v.s. O(/) g(x()) = x() + x()! 8 g(x()) = x() + x() + x3()! 0 = g3(x()) = x() + x3()! 8 curves are parallel for large Ieraions: The convergence ime of x() for a NUM problem. Consrain Violaions 0 NUM: Convergence Time of Consrain Violaions of ex() v.s. O( (0:998) ) g(ex()) = ex() + ex()! 8 g3(ex()) = ex() + ex3()! 8 g(ex()) = ex() + ex() + ex3()! 0 0: Ieraions: The convergence ime of x() for a NUM problem. VII. CONCLUSIONS This paper sudies he dual gradien mehod for srongly convex programs and shows ha he convergence ime of simple running averages is O( ɛ ). This paper also considers a variaion of he primal averages, called he sliding running averages, and shows ha if he dual funion is locally quadraic hen he convergence ime is O(log( ɛ )). Running Time (secs) Dual Gradien Mehod v.s. quadprog quadprog dual gradien mehod Problem Dimension: N VIII. ACKNOWLEDGEMENT We hank an anonymous reviewer for bringing o our aenions independen parallel works [6], [7] on similar opics and work [] on linear convergence for unconsrained opimizaion. By using he resuls from [], we improve he convergence ime from O( ) in our conference version [] ɛ /3 o O(log( ɛ )) when he dual funion is locally quadraic. REFERENCES [] H. Yu and M. J. Neely, On he convergence ime of he drif-pluspenaly algorihm for srongly convex programs, in Proceedings of IEEE Conference on Decision and Conrol (CDC), 05. [] I. Necoara and V. Nedelcu, Rae analysis of inexa dual firs-order mehods applicaion o dual decomposiion, IEEE Transaions on Auomaic Conrol, vol. 59, no. 5, pp. 3 43, May 04. [3] H. Terelius, U. Topcu, and R. M. Murray, Decenralized muli-agen opimizaion via dual decomposiion, in IFAC World Congress, 0. [4] S. H. Low and D. E. Lapsley, Opimizaion flow conrol I: basic algorihm and convergence, IEEE/ACM Transaions on Neworking, vol. 7, no. 6, pp , 999. [5] S. H. Low, A dualiy model of TCP and queue managemen algorihms, IEEE/ACM Transaions on Neworking, vol., no. 4, pp , 003. [6] M. S. Bazaraa, H. D. Sherali, and C. M. Shey, Nonlinear Programming: Theory and Algorihms. Wiley-Inerscience, 006. [7] D. P. Bersekas, Nonlinear Programming, nd ed. Ahena Scienific, 999. [8] T. Larsson, M. Pariksson, and A.-B. Srömberg, Ergodic, primal convergence in dual subgradien schemes for convex programming, Mahemaical programming, vol. 86, no., pp. 83 3, 999. [9] N. Z. Shor, Minimizaion Mehods for Non-Differeniable Funions. Springer-Verlag, 985. [0] H. D. Sherali and G. Choi, Recovery of primal soluions when using subgradien opimizaion mehods o solve lagrangian duals of linear programs, Operaions Research Leers, vol. 9, no. 3, pp. 05 3, 996. [] M. J. Neely, Sochasic Nework Opimizaion wih Applicaion o Communicaion and Queueing Sysems. Morgan & Claypool Publishers, 00. [], Disribued and secure compuaion of convex programs over a nework of conneed processors, in DCDIS Inernaional Conference on Engineering Applicaions and Compuaional Algorihms, 005. [3], A simple convergence ime analysis of drif-plus-penaly for sochasic opimizaion and convex programs, arxiv:4.079, 04. [4] A. Nedić and A. Ozdaglar, Approximae primal soluions and rae analysis for dual subgradien mehods, SIAM Journal on Opimizaion, vol. 9, no. 4, pp , 009. [5] S. Supiayapornpong, L. Huang, and M. J. Neely, Time-average opimizaion wih nonconvex decision se and is convergence, in Proceedings of IEEE Conference on Decision and Conrol (CDC), 04. [6] I. Necoara and A. Parascu, Ieraion complexiy analyisis of dual firs order mehods for conic convex programming, Opimizaion Mehod and Sofware, vol. 3, no. 3, pp , 06. [7] I. Necoara, A. Parascu, and A. Nedić, Complexiy cerificaions of firs-order inexa lagrangian mehods for general convex programming: Applicaion o real-ime mpc, in Developmens in Model-Based Opimizaion and Conrol. Springer, 05, pp [8] M. J. Neely, Dynamic power allocaion and rouing for saellie and wireless neworks wih ime varying channels, Ph.D. disseraion, Massachuses Insiue of Technology, 003. [9] H. Yu and M. J. Neely, A simple parallel algorihm wih an O(/) convergence rae for general convex programs, SIAM Journal on Opimizaion, vol. 7, no., pp , 07. [0], On he convergence ime of dual subgradien mehods for srongly convex programs, arxiv: , 05. [] I. Necoara, Y. Neserov, and F. Glineur, Linear convergence of firs order mehods for non-srongly convex opimizaion, arxiv: , 05. [] F. P. Kelly, Charging and rae conrol for elasic raffic, European Transaions on Telecommunicaions, vol. 8, no., pp , 997. Fig. 5. The average running ime for large scale quadraic programs.