Stability-based generation of scenario trees for multistage stochastic programs

Transcription

1 Stability-based generation of scenario trees for multistage stochastic programs H. Heitsch and W. Römisch Humboldt-University Berlin Institute of Mathematics Berlin, Germany Page 1 of 22 ISMP 2006, Rio de Janeiro, July 30 August 5, 2006

2 Multistage stochastic programs Let ξ = {ξ t } T t=1 be an IR d -valued discrete-time stochastic process defined on some probability space (Ω, F, IP ) and with ξ 1 deterministic. The stochastic decision x t at period t is assumed to be measurable with respect to the σ-field F t (ξ) := σ(ξ 1,..., ξ t ) (nonanticipativity). Multistage stochastic program: T min IE[ x t X t, b t (ξ t ), x t ] x t is F t (ξ) measurable, t = 1,..., T, t=1 A t,0 x t + A t,1 (ξ t )x t 1 = h t (ξ t ), t = 2,..., T where X t are nonempty and polyhedral sets, A t,0 are fixed recourse matrices and b t ( ), h t ( ) and A t,1 ( ) are affine functions depending on ξ t, where ξ varies in a polyhedral set Ξ. If the process {ξ t } T t=1 has a finite number of scenarios, they exhibit a scenario tree structure. Page 2 of 22

3 To have the multistage stochastic program well defined, we assume x t L r (Ω, F, IP ; IR m t ) and ξ t L r (Ω, F, IP ; IR d ), where r 1 and r := r r 1, if costs are random r, if only right-hand sides are random, if all technology matrices are random and r = T. Then nonanticipativity may be expressed as x N r (ξ) N r (ξ) = {x T t=1l r (Ω, F, IP ; IR m t ) : x t = IE[x t F t (ξ)], t}, i.e., as a subspace constraint, by using the conditional expectations IE[ F t (ξ)]. Page 3 of 22 For T = 2 we have N r (ξ) = IR m 1 L r (Ω, F, P ; IR m 2 ). infinite-dimensional optimization problem

4 Data process approximation by scenario trees The process {ξ t } T t=1 is approximated by a process forming a scenario tree being based on a finite set N IN of nodes. n = 1 t = 1 ξ n n n t = 2 N+ (n) N T t(n) T Page 4 of 22 Scenario tree with T = 5, N = 22 and 11 leaves n = 1 root node, n unique predecessor of node n, path(n) = {1,..., n, n}, t(n) := path(n), N + (n) set of successors to n, N T := {n N : N + (n) = } set of leaves, path(n), n N T, scenario with (given) probability π n, π n := ν N + (n) πν probability of node n, ξ n realization of ξ t(n).

5 Tree representation of the optimization model { } min π n b t(n) (ξ n ), x n xn X t(n), n N, A 1,0 x 1 = h 1 (ξ 1 ) A t(n),0 x n + A t(n),1 x n =h t(n) (ξ n ), n N n N How to solve the optimization model? - Standard software (e.g., CPLEX) - Decomposition methods for (very) large scale models (Ruszczynski/Shapiro (Eds.): Stochastic Programming, Handbook, 2003) Page 5 of 22 Questions: Under which conditions and in which sense do multistage models behave stable with respect to perturbations of ξ? Can such stability results be used to generate (multivariate) scenario trees?

6 Dynamic programming Theorem: (Evstigneev 76, Rockafellar/Wets 76) Under weak assumptions the multistage stochastic program is equivalent to the (first-stage) convex minimization problem { min f(x 1, ξ)p (dξ) : x 1 X 1 }, Ξ where f is an integrand on IR m 1 Ξ given by f(x 1, ξ):= b 1 (ξ 1 ), x 1 + Φ 2 (x 1, ξ 2 ), Φ t (x 1,..., x t 1, ξ t ):=inf{ b t (ξ t ), x t +IE [ Φ t+1 (x 1,..., x t, ξ t+1 ) F t ] : x t X t, A t,0 x t + A t,1 (ξ t )x t 1 = h t (ξ t )} for t = 2,..., T, where Φ T +1 (x 1,..., x T, ξ T +1 ) := 0. The integrand f depends on the probability measure IP and, thus, also on the probability distribution P = IP ξ 1 of ξ in a nonlinear way! Hence, earlier approaches to stability fail! Page 6 of 22

7 Quantitative Stability Let us introduce some notations. Let F denote the objective function defined on L r (Ω, F, IP ; IR s ) L r (Ω, F, IP ; IR m ) IR by F (ξ, x) := IE[ T t=1 b t(ξ t ), x t ], let X t (x t 1 ; ξ t ) := {x t X t A t,0 x t + A t,1 (ξ t )x t 1 = h t (ξ t )} denote the t-th feasibility set for every t = 2,..., T and X (ξ) := {x L r (Ω, F, IP ; IR m ) x 1 X 1, x t X t (x t 1 ; ξ t )} the set of feasible elements with input ξ. Then the multistage stochastic program may be rewritten as min{f (ξ, x) : x X (ξ) N r (ξ)}. Let v(ξ) denote its optimal value and, for any α 0, l α (F (ξ, )) := {x X (ξ) N r (ξ) : F (ξ, x) v(ξ) + α} S(ξ) := l 0 (F (ξ, )) denote the α-level set and the solution set of the stochastic program with input ξ. Page 7 of 22

8 The following conditions are imposed: (A1) ξ L r (Ω, F, IP ; IR s ) for some r 1. (A2) There exists a δ > 0 such that for any ξ L r (Ω, F, IP ; IR s ) with ξ ξ r δ, any t = 2,..., T and any x 1 X 1, x τ X τ (x τ 1 ; ξ τ ), τ = 2,..., t 1, the set X t (x t 1 ; ξ t ) is nonempty (relatively complete recourse locally around ξ). (A3) The optimal values v( ξ) are finite for all ξ L r (Ω, F, IP ; IR s ) with ξ ξ r δ and the objective function F is level-bounded locally uniformly at ξ, i.e., for some α > 0 there exists a δ > 0 and a bounded subset B of L r (Ω, F, IP ; IR m ) such that l α (F ( ξ, )) is nonempty and contained in B for all ξ Lr (Ω, F, IP ; IR s ) with ξ ξ r δ. Page 8 of 22 Norm in L r : ξ r := ( T IE[ ξ t r ]) 1 r t=1

9 Theorem: (Heitsch/Römisch/Strugarek, SIAM J. Opt. 2006) Let (A1), (A2) and (A3) be satisfied, r > 1 and X 1 be bounded. Then there exist positive constants L and δ such that v(ξ) v( ξ) L( ξ ξ r + D f (ξ, ξ)) holds for all ξ L r (Ω, F, IP ; IR s ) with ξ ξ r δ. Assume that technology matrices are non-random and the solution x of the original problem is unique. If (ξ (n) ) is a sequence in T t=1l r (Ω, F t (ξ), IP ; IR s ) such that ξ (n) ξ r and D f (ξ (n), ξ) converge to 0 and if (x (n) ) is a sequence of solutions of the approximate problems, then the sequence (x (n) ) converges to x with respect to the weak topology in L r. Page 9 of 22 Here, D f (ξ, ξ) denotes the filtration distance of ξ and ξ defined by D f (ξ, ξ)= T 1 inf max{ x t IE[x t F t ( ξ)] r, x t IE[ x t F t (ξ)] r }. x S(ξ) x S( ξ) t=2

10 Remark: The continuity property of infima in the Theorem can be supplemented by a quantitative stability property of the set S 1 (ξ) of first stage solutions. Namely, there exists a constant ˆL > 0 such that sup x S 1 ( ξ) d(x, S 1 (ξ)) Ψ 1 ξ ( ˆL( ξ ξ r + D f (ξ, ξ))), where Ψ ξ (τ) := inf {IE[f(x 1, ξ)] v(ξ) : d(x 1, S 1 (ξ)) τ, x 1 X 1 } with Ψ 1 ξ (α) := sup{τ IR + : Ψ ξ (τ) α} (α IR + ) is the growth function of the original problem near its solution set S 1 (ξ). Page 10 of 22 Remark: Simple examples show that the filtration distance is indispensable for the stability result to hold.

11 Generation of scenario trees (i) In most practical situations scenarios ξ i with known probabilities p i, i = 1,..., N, can be generated, e.g., simulation scenarios from (parametric or nonparametric) statistical models of ξ or (nearly) optimal quantizations of the probability distribution of ξ. (ii) Construction of a scenario tree out of the scenarios ξ i with probabilities p i, i = 1,..., N,. Page 11 of 22

12 Approaches for (ii): (1) Bound-based approximation methods, (Frauendorfer 96, Kuhn 05, Edirisinghe 99, Casey/Sen 05). (2) Monte Carlo-based schemes (inside or outside decomposition methods) (e.g. Shapiro 03, 06, Higle/Rayco/Sen 01, Chiralaksanakul/Morton 04). (3) the use of Quasi Monte Carlo integration quadratures (Pennanen 05, 06). (4) EVPI-based sampling schemes (inside decomposition schemes) (Corvera Poire 95, Dempster 04). (5) Moment-matching principle (Høyland/Wallace 01, Høyland/Kaut/Wallace 03). (6) (Nearly) best approximations based on probability metrics (Pflug 01, Hochreiter/Pflug 02, Mirkov/Pflug 06; Gröwe-Kuska/Heitsch/Römisch 01, 03, Heitsch/Römisch 05). Page 12 of 22 Survey: Dupačová/Consigli/Wallace 00

13 Constructing scenario trees Let ξ be the original stochastic process on some probability space (Ω, F, IP ) with parameter set {1,..., T } and state space IR d. We aim at generating a scenario tree ξ tr such that and, thus, ξ ξ tr r and D f (ξ, ξ tr ) v(ξ) v(ξ tr ) are small. To determine such a scenario tree, we start with a discrete approximation ξ f consisting of scenarios ξ i = (ξ1, i..., ξt i ) with probabilities p i, i = 1,..., N. ξ f is a fan of individual scenarios. Page 13 of 22 t = 1 t = 2 t = 3 t = 4 t = 5

14 The fan ξ f is chosen such that it is adapted to the filtration (F t (ξ)) T t=1 and ξ ξ f r ε appr. Algorithms are developed that generate a scenario tree ξ tr by deleting and bundling scenarios of ξ f (that are similar at t) such that it is also adapted to the filtration (F t (ξ)) T t=1 and satisfies Since it holds (1) ξ f ξ tr r ε r T 1 (2) inf x t IE[x t F t (ξ tr )] r ε f. x S(ξ f ) t=2 D f (ξ, ξ tr ) ε appr + T 1 inf x t IE[x t F t (ξ tr )] r, x S(ξ f ) t=2 if ξ f is sufficiently close to ξ, we obtain in case ε appr + ε r δ that v(ξ) v(ξ tr ) L(2ε appr + ε r + ε f ). Page 14 of 22

15 (1) Forward tree generation Let scenarios ξ i with probabilities p i, i = 1,..., N, fixed root ξ1 IR d, r 1, and tolerances ε r, ε t, t = 2,..., T, be given such that T ε t ε r. t=2 Step 1: Set ˆξ 1 := ξ f and C 1 = {I = {1,..., N}}. Step t: Let C t 1 = {Ct 1, 1..., C K t 1 t 1 }. Determine disjoint index sets It k and Jt k of remaining and deleted scenarios such that It k Jt k = Ct 1, k a mapping α t : I I { i k α t (j) = t (j), j Jt k, k = 1,..., K t 1, j, otherwise, where i k t (j) I k t such that i k t (j) arg min i I k t ˆξ t 1,i ˆξ t 1,j t, Page 15 of 22

16 a stochastic process ˆξ t ˆξ t,i τ = { ξ α τ (i) τ, τ t,, otherwise, ξ i τ such that ˆξ t ˆξ t 1 r,t ε t. Set I t := K t 1 k=1 Ik t and C t := {α 1 t (i) : i I k t, k = 1,..., K t 1 }. Step T+1: Let C T = {CT 1,..., CK T T }. Construct a stochastic process ξ tr having K T scenarios ξtr k such that ξtr,t k := ξ α t(i) t with probabilities πt i = p j if i CT k, k = 1,..., K T, t = 2,..., T. Proposition: j C k T ξ f ξ tr r T ε t ε r. t=2 Page 16 of 22

17 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 Page 17 of 22 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 Illustration of the forward tree construction for an example including T=5 time periods starting with a scenario fan containing N=58 scenarios <Start Animation>

18 (2) Bounding approximate filtration distances T 1 Aim: (ξ f, ξ tr ) := inf x t IE[x t F t (ξ tr )] r ε f x S(ξ f ) t=2 Two possibilities: (i) Estimates in terms of some solutions with input ξ f, which would require to solve a two-stage model. (ii) Estimates in terms of the input ξ f. Proposition: Let (A2) and (A3) be satisfied and X 1 be bounded. Assume that the technology matrices A t,1 are non-random, 1 r < and ξ f is sufficiently close to ξ. Then there exists a constant ˆL > 0 such that Condition: (ξ f, ξ tr ) ˆL ( i I 2 i I 2 j I 2,i p j ξ j ξ i r j I 2,i p j ξ j ξ i r ε r f ) 1 r Page 18 of 22

19 Numerical experience We consider the electricity portfolio management of a municipal power company. Data was available on the electrical load demand and on electricity prices at the market place EEX. A multivariate statistical model is developed for the yearly demandprice process ξ that allowed to generate yearly demand-price scenarios ξ i, with probabilities p i = 1 N, i = 1,..., N. These scenarios are assumed to form the process ξ f. Branching in ξ tr was allowed at most monthly. The tolerances ε t at branching points were chosen such that Page 19 of 22 ε t = ε T [1 + q(1 2 t )], t = 2,..., T, T where the parameter q [0, 1] affects the branching structure of the constructed trees. For the test runs we used q = 0.6. The test runs were performed on a PC with a 3 GHz Intel Pentium CPU and 1 GByte main memory.

20 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec a) Forward tree construction with relative filtration tolerance ε rel,f = 0.35 Page 20 of 22 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec b) Forward tree construction with relative filtration tolerance ε rel,f = 0.45 Yearly demand-price scenario trees with relative tolerance ε rel,r = 0.25

21 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec a) Forward tree construction with relative filtration tolerance ε rel,f = 0.6 Page 21 of 22 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec b) Forward tree construction with relative filtration tolerance ε rel,f = 0.7 Yearly demand-price scenario trees with relative tolerance ε rel,r = 0.5

22 ε rel,r ε rel,f Scenarios Nodes Stages Time (sec) Numerical results for yearly demand-price scenario trees Page 22 of 22