This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

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1 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 wih colleagues. Oher uses, including reproducion and disribuion, or selling or licensing copies, or posing o personal, insiuional or hird pary websies are prohibied. In mos cases auhors are permied o pos heir version of he aricle (e.g. in Word or ex form) o heir personal websie or insiuional reposiory. Auhors requiring furher informaion regarding Elsevier s archiving and manuscrip policies are encouraged o visi: hp://

2 Auhor's personal copy Operaions Research Leers 40 (202) Conens liss available a SciVerse ScienceDirec Operaions Research Leers journal homepage: SDDP for mulisage sochasic linear programs based on specral risk measures Vincen Guigues a,b, Werner Römisch c, a IMPA, Insiuo de Maemáica Pura e Aplicada, 0 Esrada Dona Casorina, Jardim Boanico, Rio de Janeiro, Brazil b UFRJ, Escola Poliécnica, Deparameno de Engenharia Indusrial, Ilha do Fundão, C, Bloco F, Rio de Janeiro, Brazil c Humbold-Universiy Berlin, Insiue of Mahemaics, 0099 Berlin, Germany a r i c l e i n f o a b s r a c Aricle hisory: Received 8 February 202 Received in revised form 29 March 202 Acceped 0 April 202 Available online 5 May 202 Keywords: Specral risk measure Sochasic programming Risk-averse opimizaion Decomposiion algorihms Mone Carlo sampling 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. his 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 cu coefficiens have analyic formulas. 202 Elsevier B.V. All righs reserved.. 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. he 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 [5,8] for overviews and [3, Par III] for Corresponding auhor. el.: addresses: vguigues@impa.br (V. Guigues), romisch@mah.hu-berlin.de (W. Römisch). 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 such as hose in [,0,2,2] provide approaches and condiions implying he 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. he 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 [9] 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 [7]), and Sochasic Dual Dynamic Programming (SDDP), iniiaed in [3], revisied in [6,22], and also sudied in he presen paper. We consider risk-averse formulaions of mulisage sochasic linear programs of he form inf d x + θ E d x + θ ρ φ d k x k x,...,x C x = ξ D x, =2 =2 x 0, x is F -measurable, =,...,, k=2 () /$ see fron maer 202 Elsevier B.V. All righs reserved. doi:0.06/j.orl

3 Auhor's personal copy 34 V. Guigues, W. Römisch / Operaions Research Leers 40 (202) where x 0 is given, parameers d, C, D are deerminisic, (ξ ) = is a sochasic process, F is he sigma-algebra F := σ (ξ j, j ), (θ ) = are nonnegaive weighs summing o, and ρ φ is a specral risk measure [] or disorion risk measure [4,5] depending on a risk specrum φ L ([0, ]). 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 (), which means ha, for any feasible sequence of decisions (x,..., x ) o any -sage scenario (ξ, ξ 2,..., ξ ), here exiss a sequence of feasible decisions (x +,..., x ) wih probabiliy. A non-risk-averse model amouns o aking θ = and θ = 0 for = 2,...,. 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. he main conribuion of his paper is o provide analyic formulas for some cu coefficiens, independen of he sampled scenarios, ha can be useful for implemenaion. We also specialize he SDDP algorihm and especially he compuaion of he cus for he paricular riskaverse model (). We sar by seing down some noaion. e will denoe a column vecor of all s; for x, y R n, he vecor x y R n is defined by (x y)(i) = x(i)y(i), i =,..., n; for x R n, he vecor x + R n is defined by x + (i) = max(x(i), 0), i =,..., n; he available hisory of he process a sage is denoed by ξ [] := (ξ j, j ); for vecors x,..., x n, he noaion x n :n 2 sands for he concaenaion (x n, x n +,..., x n2 ) for n n 2 n; δ ij is he Kronecker dela defined for i, j inegers by δ ij = 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 ([0, ]), he specral risk measure ρ φ generaed by φ is (see []): ρ φ (Z) = F Z 0 (p)φ(p)dp. Specral risk measures have been used in various applicaions (porfolio selecion by Acerbi and Simonei [2]; insurance by Coer and Dowd [4]). he condiional value-a-risk (CVaR) of level 0 < ε <, denoed by CVaR ε, is a paricular specral risk measure obained aking φ(u) = ε 0 u<ε (see Acerbi []). In wha follows, we consider more generally a piecewise consan risk funcion φ( ) wih J jumps a 0 < p < p 2 < < p J <. We se φ k = φ(p + k ) φ(p k ) = φ(p k) φ(p k ), for k =,..., J, wih p 0 = 0, and we assume ha (i) φ( ) is posiive, (ii) φ k < 0, k =,..., J, (iii) 0 φ(u)du =. 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; see Acerbi and Simonei [2]: ρ φ (Z) = inf w R J J φ k [p k w k E[w k Z] + ] φ()e[z]. (2) k= Using his formulaion for ρ φ, dynamic programming equaions are given in [8] for risk-averse formulaion (). More precisely, problem () can be expressed as inf d x + x, w 2: θ c w + Q 2 (x, ξ [], z, w 2,..., w ), =2 C x = ξ D x 0, x 0, w R J, = 2,...,, wih z = 0, vecor c = φ p, and where, for = 2,...,, Q (x, ξ [], z, w : ) inf f (z, w ) + Q + (x, ξ [], z, w +: ) = E x ξ ξ,z [], z = z d x (4), C x = ξ D x, x 0 wih f (z, w ) = (δ θ + φ()θ )z θ φ (w z e) +, (5) and Q + 0. Funcion Q + represens a sage a cos-ogo or recourse funcion which is risk averse. As shown in he nex secion, i can be approximaed by cuing planes by some polyhedral funcion Q +. hese approximae recourse funcions are useful for defining a feasible approximae policy obained by solving inf f (z, w ) + Q + (x, ξ [], z, w +: ) x,z C x = ξ D x, x 0, z = z d x, a sage = 2,...,, knowing x, z, firs-sage decision variables w :, and ξ. Firs-sage decision variables x and w 2: are he soluion o (3) wih Q 2 replaced by he approximaion Q Algorihmic issues he DP 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. (A) he suppors of he disribuions of ξ,..., ξ are discree and finie. (A2) Process (ξ ) is inersage independen. (A3) For =,...,, for any feasible x, and for any realizaion ξ of ξ, he se {x : x 0, C x = ξ D x } is bounded and nonempy. In wha follows, we assume ha Assumpions (A) (A3) hold. In paricular, we denoe he realizaions of ξ by ξ i, i =,..., q < +, and se p(, i) = P(ξ = ξ i ). Since he suppors of he disribuions of he random vecors ξ 2,..., ξ are discree and finie, opimizaion problem () 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 ha can happen in he fuure for ξ 2,..., ξ. he roo node of he scenario ree corresponds o he firs ime sep wih ξ deerminisic. For a given sage, o each node of he scenario ree here corresponds an hisory ξ []. he children nodes of a node a sage are he nodes ha can happen a sage + if we are a his node a. A sampled scenario (ξ,..., ξ ) corresponds o a paricular succession of nodes such ha ξ is a possible value for he process a and ξ + is a child of ξ. A given node in he ree a sage is idenified wih a scenario (ξ,..., 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,..., +. Each (3) (6)

4 Auhor's personal copy V. Guigues, W. Römisch / Operaions Research Leers 40 (202) 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: Q i (x, z, w : ) = max j=0,,...,ih + + τ= E j x Z j z W j,τ w +τ + e j, (7) knowing ha he algorihm sars aking for Q 0 a known lower bounding affine approximaion of Q while Q i + 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,..., +. A ieraion i, cus for Q, = 2,...,, are buil a some poins x k, zk, wi :, k = (i )H +,..., ih, compued in he forward pass replacing he recourse funcions Q + by Q i + (noe ha, since variables w 2: are firs-sage decision variables, hey jus depend on he ieraion). More precisely, he cus are compued for ime sep + down o ime sep 2. For ime sep +, since Q i = Q + + = 0, he cus for Q + are obained aking null vecors for E k, Z k,, and e k for k = (i )H +,..., ih. For = 2,...,, using he lower bounding approximaion Q i + of Q +, we can bound from below Q (x, z, w : ) by E ξ [Q i (x, z, w :, ξ )] wih Q i (x, z, w :, ξ ) given as he opimal value of he following linear program: inf x,z,v, θ (δ θ + φ()θ )z θ φ v + θ v 0, v w z e, x 0, z + d x = z C x = ξ D x τ= (8a) (8b) E i x + Z i z i,τ + θ e W w +τ + e i, (8c) where E i (respecively, Z i, W i,τ, and e i ) is he marix whose (j + )h line is E j (respecively, Z, j W j,τ, and e j ) for j = 0,..., ih. In he backward pass of ieraion i, he above problem is solved wih (x, z, w :, ξ ) respecively replaced by (x k, zk, wi, ξ j : ) for k = (i )H +,..., ih and j =,..., 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, (8a), (8b) and (8c) for he problem defining Q i (xk, zk, wi, ξ j : ) for k = (i )H +,..., ih and j =,..., q. he following proposiion provides he cus compued for Q, = 2,...,, a ieraion i. Proposiion 3. (Opimaliy Cus). Le Q, = 2,..., +, 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 = +, E k, Z k,, and are null for k = (i )H +,..., ih. For = 2,..., and e k k = (i )H +,..., ih, E k is given by q Z k = q = q p(, j)µ k,j, W k, = q k,j p(, j)π D, and p(, j)σ k,j, (9) p(, j)ρ k,j i,τ W, τ = 2,..., +. (0) Furher, e k is given by q p(, j) Q i (xk, zk, wi, ξ j : ) µ k,j σ k,j w i τ= ρ k,j W i,τ z k w+τ i + π k,j D x k Proof. Since a dual soluion of he problem defining Q i (xk, z k, wi, ξ j : ) is a subgradien of he value funcion for problem (8), we obain ha Q i (x, z, w :, ξ) j is bounded from below by Q i (xk, zk, wi, ξ j : ) + µ k,j (z z k ) + σ k,j (w w i ) + + τ=2 ρ k,j i,τ W (w +τ w i ) +τ π k,j D (x x k ). Using he above lower bound and he fac ha Q (x, z, w : ) q is bounded from below by i p(, j)q (x, z, w :, ξ), j we obain he announced cus. he sopping crierion is discussed in [22] for a non-risk-averse model. he definiion of a sound sopping crierion for he riskaverse model from [22] (based on a nesed formulaion of he problem defined in erms of condiional risk mappings) is a more delicae issue, and is sill open for discussion. However, since problem () can be expressed as a non-risk-averse problem wih modified objecive, variables, and consrains, in our riskaverse 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 by replacing he 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 (). In he forward pass of ieraion i, we can compue he oal cos C k on each scenario k = (i )H +,..., ih: C k = d xk + θ c wi + =2 =2 f (z k, wi ). () If hese H scenarios were represening all possible evoluions of (ξ,..., ξ ), hen ih C = H k=(i )H+ C k would be an upper bound on he opimal value of () (recall ha he approximae policy is feasible and ha he objecive funcion of () 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,..., C H ), σ = H ih k=(i )H+ ( C C k ) 2, we can compue he ( α)-confidence upper bound σ C + α,h H (2) on he approximae policy mean value, where α,h is he ( α)-quanile of he Suden -disribuion wih H degrees of freedom. Since he opimal value of () is less han or equal o he.

5 Auhor's personal copy 36 V. Guigues, W. Römisch / Operaions Research Leers 40 (202) Fig.. SDDP algorihm wih relaively complee recourse for he risk-averse inersage independen sochasic linear program (). approximae policy mean value, (2) gives an upper bound for he opimal value of () wih confidence a leas α. Consequenly, σ we can sop he algorihm when C + α,h H z inf ε for some ε > 0. Using he previous developmens, he SDDP algorihm for solving () can be formulaed as in Fig.. We now give, for some paricular choices of he firs-sage variables w 2:, he exac expressions (independen of he sampled scenarios) of Z k k,τ and W for every = 2,...,, k =,..., H, and τ =,..., +. hough he firs-sage feasible se for (3) is no bounded, i can be easily shown ha he opimal values of w 2: are bounded (see [8], for insance). As a resul, well-chosen box consrains on w, = 2,..., 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,...,, where w are sae variables). Le us define, for =,...,, x = (x,..., x ), ξ = (ξ,..., 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, τ =,..., }.

6 Auhor's personal copy V. Guigues, W. Römisch / Operaions Research Leers 40 (202) Since (A3) holds, he ses χ are compac, and, since g (x ) = τ=2 d τ x τ is coninuous, we can inroduce he pairs (C u, C l ) R2 defined by C u max g = (x ) x χ, C l = min g (x ) x χ. he 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 xk, zk, w, = 2,..., (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 w 2: such ha (i) w > C l e for = 2,..., (respecively, such ha (ii) w < C u e for = 2,..., ), hen Z k k,τ and W, for k =,..., H, can be compued using Proposiion 3.2(i) (respecively, Proposiion 3.2(ii)), which follows. For insance, if he coss are posiive, hen iem (i) is fulfilled wih w = 0 and iem (ii) by aking for each componen of w 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.. Moreover, in he following wo cases, we have closed-form expressions for Z k k,τ and W (independen of he sampled scenarios). (i) If, for = 2,...,, w > C l e, hen, for = 2,...,, P () k =,..., H, Z k = θ + φ(0) θ l, P () : l= k =,..., H, = θ +τ φ, τ =,..., +. (ii) If, for = 2,...,, w k =,..., H, P () : k =,..., H, τ =,..., +, < C u e, hen, for = 2,...,, P () Z k = θ + φ() = 0. θ l, Proof. Le us fix {2,..., }, k {,..., H}, and j {,..., q }. We denoe by x, z, v, θ an opimal soluion o he problem defining Q (xk, zk, w, ξ j : ), i.e., problem (8) wrien for i =, and wih (x, z, w :, ξ ) replaced by (x k, zk, wi, ξ j : ) (he dependence of he soluion wih respec o k, j is suppressed, o alleviae noaion). he Karush Kuhn ucker (KK) condiions for his problem imply ha δ θ φ()θ µ k,j θ φ σ k,j σ k,j l= e ρ k,j Z = 0, (3) σ k,j = 0, (4) σ k,j ( z e + w v ) = 0, (5) σ k,j v = 0, (6) where, for =, 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. Furher, v = max(0, w z e) = w z e > 0. Using (4) and (6), we hen ge σ k,j = 0 and σ k,j = θ φ. (7) Le us now firs show (i) by backward inducion on. Plugging he value of σ k,j given in (7) ino (3), we obain µ k,j = θ φ()θ + θ e φ J = θ + θ φ() + [φ(p l ) φ(p l )] = θ θ φ(0). l= Using he above relaion and (9) yields Z k = q p(, j)µk,j = θ φ(0) + θ. Furher, using once again (9), we obain W k, = q = p(, j)σ k,j q p(, j)θ φ = θ φ. (8) his shows P (). Le us now assume ha P ( + ) holds for some {2,..., }, and le us show ha P () holds. Firs, noice ha (8) sill holds wih subsiued wih, i.e., W k, = θ φ. Furher, for τ = 2,..., +, = Also, q q = p(, j)ρ k,j,τ W, from (0), p(, j)ρ k,j θ +τ eφ, using P ( + ), q = p(, j)θ +τ φ = θ +τ φ. Z k = q q = p(, j)µ k,j, from (9), since ρ k,j e =, p(, j)( φ()θ + θ φ e ρ k,j Z ), using (3) and (7), q = p(, j)( φ(0)θ ρ k,j Z ), using he definiion of φ, q = φ(0)θ + p(, j)ρ k,j θ + φ(0) θ l e, l=+ using P ( + ), = θ + φ(0) θ l since ρ k,j e =. l= We have hus shown P () which achieves he proof of (i). Le us now assume ha w < C u e for = 2,...,, and le us show (ii). Le us fix {2,..., }, k {,..., H}, and j {,..., q }. As before, we denoe by x, z, v, θ an opimal soluion o he problem defining Q (xk, zk, w, ξ j : ). In his case, z e C u e > w and v = max(0, w z e) = 0. Using

7 Auhor's personal copy 38 V. Guigues, W. Römisch / Operaions Research Leers 40 (202) (4) and (5), we see ha σ k,j = θ φ and σ k,j = 0. (9) Using (9), we ge W k, = 0. We show (ii) by backward inducion. For =, plugging he value of σ k,j ino (3) gives µ k,j = θ φ()θ, which, ogeher wih (9), gives Z k = θ + φ()θ. We have already proved ha W k, = 0, and hus P () holds. Le us now assume ha P ( + ) holds for some {2,..., }, and le us show ha P () holds. Since W,τ = 0, we obain = q,τ p(, j)ρk,j W = 0 for τ = 2,..., +. Plugging σ k,j = 0 ino (3) and using (9) gives Z k = = q q p(, j)(φ()θ + ρ k,j Z ), p(, j) θ + φ() θ l, l= using P ( + ) and ρ k,j e =, = θ + φ() θ l. l= his shows P () and achieves he proof of (ii). Proposiion 3.2 can be used as a debugging ool o check he implemenaion of he SDDP algorihm for risk-averse problem (). More precisely, we can check ha, in cases (i) and (ii), implemening he formulas for Z k Proposiion 3. will give he same resuls as implemening he formulas from Proposiion 3.2. A sage, if insead of ρ φ in () we use CVaR ε, problem () becomes inf d x + θ E x,...,x C x = ξ D x, d x + =2 θ CVaR ε =2 x 0, x is F -measurable, =,...,. k,τ and W given in d k x k For his model, we obain a resul analogous o Proposiion 3.2. k=2 (20) Proposiion 3.3. Le us consider he risk-averse recourse funcions Q for model (20) and heir approximaions Q i of form (7), obained by applying he SDDP algorihm o he corresponding DP equaions. In he following wo cases, we obain closed-form expressions for Z k and (independen of he sampled scenarios). > C l, hen, for = 2,...,, P () (i) If, for = 2,...,, w k =,..., H, Z k = θ + l= P () : k =,..., H, = θ +τ. ε +τ τ =,..., +, (ii) If, for = 2,...,, w θ l ε l, < C u, hen, for = 2,...,, P () P () : k =,..., H, Z k = θ, and τ =,..., +, = 0. Proof. he proof is similar o he proof of Proposiion 3.2. Remark 3.4. In he paricular case when he CVaR levels ε = ε (0, ) are he same a each ime sep, Proposiion 3.3 is a paricular case of Proposiion 3.2 wih φ() = 0, φ(0) = ε, and φ = /ε R. Numerical simulaions for a real-life applicaion modeled as (20) are repored in [7]. When Assumpion (A) does no hold, as saed in [22], a feasible nonanicipaive policy can sill be proposed using approximae recourse funcions Q obained applying he SDDP algorihm on a sample average approximaion (SAA) of he original problem (). References [] C. Acerbi, Specral measures of risk: a coheren represenaion of subjecive risk aversion, Journal of Banking and Finance 7 (2002) [2] C. Acerbi, P. Simonei, Porfolio opimizaion wih specral measures of risk, Abaxbank inernal repor, Available a hp:// [3] M. Berocchi, G. Consigli, M.A.H. Dempser (Eds.), Sochasic Opimizaion Mehods in Finance and Energy, Springer, New York, 20. [4] J. Coer, K. 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