SDDP FOR MULTISTAGE STOCHASTIC LINEAR PROGRAMS BASED ON SPECTRAL RISK MEASURES
|
|
- Marjory Morgan
- 5 years ago
- Views:
Transcription
1 SDDP FOR MULTISTAGE STOCHASTIC LINEAR PROGRAMS BASED ON SPECTRAL RISK MEASURES VINCENT GUIGUES AND WERNER RÖMISCH Absrac. We consider risk-averse formulaions of mulisage sochasic linear programs. For hese formulaions, based on convex combinaions of specral risk measures, risk-averse dynamic programming equaions can be wrien. As a resul, he Sochasic Dual Dynamic Programming (SDDP) algorihm can be used o obain approximaions of he corresponding risk-averse recourse funcions. This allows us o define a risk-averse nonanicipaive feasible policy for he sochasic linear program. Formulas for he cus ha approximae he recourse funcions are given. In paricular, we show ha some of he cu coefficiens have analyic formulas. AMS subjec classificaions: 90C15, 91B Inroducion Mulisage sochasic programs play a cenral role when developing opimizaion models under sochasic uncerainy in engineering, ransporaion, finance and energy. Furhermore, since measuring, bounding or minimizing he risk of decisions becomes more and more imporan in applicaions, risk-averse formulaions of such opimizaion models are needed and have o be solved. Several risk-averse model varians allow for a reformulaion as a classical mulisage model as in [6, 8] and he presen paper. From a mahemaical poin of view mulisage sochasic opimizaion mehods represen infinie-dimensional models in spaces of random vecors saisfying cerain momen condiions and conain high-dimensional inegrals. Hence, heir numerical soluion is a challenging ask. Each soluion approach consiss a leas of wo ingrediens: (i) numerical inegraion mehods for compuing he expecaion funcionals and (ii) algorihms for solving he resuling finie-dimensional opimizaion models. The favorie approach for (i) is o generae possible scenarios (i.e., realizaions) of he random vecor involved and o use hem as grid poins for he numerical inegraion. Scenario generaion can be done by Mone Carlo, Quasi-Mone Carlo or opimal quanizaion mehods (see [5, 18] for overviews and [3, Par III] for furher informaion). Scenarios for mulisage sochasic programs have o be ree srucured o model he increasing chain of σ-fields. Exising sabiliy and convergence resuls like [11, 10], [12], and [21] provide approaches and condiions implying convergence of such schemes, in paricular, for he deerminisic firs-sage soluions. Hence, hey jusify rolling horizon approaches based on repeaed solving of mulisage models, see [9] for insance. The algorihms employed for (ii) depend on srucural properies of he basic opimizaion model and on he inheren srucure induced by he scenario ree approximaion (see he survey [19] on decomposiion mehods). Some algorihmic approaches incorporae he scenario generaion mehod (i) as an algorihmic sep of he soluion mehod. Such approaches are, for example, sochasic decomposiion mehods for mulisage models (see [20]), approximae dynamic programming (see [17]) and Sochasic Dual Key words and phrases. Specral risk measure and Sochasic programming and Risk-averse opimizaion and Decomposiion algorihms and Mone-Carlo sampling. 1
2 2 VINCENT GUIGUES AND WERNER RÖMISCH Dynamic Programming (SDDP) iniiaed in [13], revisied in [16, 22] and also sudied in he presen paper. We consider risk-averse formulaions of mulisage sochasic linear programs of he form (1) T T inf d x 1,...,x T 1x 1 +θ 1 E[ d x ]+ θ ρ φ ( d kx k ) =2 =2 k=2 C x = ξ D x 1, x 0, x is F -measurable, = 1,...,T, where x 0 is given, parameers d,c,d are deerminisic, (ξ ) T =1 is a sochasic process, F is he sigma-algebra F := σ(ξ j,j ), (θ ) T =1 are nonnegaive weighs summing o one, and ρ φ is a specral risk measure [1] or disorion risk measure [14, 15] depending on a risk specrum φ L 1 ([0,1]). In he above formulaion, we have assumed ha he (one-period) specral risk measure akes as argumen a random income and ha he rajecory of he process is known unil he firs sage. We assume relaively complee recourse for (1), which means ha for any feasible sequence of decisions (x 1,...,x ) o any -sage scenario (ξ 1,ξ 2,...,ξ ), here exiss a sequence of feasible decisions (x +1,...,x T ) wih probabiliy one. A non-risk-averse model amouns o aking θ 1 = 1 and θ = 0 for = 2,...,T. A more general risk-averse formulaion for mulisage sochasic programs is considered in [8]. For hese models, dynamic programming (DP) equaions are wrien in [8] and an SDDP algorihm is deailed o obain approximaions of he corresponding recourse funcions in he form of cus. The main conribuion of his paper is o provide analyic formulas for some cu coefficiens, independen of he sampled scenarios and ha can be useful for implemenaion. We also specialize he SDDP algorihm and especially he compuaion of he cus for he paricular risk-averse model (1). We sar by seing down some noaion: e will denoe a column vecor of all ones; for x,y R n, he vecor x y R n is defined by (x y)(i) = x(i)y(i), i = 1,...,n; for x R n, he vecor x + R n is defined by x + (i) = max(x(i),0), i = 1,...,n; he available hisory of he process a sage is denoed by ξ [] := (ξ j, j ); for vecors x 1,...,x n, he noaion x n1:n 2 sands for he concaenaion (x n1,x n1+1,...,x n2 ) for 1 n 1 n 2 n; δ ij is he Kronecker dela defined for i,j inegers by δ ij = 1 if i = j and 0 oherwise. 2. Risk-averse dynamic programming Le F Z (x) = P(Z x) be he cumulaive disribuion funcion of an essenially bounded random variable Z and le F Z (p) = inf{x : F Z(x) p} be he generalized inverse of F Z. Given a risk specrum φ L 1 ([0,1]) he specral risk measure ρ φ generaed by φ is given by Acerbi [1] 1 ρ φ (Z) = 0 F Z (p)φ(p)dp. Specral risk measures have been used in various applicaions (porfolio selecion Acerbi and Simonei [2], insurance Coer and Kevin [4]). The Condiional Value-a-Risk (CVaR) of level 0 < ε < 1, denoed by CVaR ε, is a paricular specral risk measure obained aking φ(u) = 1 ε 1 0 u<ε (Acerbi [1]). In wha follows, we consider more generally a piecewise consan risk funcion φ( ) wih J jumps a 0 < p 1 < p 2 <... < p J < 1. We se φ k = φ(p + k ) φ(p k ) = φ(p k) φ(p k 1 ), for k = 1,...,J,
3 SDDP FOR MULTISTAGE STOCHASTIC PROGRAMS BASED ON SPECTRAL RISK MEASURES 3 wih p 0 = 0, and we assume ha (i) φ( ) is posiive, (ii) φ k < 0, k = 1,...,J, (iii) 1 0 φ(u)du = 1. In his conex, ρ φ can be expressed as a linear combinaion of Condiional Value-a-Risk measures. Wih his choice of risk funcion φ, he specral risk measure ρ φ (Z) can be expressed as he opimal value of a linear program, Acerbi and Simonei [2]: (2) ρ φ (Z) = inf w R J J φ k [p k w k E [w k Z] + ] φ(1)e[z]. k=1 Using his formulaion for ρ φ, dynamic programming equaions are wrien in [8] for risk-averse formulaion (1). More precisely, problem (1) can be expressed as T inf d (3) x 1 x 1 + θ c 1 w +Q 2 (x 1,ξ [1],z 1,w 2,...,w T ), 1,w 2:T =2 C 1 x 1 = ξ 1 D 1 x 0, x 1 0, w R J, = 2,...,T, wih z 1 = 0, vecor c 1 = φ p, and where for = 2,...,T, (4) Q (x 1,ξ [ 1],z 1,w :T ) = E ξ ξ [ 1] ( inf x,z f (z,w )+Q +1 (x,ξ [],z,w +1:T ) z = z 1 d x, C x = ξ D x 1, x 0 wih (5) f (z,w ) = (δ T θ 1 +φ(1)θ )z θ φ (w z e) +, and Q T+1 0. Funcion Q +1 represens a sage a cos-o-go or recourse funcion which is riskaverse. As shown in he nex secion, i can be approximaed by cuing planes by some polyhedral funcion Q +1. These approximae recourse funcions are useful for defining a feasible approximae policy obained solving (6) inf f (z,w )+Q +1 (x,ξ [],z,w +1:T ) x,z C x = ξ D x 1, x 0, z = z 1 d x, a sage = 2,...,T, knowing x 1,z 1, firs sage decision variables w :T, and ξ. Firs sage decision variables x 1 and w 2:T are soluion o (3) wih Q 2 replaced by he approximaion Q Algorihmic issues Dynamic programming equaions (3)-(4) make possible he use of decomposiion algorihms such as SDDP o obain approximaions of he corresponding recourse funcions. When applied o DP equaions (3)-(4), he convergence of his algorihm is proved in [8] under he following assumpions: (A1) The suppors of he disribuions of ξ 1,...,ξ T, are discree and finie. (A2) Process (ξ ) is inersage independen. (A3) For = 1,...,T, for any feasible x 1 and for any realizaion ξ of ξ, he se {x : x 0, C x = ξ D x 1 } is bounded and nonempy. In he sequel, we assume ha Assumpions (A1), (A2), and (A3) hold. In paricular, we denoe he realizaions of ξ by ξ i, i = 1,...,q < + and se p(,i) = P(ξ = ξ i ). Since he suppors of he disribuions of he random vecors ξ 2,...,ξ T are discree and finie, opimizaion problem (1) is finie dimensional and he evoluion of he uncerain parameers over he opimizaion period can be represened by a scenario ree having a finie number of scenarios )
4 4 VINCENT GUIGUES AND WERNER RÖMISCH ha can happen in he fuure for ξ 2,...,ξ T. The roo node of he scenario ree corresponds o he firs ime sep wih ξ 1 deerminisic. For a given sage, o each node of he scenario ree corresponds an hisory ξ []. The children nodes of a node a sage 1 are he nodes ha can happen a sage +1 if we are a his node a. A sampled scenario (ξ 1,...,ξ T ) corresponds o a paricular succession of nodes such ha ξ is a possible value for he process a and ξ +1 is a child of ξ. A given node in he ree a sage is idenified wih a scenario (ξ 1,...,ξ ) going from he roo node o his node. In his conex, he SDDP algorihm builds polyhedral lower bounding approximaions Q of Q for = 2,...,T + 1. Each ieraion of his algorihm is made of a forward pass followed by a backward pass. Approximaion Q i for Q available a he end of ieraion i can be expressed as a maximum of cus (hyperplanes lying below he recourse funcions) buil in he backward passes: T +1 (7) Q i (x 1,z 1,w :T ) = max j=0,1,...,ih [ Ej 1 x 1 Z j 1 z 1 + W j,τ 1 w +τ 1 +e j 1 ], knowing ha he algorihm sars aking for Q 0 a known lower bounding affine approximaion of Q while Q i T+1 0. In he above expression, we have assumed ha H cus are buil a each ieraion. If he algorihm runs for K ieraions, we end up wih approximae recourse funcions Q = Q K, = 2,...,T +1. A ieraion i, cus for Q, = 2,...,T, are buil a some poins x k 1,zk 1,wi :T, k = (i 1)H + 1,...,iH, compued in he forward pass replacing he recourse funcions Q +1 by Q i 1 +1 (noe ha since variables w 2:T are firs sage decision variables, hey jus depend on he ieraion). More precisely, he cus are compued for ime sep T + 1 down o ime sep 2. For ime sep T + 1, since Q i T+1 = Q T+1 = 0, he cus for Q T+1 are obained aking null vecors for ET k, Zk T, W k,τ T, ek T for k = (i 1)H + 1,...,iH. For = 2,...,T, using lower bounding approximaion Q i +1 of Q +1, we can bound from below Q (x 1,z 1,w :T ) by E ξ [Q i (x 1,z 1,w :T,ξ )] wih Q i (x 1,z 1,w :T,ξ ) given as he opimal value of he following linear program: τ=1 (8) inf (δ T θ 1 +φ(1)θ )z θ φ v + θ x,z,v, θ v 0, v w z e, x 0, z +d x = z 1 C x = ξ D x 1 E i x + Z i z + θ e T τ=1 W i,τ w +τ + e i (a) (b) (c) where E i (resp. Z i, W i,τ, and e i ) is he marix whose (j + 1)h line is Ej (resp. Z j, Wj,τ, and e j ) for j = 0,...,iH. In he backward pass of ieraion i, he above problem is solved wih (x 1,z 1,w :T,ξ ) respecively replaced by (x k 1,zk 1,wi :T,ξj ) for k = (i 1)H +1,...,iH and j = 1,...,q. Le σ k,j, σ k,j,µ k,j,π k,j, and ρ k,j, be he (row vecors) opimal Lagrange mulipliers respecively for he consrains v w i z e, v 0, (8)-(a), (8)-(b), and (8)-(c) for he problem defining Q i (xk 1,zk 1,wi :T,ξj ) for k = (i 1)H + 1,...,iH and j = 1,...,q. The following proposiion provides he cus compued for Q, = 2,...,T, a ieraion i: Proposiion 3.1. [Opimaliy cus] Le Q, = 2,...,T +1, be he risk-averse recourse funcions given by (4). In he backward pass of ieraion i of he SDDP algorihm, he following cus are compued for hese recourse funcions. For = T + 1, E 1 k, Zk 1, Wk,τ 1, and ek 1 are null for k = (i 1)H + 1,...,iH. For = 2,...,T and k = (i 1)H + 1,...,iH, E 1 k is given by
5 SDDP FOR MULTISTAGE STOCHASTIC PROGRAMS BASED ON SPECTRAL RISK MEASURES 5 q p(,j)πk,j D, and (9) (10) Furher, e k 1 is given by [ q p(, j) q Z 1 k = W k,τ 1 = q q p(,j)µ k,j, W k,1 1 = p(,j)ρ k,j W i,τ 1 Q i (x k 1,z k 1,w i :T,ξ j ) µ k,j z k 1 σ k,j p(,j)σ k,j,, τ = 2,...,T +1. T w i τ=1 ρ k,j ] W i,τ w+τ i +π k,j D x k 1. Proof. Since a dual soluion of he problem defining Q i (x k 1,z k 1,w i :T,ξj ) is a subgradien of he value funcion for problem (8), we obain ha Q i (x 1,z 1,w :T,ξ j ) is bounded from below by Q i (x k 1,z 1,w k :T i,ξj )+µ k,j (z 1 z 1)+σ k k,j (w w) i + T +1 W i,τ 1 (w +τ 1 w+τ 1) π i k,j D (x 1 x k 1). τ=2 ρ k,j Using he above lower bound and he fac ha Q (x 1,z 1,w :T ) is bounded from below by q p(,j)qi (x 1,z 1,w :T,ξ j ), we obain he announced cus. The sopping crierion is discussed in [22] for a non-risk-averse model. The definiion of a sound sopping crierion for he risk-averse model from [22] (based on a nesed formulaion of he problem defined in erms of condiional risk mappings) is a more delicae issue and sill open for discussion. However, since problem (1) can be expressed as a non-risk-averse problem wih modified objecive, variables, and consrains, in our risk-averse conex he sopping crierion is a simple adapaion of he sopping crierion for he non-risk-averse case. More specifically, in he backward pass of ieraion i, for he firs ime sep, firs sage problem (3) is solved replacing recourse funcion Q 2 by Q i 2 Q 2. As a resul, he opimal value of his problem gives a lower bound z inf on he opimal value of (1). In he forward pass of ieraion i, we can compue he oal cos C k on each scenario k = (i 1)H +1,...,iH: (11) C k = d 1x k 1 + T θ c 1w i + =2 T f (z,w k ). i If hese H scenarios were represening all possible evoluions of (ξ 1,...,ξ T ), hen C = 1 H ih k=(i 1)H+1 would be an upper bound on he opimal value of (1) (recall ha he approximae policy is feasible and ha he objecive funcion of (1) can be wrien as an expecaion). Since we only have a sample of all he possible scenarios, C is an esimaion of an upper bound on his opimal value. Inroducing he empirical sandard deviaion σ of he sample (C 1,...,C H ): σ = 1 ih ( C C k ) H 1 2, k=(i 1)H+1 =2 C k
6 6 VINCENT GUIGUES AND WERNER RÖMISCH we can compue he (1 α)-confidence upper bound (12) C +1 α,h 1 σ H on he approximae policy mean value where 1 α,h 1 is he (1 α)-quanile of he Suden s - disribuion wih H 1 degrees of freedom. Since he opimal value of (1) is less han or equal o he approximae policy mean value, (12) gives an upper bound for he opimal value of (1) wih confidence a leas 1 α. Consequenly, we can sop he algorihm when C+ σ 1 α,h 1 H z inf ε for some ε > 0. Using he previous developmens, he SDDP algorihm for solving (1) can be formulaed as in Figure 1. We now give for some paricular choices of he firs sage variables w2:t 1, he exac expressions (independen of he sampled scenarios) of Z 1 k and W k,τ 1 for every = 2,...,T, k = 1,...,H, and τ = 1,...,T + 1. Though he firs sage feasible se for (3) is no bounded, i can be easily shown ha he opimal values of w 2:T are bounded (see [8] for insance). As a resul, well-chosen box consrains on w, = 2,...,T can be added (a he firs sage, and ha do no modify he opimal value of (3)) wihou changing he cu calculaions (since hese laer are performed for sages = 2,...,T, where w are sae variables). Le us define for = 1,...,T, x = (x 1,...,x ), ξ = (ξ 1,...,ξ ), and le us inroduce he se χ of admissible decisions up o ime sep : χ = {x : ξ realizaion of ξ : x τ 0 and C τ x τ = ξ τ D τ x τ 1,τ = 1,...,}. Since (A3) holds, he ses χ are compac and since g (x ) = τ=2 d τx τ is coninuous, we can inroduce he pairs (C u,cl ) R2 defined by C u = { max g (x ) x χ, C l = { min g (x ) x χ. The objecive of he forward pass is o build saes where cus are compued in he backward pass. A he firs ieraion, insead of building hese saes using he approximae recourse funcions Q 0, we can choose arbirary feasible saes x k 1,z 1,w k, 1 = 2,...,T, (which is a simple ask since relaively complee recourse holds). Wih his varian of he firs ieraion, we have ih cus for Q i a he end of ieraion i. If we choose firs sage variables w1 2:T such ha (i) w1 > Cl e for = 2,...,T (resp. such ha (ii) w 1 < C u e for = 2,...,T) hen Z 1 k and W k,τ 1 for k = 1,...,H, can be compued using Proposiion 3.2-(i) (resp. Proposiion 3.2-(ii)) which follows. For insance, if he coss are posiive hen iem (i) is fulfilled wih w 1 = 0 and iem (ii) aking for each componen of w 1 he opposie of a sric upper bound on he wors cos. Proposiion 3.2. [Cus calculaion a he firs ieraion] Le us consider he risk-averse recourse funcions Q given by (4). Valid cus for Q are given by Proposiion 3.1. Moreover, in he following wo cases, we have closed-form expressions for Z 1 k and Wk,τ 1 (independen of he sampled scenarios): (i) If for = 2,...,T, w 1 > Ce, l hen for = 2,...,T, P() holds where P() : { k = 1,...,H,Z k 1 = θ 1 +φ(0) T l= θ l, k = 1,...,H,W k,τ 1 = θ +τ 1 φ, τ = 1,...,T +1.
7 SDDP FOR MULTISTAGE STOCHASTIC PROGRAMS BASED ON SPECTRAL RISK MEASURES 7 Sep 0: INITIALIZATION. Se i = 1 (ieraion number) and selec confidence levels α (1/2,1) and ε > 0. Take null values for E 1 0,Z0 1, W0,τ 1, = 2,...,T +1. Take e 0 T = 0 and for e0 1 a lower bound on Q for = 2,...,T. Go o Sep 1. Sep 1: FORWARD PASS. Sample H scenarios (ξ 1,ξ2 k,...,ξk T ),k = (i 1)H +1,...,iH. C=0, C Sq=0. Solve he firs sage problem T inf d x 1 x 1 + θ c 1 w +Q i 1 2 (x 1,z 1,w 2,...,w T ), 1,w 2:T =2 C 1 x 1 = ξ 1 D 1 x 0, x 1 0, w R J, = 2,...,T, and sore an opimal soluion (x 1,wi 2:T ). For k = (i 1)H +1,...,iH, Se x k 1 = x 1. For = 2,...,T, Solve inf f (z,w x i )+Qi 1 +1 (x,z,w+1:t i ),z C x = ξ k D x k 1, x 0, z = z 1 k d x, and sore an opimal soluion (x k,z). k End For Compue C k given by (11), C=C+C k, C Sq=C Sq+Ck 2. End For C = C H, σ = 1 H 1 (C Sq H C 2 ), z sup = C σ + 1 α,h 1 H. Go o Sep 2. Sep 2: BACKWARD PASS. For = T +1 down o 2, For k = (i 1)H +1,...,iH, If ( = T +1) hen se E 1 k,zk 1, Wk,τ 1, and ek 1 o 0. Else For j = 1,...,q, Compue Q i (xk 1,zk 1,wi :T,ξj ), i.e., solve (8) replacing (x 1,z 1,w :T,ξ ) by (x k 1,z 1,w k :T i,ξj ) and sore a dual soluion. End For Build a cu for Q, i.e., compue E 1 k,zk 1,Wk,τ 1, and ek 1 using he formulas from Proposiion 3.1. End If End For End For Se z inf o he opimal value of he firs sage problem. Go o Sep 3. Sep 3: STOPPING RULE. If z sup z inf ε hen sop. Else i i+1 and go o Sep 1. End If Figure 1. SDDP algorihm wih relaively complee recourse for risk-averse inersage independen sochasic linear program (1).
8 8 VINCENT GUIGUES AND WERNER RÖMISCH (ii) If for = 2,...,T, w 1 < C u e, hen for = 2,...,T, P() holds where { k = 1,...,H,Z P() : 1 k = θ 1 +φ(1) T l= θ l, k = 1,...,H,W k,τ 1 = 0, τ = 1,...,T +1. Proof. Le us fix {2,...,T}, k {1,...,H}, and j {1,...,q }. We denoe by x,z,v, θ an opimal soluion o he problem defining Q 1 (x k 1,zk 1,w1 :T,ξj ), i.e., problem (8) wrien for i = 1 and wih (x 1,z 1,w :T,ξ ) replaced by (x k 1,zk 1,wi :T,ξj ) (he dependence of he soluion wih respec o k, j is suppressed o alleviae noaion). The KKT condiions for his problem imply δ T θ 1 φ(1)θ µ k,j σ k,j e ρ k,j Z 1 (13) = 0, (14) θ φ σ k,j σ k,j = 0, (15) σ k,j ( z e+w 1 v ) = 0, (16) σ k,j v = 0, where for = T we have se ρ k,j = 0. Nex, since z can be wrien as z = g (x ) for some x χ, in case (i), we have z e C l e < w 1. Furher v = max(0,w 1 z e) = w 1 z e > 0. Using (14) and (16) we hen ge (17) σ k,j = 0 and σ k,j = θ φ. Le us now firs show (i) by backward inducion on. Plugging he value of σ k,j T given in (17) ino (13) we obain µ k,j T = θ 1 φ(1)θ T +θ T e φ = θ 1 +θ T ( φ(1)+ J [φ(p l ) φ(p l 1 )]) = θ 1 θ T φ(0). Using he above relaion and (9) yields ZT 1 k = q T p(t,j)µk,j T = θ T φ(0) +θ 1. Furher, using once again (9), we obain qt qt (18) W k,1 T 1 = p(t,j)σ k,j T = p(t,j)θ T φ = θ T φ. This shows P(T). Le us now assume ha P(+1) holds for some {2,...,T 1} and le us show ha P() holds. Firs noice ha (18) sill holds wih T subsiued wih, i.e., W k,1 1 = θ φ. Furher, for τ = 2,...,T +1, W k,τ 1 = q p(,j)ρk,j W 1,τ 1, from (10), = q p(,j)ρk,j θ +τ 1 e φ, using P(+1), = q p(,j)θ +τ 1 φ = θ +τ 1 φ, since ρ k,j e = 1. Also Z 1 k = q p(,j)µk,j, from (9), = q p(,j)( φ(1)θ +θ φ e ρ k,j Z 1 ), using (13) and (17), = q Z 1 ), using he definiion of φ, p(,j)( φ(0)θ ρ k,j = φ(0)θ + q p(,j)ρk,j (θ 1 +φ(0) T = θ 1 +φ(0) T l= θ l We have hus shown P() which achieves he proof of (i). l=1 l=+1 θ l)e, using P(+1), since ρ k,j e = 1.
9 SDDP FOR MULTISTAGE STOCHASTIC PROGRAMS BASED ON SPECTRAL RISK MEASURES 9 Le us now assume ha w 1 < C u e for = 2,...,T and le us show (ii). Le us fix {2,...,T}, k {1,...,H}, and j {1,...,q }. As before, we denoe by x,z,v, θ an opimal soluion o he problemdefiningq 1 (xk 1,zk 1,w1 :T,ξj ). In hiscase, z e C u e > w1 andv = max(0,w 1 z e) = 0. Using (14) and (15), we see ha (19) σ k,j = θ φ and σ k,j = 0. Using (9), we ge W k,1 1 = 0. We show (ii) by backward inducion. For = T, plugging he value of σ k,j T ino (13) gives µ k,j T = θ 1 φ(1)θ T, which, ogeher wih (9), gives Z k T 1 = θ 1 + φ(1)θ T. We have already proved ha W k,1 T 1 = 0 and hus P(T) holds. Le us now assume ha P( + 1) holds for some {2,...,T 1} and le us show ha P() holds. Since W 1,τ 1 = 0, we obain W k,τ 1 = q p(,j)ρk,j (9) gives W 1,τ 1 = 0 for τ = 2,...,T +1. Plugging σ k,j = 0 ino (13) and using Z 1 k = q p(,j)(φ(1)θ +ρ k,j Z 1 ), = q p(,j)(θ 1 +φ(1) T l= θ l), using P(+1) and ρ k,j e = 1, = θ 1 +φ(1) T l= θ l. This shows P() and achieves he proof of (ii). Proposiion 3.2 can be used as a debugging ool o check he implemenaion of SDDP for riskaverse problem(1). More precisely, we can check ha in cases(i) and (ii), implemening he formulas for Z 1 k and Wk,τ 1 given in Proposiion 3.1 will give he same resuls as implemening he formulas from Proposiion 3.2. A sage, if insead of ρ φ in (1) we use CVaR ε, problem (1) becomes (20) T T inf d x 1,...,x T 1x 1 +θ 1 E[ d x ]+ θ CVaR ε ( d kx k ) =2 =2 k=2 C x = ξ D x 1, x 0, x is F -measurable, = 1,...,T. For his model, we obain a resul analogous o Proposiion 3.2: Proposiion 3.3. Le us consider he risk-averse recourse funcions Q for model (20) and heir approximaions Q i of form (7), obained applying SDDP o he corresponding DP equaions. In he following wo cases, we obain closed-form expressions for Z 1 k and Wk,τ 1 (independen of he sampled scenarios): (i) If for = 2,...,T, w 1 > Cl, hen for = 2,...,T, P() holds where { k = 1,...,H,Z k P() : 1 = θ 1 + T θ l l= ε l, k = 1,...,H,W k,τ 1 = θ+τ 1 ε +τ 1, τ = 1,...,T +1. (ii) If for = 2,...,T, w 1 < Cu, hen for = 2,...,T, P() holds where P() : k = 1,...,H,Z 1 k = θ 1, and W k,τ 1 = 0, τ = 1,...,T +1. Proof. The proof is similar o he proof of Proposiion 3.2. Remark 3.4. In he paricular case when he CVaR levels ε = ε (0,1) are he same a each ime sep, Proposiion 3.3 is a paricular case of Proposiion 3.2 wih φ(1) = 0,φ(0) = 1 ε, and φ = 1/ε R.
10 10 VINCENT GUIGUES AND WERNER RÖMISCH Numerical simulaions for a real-life applicaion modeled as (20) are repored in [7]. When Assumpion (A1) does no hold, as saed in [22], a feasible nonanicipaive policy can sill be proposed using approximae recourse funcions Q obained applying SDDP on a Sample Average Approximaion (SAA) of he original problem (1). References [1] C. Acerbi. Specral measures of risk: a coheren represenaion of subjecive risk aversion. J. of Banking and Finance, 7: , [2] C. Acerbi and P. Simonei. Porfolio opimizaion wih specral measures of risk. Abaxbank inernal repor. Available a hp:// [3] M. Berocchi, G. Consigli, and M. A. H. Dempser, ediors. Sochasic Opimizaion Mehods in Finance and Energy. Springer, New York, [4] J. Coer and K. Dowd. Exreme specral risk measures: An applicaion o fuures clearinghouse variaion margin requiremens. Journal of Banking and Finance, 30: , [5] J. Dupačová, G. Consigli, and S. W. Wallace. Scenarios for mulisage sochasic programs. Annals of Operaions Research, 100:25 53, [6] A. Eichhorn and W. Römisch. Polyhedral risk measures in sochasic programming. SIAM Journal on Opimizaion, 16:69 95, [7] V. Guigues. SDDP for some inersage dependen risk-averse problems and applicaion o hydro-hermal planning. Submied o Compuaional Opimizaion and Applicaions, [8] V. Guigues and W. Römisch. Sampling-based decomposiion mehods for mulisage sochasic programs based on exended polyhedral risk measures. SIAM Journal on Opimizaion, o appear. [9] V. Guigues and C. Sagasizábal. The value of rolling horizon policies for risk-averse hydro-hermal planning. European Journal of Operaional Research, 217: , [10] H. Heisch and W. Römisch. Scenario ree modeling for mulisage sochasic programs. Mahemaical Programming, 118: , [11] H. Heisch, W. Römisch, and C. Srugarek. Sabiliy of mulisage sochasic programs. SIAM Journal on Opimizaion, 17: , [12] T. Pennanen. Epi-convergen discreizaions of mulisage sochasic programs via inegraion quadraures. Mahemaical Programming, 116: , [13] M.V.F. Pereira and L.M.V.G Pino. Muli-sage sochasic opimizaion applied o energy planning. Mahemaical Programming, 52: , [14] G. Ch. Pflug. On disorion funcionals. Saisics and Decisions, 24:45 60, [15] G. Ch. Pflug and W. Römisch. Modeling, Measuring, and Managing Risk. World Scienific, Singapore, [16] A. B. Philpo and Z. Guan. On he convergence of sochasic dual dynamic programming and relaed mehods. Operaions Research Leers, 36: , [17] W.B. Powell. Approximae Dynamic Programming - Solving he curses of dimensionaliy. Wiley, 2nd ediion, Hoboken, New Jersey, [18] W. Römisch. Scenario generaion. In J. J. Cochran, edior, Encyclopedia of Operaions Research and Managemen Science. Wiley, [19] A. Ruszczyński. Decomposiion mehods. In Sochasic Programming, Handbooks in Operaions Research and Managemen Science, chaper 3, pages Elsevier, Amserdam, [20] S. Sen and Zhihong Zhou. Mulisage sochasic decomposiion: A bridge beween sochasic programming and approximae dynamic programming. SIAM Journal on Opimizaion, o appear. [21] A. Shapiro. Inference of saisical bounds for mulisage sochasic programming problems. Mahemaical Mehods of Operaions Research, 58:57 68, [22] A. Shapiro. Analysis of sochasic dual dynamic programming mehod. European Journal of Operaional Research, 209:63 72, Vincen Guigues: Fundação Geulio Vargas, Escola de Maemáica Aplicada, praia de Boafogo, Rio de Janeiro, Brazil, IMPA, Insiuo de Maemáica Pura e Aplicada, 110 Esrada Dona Casorina, Jardim Boanico, Rio de Janeiro, Brazil, vguigues@impa.br and, UFRJ, Escola Poliécnica, Deparameno de Engenharia Indusrial, Ilha do Fundão, CT, Bloco F, Rio de Janeiro, Brazil and Werner Römisch: Humbold-Universiy Berlin, Insiue of Mahemaics, Berlin, Germany, romisch@mah.hu-berlin.de
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
his aricle appeared in a journal published by Elsevier. he aached copy is furnished o he auhor for inernal non-commercial research and educaion use, including for insrucion a he auhors insiuion and sharing
More informationAn introduction to the theory of SDDP algorithm
An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking
More informationApplication of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing
Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationRobust estimation based on the first- and third-moment restrictions of the power transformation model
h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,
More informationINEXACT CUTS FOR DETERMINISTIC AND STOCHASTIC DUAL DYNAMIC PROGRAMMING APPLIED TO CONVEX NONLINEAR OPTIMIZATION PROBLEMS
INEXACT CUTS FOR DETERMINISTIC AND STOCHASTIC DUAL DYNAMIC PROGRAMMING APPLIED TO CONVEX NONLINEAR OPTIMIZATION PROBLEMS Vincen Guigues School of Applied Mahemaics, FGV Praia de Boafogo, Rio de Janeiro,
More informationScenario tree reduction for multistage stochastic programs
Compu Manag Sci (009) 6:7 33 DOI 0.007/s087-008-0087-y ORIGINAL PAPER Scenario ree reducion for mulisage sochasic programs Holger Heisch Werner Römisch Published online: 0 December 008 Springer-Verlag
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationOn Boundedness of Q-Learning Iterates for Stochastic Shortest Path Problems
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 2, May 2013, pp. 209 227 ISSN 0364-765X (prin) ISSN 1526-5471 (online) hp://dx.doi.org/10.1287/moor.1120.0562 2013 INFORMS On Boundedness of Q-Learning Ieraes
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationGENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT
Inerna J Mah & Mah Sci Vol 4, No 7 000) 48 49 S0670000970 Hindawi Publishing Corp GENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT RUMEN L MISHKOV Received
More informationRisk-Averse Stochastic Dual Dynamic Programming
Risk-Averse Sochasic Dual Dynamic Programming Václav Kozmík Deparmen of Probabiliy and Mahemaical Saisics Charles Universiy in Prague Prague, Czech Republic David P. Moron Graduae Program in Operaions
More informationA Primal-Dual Type Algorithm with the O(1/t) Convergence Rate for Large Scale Constrained Convex Programs
PROC. IEEE CONFERENCE ON DECISION AND CONTROL, 06 A Primal-Dual Type Algorihm wih he O(/) Convergence Rae for Large Scale Consrained Convex Programs Hao Yu and Michael J. Neely Absrac This paper considers
More informationMulti-scale 2D acoustic full waveform inversion with high frequency impulsive source
Muli-scale D acousic full waveform inversion wih high frequency impulsive source Vladimir N Zubov*, Universiy of Calgary, Calgary AB vzubov@ucalgaryca and Michael P Lamoureux, Universiy of Calgary, Calgary
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationNotes on Kalman Filtering
Noes on Kalman Filering Brian Borchers and Rick Aser November 7, Inroducion Daa Assimilaion is he problem of merging model predicions wih acual measuremens of a sysem o produce an opimal esimae of he curren
More informationOnline Appendix to Solution Methods for Models with Rare Disasters
Online Appendix o Soluion Mehods for Models wih Rare Disasers Jesús Fernández-Villaverde and Oren Levinal In his Online Appendix, we presen he Euler condiions of he model, we develop he pricing Calvo block,
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationAn Optimal Approximate Dynamic Programming Algorithm for the Lagged Asset Acquisition Problem
An Opimal Approximae Dynamic Programming Algorihm for he Lagged Asse Acquisiion Problem Juliana M. Nascimeno Warren B. Powell Deparmen of Operaions Research and Financial Engineering Princeon Universiy
More informationSZG Macro 2011 Lecture 3: Dynamic Programming. SZG macro 2011 lecture 3 1
SZG Macro 2011 Lecure 3: Dynamic Programming SZG macro 2011 lecure 3 1 Background Our previous discussion of opimal consumpion over ime and of opimal capial accumulaion sugges sudying he general decision
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationBU Macro BU Macro Fall 2008, Lecture 4
Dynamic Programming BU Macro 2008 Lecure 4 1 Ouline 1. Cerainy opimizaion problem used o illusrae: a. Resricions on exogenous variables b. Value funcion c. Policy funcion d. The Bellman equaion and an
More informationEnergy Storage Benchmark Problems
Energy Sorage Benchmark Problems Daniel F. Salas 1,3, Warren B. Powell 2,3 1 Deparmen of Chemical & Biological Engineering 2 Deparmen of Operaions Research & Financial Engineering 3 Princeon Laboraory
More informationSubway stations energy and air quality management
Subway saions energy and air qualiy managemen wih sochasic opimizaion Trisan Rigau 1,2,4, Advisors: P. Carpenier 3, J.-Ph. Chancelier 2, M. De Lara 2 EFFICACITY 1 CERMICS, ENPC 2 UMA, ENSTA 3 LISIS, IFSTTAR
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationSimulation of BSDEs and. Wiener Chaos Expansions
Simulaion of BSDEs and Wiener Chaos Expansions Philippe Briand Céline Labar LAMA UMR 5127, Universié de Savoie, France hp://www.lama.univ-savoie.fr/ Workshop on BSDEs Rennes, May 22-24, 213 Inroducion
More informationA Hop Constrained Min-Sum Arborescence with Outage Costs
A Hop Consrained Min-Sum Arborescence wih Ouage Coss Rakesh Kawara Minnesoa Sae Universiy, Mankao, MN 56001 Email: Kawara@mnsu.edu Absrac The hop consrained min-sum arborescence wih ouage coss problem
More informationInequality measures for intersecting Lorenz curves: an alternative weak ordering
h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 Absrac Inequaliy measures for
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More informationTechnical Report Doc ID: TR March-2013 (Last revision: 23-February-2016) On formulating quadratic functions in optimization models.
Technical Repor Doc ID: TR--203 06-March-203 (Las revision: 23-Februar-206) On formulaing quadraic funcions in opimizaion models. Auhor: Erling D. Andersen Convex quadraic consrains quie frequenl appear
More informationMATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018
MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren
More informationLecture 33: November 29
36-705: Inermediae Saisics Fall 2017 Lecurer: Siva Balakrishnan Lecure 33: November 29 Today we will coninue discussing he boosrap, and hen ry o undersand why i works in a simple case. In he las lecure
More informationGlobal Optimization for Scheduling Refinery Crude Oil Operations
Global Opimizaion for Scheduling Refinery Crude Oil Operaions Ramkumar Karuppiah 1, Kevin C. Furman 2 and Ignacio E. Grossmann 1 (1) Deparmen of Chemical Engineering Carnegie Mellon Universiy (2) Corporae
More informationScheduling of Crude Oil Movements at Refinery Front-end
Scheduling of Crude Oil Movemens a Refinery Fron-end Ramkumar Karuppiah and Ignacio Grossmann Carnegie Mellon Universiy ExxonMobil Case Sudy: Dr. Kevin Furman Enerprise-wide Opimizaion Projec March 15,
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationSimulation of BSDEs and. Wiener Chaos Expansions
Simulaion of BSDEs and Wiener Chaos Expansions Philippe Briand Céline Labar LAMA UMR 5127, Universié de Savoie, France hp://www.lama.univ-savoie.fr/ Sochasic Analysis Seminar Oxford, June 1, 213 Inroducion
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationChapter 2. Models, Censoring, and Likelihood for Failure-Time Data
Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based
More informationSUFFICIENT CONDITIONS FOR EXISTENCE SOLUTION OF LINEAR TWO-POINT BOUNDARY PROBLEM IN MINIMIZATION OF QUADRATIC FUNCTIONAL
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, Series A, OF HE ROMANIAN ACADEMY Volume, Number 4/200, pp 287 293 SUFFICIEN CONDIIONS FOR EXISENCE SOLUION OF LINEAR WO-POIN BOUNDARY PROBLEM IN
More informationNavneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
More informationMODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE
Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS
More informationIntroduction to Probability and Statistics Slides 4 Chapter 4
Inroducion o Probabiliy and Saisics Slides 4 Chaper 4 Ammar M. Sarhan, asarhan@mahsa.dal.ca Deparmen of Mahemaics and Saisics, Dalhousie Universiy Fall Semeser 8 Dr. Ammar Sarhan Chaper 4 Coninuous Random
More informationLECTURE 1: GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS
LECTURE : GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS We will work wih a coninuous ime reversible Markov chain X on a finie conneced sae space, wih generaor Lf(x = y q x,yf(y. (Recall ha q
More informationOptimal approximate dynamic programming algorithms for a general class of storage problems
Opimal approximae dynamic programming algorihms for a general class of sorage problems Juliana M. Nascimeno Warren B. Powell Deparmen of Operaions Research and Financial Engineering Princeon Universiy
More informationAn Introduction to Stochastic Programming: The Recourse Problem
An Inroducion o Sochasic Programming: he Recourse Problem George Danzig and Phil Wolfe Ellis Johnson, Roger Wes, Dick Cole, and Me John Birge Where o look in he ex pp. 6-7, Secion.2.: Inroducion o sochasic
More informationOptima and Equilibria for Traffic Flow on a Network
Opima and Equilibria for Traffic Flow on a Nework Albero Bressan Deparmen of Mahemaics, Penn Sae Universiy bressan@mah.psu.edu Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 1 / 1 A Traffic
More informationDecentralized Stochastic Control with Partial History Sharing: A Common Information Approach
1 Decenralized Sochasic Conrol wih Parial Hisory Sharing: A Common Informaion Approach Ashuosh Nayyar, Adiya Mahajan and Demoshenis Tenekezis arxiv:1209.1695v1 [cs.sy] 8 Sep 2012 Absrac A general model
More informationGeneralized Chebyshev polynomials
Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 86 Roma, Ialy email: c.cesarano@unineunouniversiy.ne ABSTRACT
More informationLecture Notes 2. The Hilbert Space Approach to Time Series
Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship
More informationSupplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence
Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given
More informationInstitute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler
MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion
More informationCorrespondence should be addressed to Nguyen Buong,
Hindawi Publishing Corporaion Fixed Poin Theory and Applicaions Volume 011, Aricle ID 76859, 10 pages doi:101155/011/76859 Research Aricle An Implici Ieraion Mehod for Variaional Inequaliies over he Se
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3
More informationLecture 2 October ε-approximation of 2-player zero-sum games
Opimizaion II Winer 009/10 Lecurer: Khaled Elbassioni Lecure Ocober 19 1 ε-approximaion of -player zero-sum games In his lecure we give a randomized ficiious play algorihm for obaining an approximae soluion
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationTHE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX
J Korean Mah Soc 45 008, No, pp 479 49 THE GENERALIZED PASCAL MATRIX VIA THE GENERALIZED FIBONACCI MATRIX AND THE GENERALIZED PELL MATRIX Gwang-yeon Lee and Seong-Hoon Cho Reprined from he Journal of he
More informationarxiv: v1 [math.oc] 27 Jul 2009
PARTICLE METHODS FOR STOCHASTIC OPTIMAL CONTROL PROBLEMS PIERRE CARPENTIER GUY COHEN AND ANES DALLAGI arxiv:0907.4663v1 [mah.oc] 27 Jul 2009 Absrac. When dealing wih numerical soluion of sochasic opimal
More informationZápadočeská Univerzita v Plzni, Czech Republic and Groupe ESIEE Paris, France
ADAPTIVE SIGNAL PROCESSING USING MAXIMUM ENTROPY ON THE MEAN METHOD AND MONTE CARLO ANALYSIS Pavla Holejšovsá, Ing. *), Z. Peroua, Ing. **), J.-F. Bercher, Prof. Assis. ***) Západočesá Univerzia v Plzni,
More informationScenario tree modelling for multistage stochastic programs
Mahemaical Programming manuscrip No. (will be insered by he edior) Holger Heisch Werner Römisch Scenario ree modelling for mulisage sochasic programs he dae of receip and accepance should be insered laer
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationA New Perturbative Approach in Nonlinear Singularity Analysis
Journal of Mahemaics and Saisics 7 (: 49-54, ISSN 549-644 Science Publicaions A New Perurbaive Approach in Nonlinear Singulariy Analysis Ta-Leung Yee Deparmen of Mahemaics and Informaion Technology, The
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationDistributionally Robust Stochastic Control with Conic Confidence Sets
Disribuionally Robus Sochasic Conrol wih Conic Confidence Ses Insoon Yang Absrac The heory of (sandard) sochasic opimal conrol is based on he assumpion ha he probabiliy disribuion of uncerain variables
More informationAsymptotic instability of nonlinear differential equations
Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp. 1 7. ISSN: 172-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) 147.26.13.11 or 129.12.3.113 Asympoic insabiliy
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationApplying Genetic Algorithms for Inventory Lot-Sizing Problem with Supplier Selection under Storage Capacity Constraints
IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 18 Applying Geneic Algorihms for Invenory Lo-Sizing Problem wih Supplier Selecion under Sorage Capaciy
More informationThe Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite
American Journal of Operaions Research, 08, 8, 8-9 hp://wwwscirporg/journal/ajor ISSN Online: 60-8849 ISSN Prin: 60-8830 The Opimal Sopping Time for Selling an Asse When I Is Uncerain Wheher he Price Process
More informationRandom Walk with Anti-Correlated Steps
Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More informationAn Introduction to Backward Stochastic Differential Equations (BSDEs) PIMS Summer School 2016 in Mathematical Finance.
1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca This inroducion is based on Touzi [14], Bouchard
More informationRecursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems
8 Froniers in Signal Processing, Vol. 1, No. 1, July 217 hps://dx.doi.org/1.2266/fsp.217.112 Recursive Leas-Squares Fixed-Inerval Smooher Using Covariance Informaion based on Innovaion Approach in Linear
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationRANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY
ECO 504 Spring 2006 Chris Sims RANDOM LAGRANGE MULTIPLIERS AND TRANSVERSALITY 1. INTRODUCTION Lagrange muliplier mehods are sandard fare in elemenary calculus courses, and hey play a cenral role in economic
More information1 Subdivide the optimization horizon [t 0,t f ] into n s 1 control stages,
Opimal Conrol Formulaion Opimal Conrol Lecures 19-2: Direc Soluion Mehods Benoî Chachua Deparmen of Chemical Engineering Spring 29 We are concerned wih numerical soluion procedures for
More informationOrdinary Differential Equations
Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described
More informationFractional Method of Characteristics for Fractional Partial Differential Equations
Fracional Mehod of Characerisics for Fracional Parial Differenial Equaions Guo-cheng Wu* Modern Teile Insiue, Donghua Universiy, 188 Yan-an ilu Road, Shanghai 51, PR China Absrac The mehod of characerisics
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationModal identification of structures from roving input data by means of maximum likelihood estimation of the state space model
Modal idenificaion of srucures from roving inpu daa by means of maximum likelihood esimaion of he sae space model J. Cara, J. Juan, E. Alarcón Absrac The usual way o perform a forced vibraion es is o fix
More informationStochastic models and their distributions
Sochasic models and heir disribuions Couning cusomers Suppose ha n cusomers arrive a a grocery a imes, say T 1,, T n, each of which akes any real number in he inerval (, ) equally likely The values T 1,,
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More informationA problem related to Bárány Grünbaum conjecture
Filoma 27:1 (2013), 109 113 DOI 10.2298/FIL1301109B Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma A problem relaed o Bárány Grünbaum
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationGMM - Generalized Method of Moments
GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................
More information1 Review of Zero-Sum Games
COS 5: heoreical Machine Learning Lecurer: Rob Schapire Lecure #23 Scribe: Eugene Brevdo April 30, 2008 Review of Zero-Sum Games Las ime we inroduced a mahemaical model for wo player zero-sum games. Any
More informationInventory Control of Perishable Items in a Two-Echelon Supply Chain
Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More information