SDDP FOR MULTISTAGE STOCHASTIC LINEAR PROGRAMS BASED ON SPECTRAL RISK MEASURES

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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