Stability of Stochastic Programming Problems

Transcription

1 Stability of Stochastic Programming Problems W. Römisch Humboldt-University Berlin Institute of Mathematics Berlin, Germany Page 1 of 35 Spring School Stochastic Programming, Bergamo, April 13, 2007

2 : 1. Introduction 2. General quantitative stability results 3. Two-stage stochastic programs Stability with respect to Fortet-Mourier metrics Consequences for discrete approximations Consequences for scenario reduction 4. Two-stage mixed-integer stochastic programs 5. Chance constrained stochastic programs 6. Multi-stage stochastic programs Page 2 of 35

3 1. Introduction Consider the stochastic programming model { } min f 0 (ξ, x)p (dξ) : x M(P ) M(P ) := { x X : } f j (ξ, x)p (dξ) 0, j = 1,..., d where f j from R m to the extended reals R are normal integrands, X is a nonempty closed subset of R m, is a closed subset of R s and P is a Borel probability measure on. (f is a normal integrand if it is Borel measurable and f(ξ,.) is lower semicontinuous ξ.) Let P() the set of all Borel probability measures on and by v(p ) = inf f 0 (ξ, x)p (dξ) (optimal value) x M(P ) { } S ε (P ) = x M(P ) : f 0 (ξ, x)p (dξ) v(p ) + ε S(P ) = S 0 (P ) = arg min f 0 (ξ, x)p (dξ) (solution set). x M(P ) Page 3 of 35

4 The underlying probability distribution P is often incompletely known in applied models and/or has to be approximated (estimated, discretized). stability behaviour of stochastic programs becomes important when changing (perturbing, estimating, approximating) P P(). Here, stability refers to (quantitative) continuity properties of the optimal value function v(.) and of the set-valued mapping S ε (.) at P, where both are regarded as mappings given on certain subset of P() equipped with some convergence of probability measures and some probability metric, respectively. (The corresponding subset of probability measures is determined such that certain moment conditions are satisfied that are related to growth properties of the integrands f j with respect to ξ.) Page 4 of 35 Examples: Two-stage stochastic programs, chance constrained stochastic programs. Survey: W. Römisch: Stability of stochastic programming problems, in: Stochastic Programming (A. Ruszczynski, A. Shapiro eds.), Handbook, Elsevier, 2003.

5 Weak convergence in P() P n w P iff f(ξ)p n (dξ) f(ξ)p (dξ) ( f C b ()), iff P n ({ξ z}) P ({ξ z}) at continuity points z of P ({ξ }). Probability metrics on P() (Monographs: Rachev 91, Rachev/Rüschendorf 98) Metrics with ζ-structure: { } d F (P, Q) = sup f(ξ)p (dξ) f(ξ)q(dξ) : f F where F is a suitable set of measurable functions from to R and P, Q are probability measures in some set P F on which d F is finite. Page 5 of 35 Examples (of F): Sets of locally Lipschitzian functions on or of piecewise (locally) Lipschitzian functions. There exist canonical sets F and metrics d F for each specific class of stochastic programs!

6 2. General quantitative stability results To simplify matters, let X be compact (otherwise, consider localizations). F := {f j (., x) : x X, j = 0,..., d}, P F := {Q P() : inf f j(ξ, x)q(dξ) >, x X f j (ξ, x)q(dξ) <, j = 0,..., d}, sup x X and the probability (semi-) metric on P F : d F (P, Q) = sup max x X j=0,...,d f j (ξ, x)(p Q)(dξ). Lemma: The functions (x, Q) on X P F. f j (ξ, x)q(dξ) are lower semicontinuous Page 6 of 35

7 Theorem: (Rachev-Römisch 02) If d 1, let the function x f 0(ξ, x)p (dξ) be Lipschitz continuous on X, and, let the function (x, y) d ( x, { x X : }) f j (ξ, x)p (dξ) y j, j = 1,..., d be locally Lipschitz continuous around ( x, 0) for every x S(P ) (regularity condition). Then there exist constants L, δ > 0 such that v(p ) v(q) Ld F (P, Q) S(Q) S(P ) + Ψ P (Ld F (P, Q))B holds for all Q P F with d F (P, Q) < δ. Page 7 of 35 Here Ψ P (η) := η + ψ 1 (η) and ψ : R + R + { } ψ(τ):=min f 0 (ξ, x)p (dξ) v(p ):d(x, S(P )) τ, x M(P ). (Proof by appealing to general perturbation results of Klatte 94 and Rockafellar/Wets 98.)

8 Convex case and d := 0: Assume that f 0 (ξ, ) is convex on R m ξ. Theorem: (Römisch-Wets 06) Then there exist constants L, ε > 0 such that dl (S ε (P ), S ε (Q)) L ε d F(P, Q) for every ε (0, ε) and Q P F such that d F (P, Q) < ε. Here, dl is the Pompeiu-Hausdorff distance of nonempty closed subsets of R m, i.e., dl (C, D) = inf{η 0 : C D + ηb, D C + ηb}. Page 8 of 35 Proof using a perturbation result by Rockafellar/Wets 98.

9 The (semi-) distance d F plays the role of a minimal probability metric implying quantitative stability. Furthermore, the result remains valid when bounding d F from above by another distance and when reducing the set P F to a subset on which this distance is defined and finite. Idea: Enlarge F, but maintain the analytical (e.g., (dis)continuity) properties of f j (, x), j = 0,..., d! This idea may lead to well-known probability metrics, for which a well developed theory is available! Page 9 of 35

10 Example: (Fortet-Mourier-type metrics) We consider the following classes of locally Lipschitz continuous functions (on ) F H := {f : R : f(ξ) f( ξ) max{1, H( ξ ), H( ξ )} ξ ξ, ξ, ξ } are of particular interest, where H : R + R + is nondecreasing, H(0) = 0. The corresponding distances are d FH (P, Q) = sup f(ξ)p (dξ) f(ξ)q(dξ) =: ζ H(P, Q) f F H are so-called Fortet-Mourier-type metrics defined on P H ():={Q P(): max{1, H( ξ )} ξ Q(dξ) < } Important special case: H(t) := t p 1 for p 1. The corresponding classes of functions and measures, and the distances are denoted by F p, P p () and ζ p, respectively. (Convergence with respect to ζ p means weak conergence of the probability measures and convergence of the p-th order moments (Rachev 91)) Page 10 of 35

11 Application: Convergence of empirical estimates Let P P() and let ξ 1, ξ 2,..., ξ n,... be independent, identically distributed -valued random variables on some probability space (Ω, A, P) having the common distribution P. Let P n (ω) := 1 n n i=1 δ ξ i (ω) be the empirical measures n N. We consider the empirical estimates or sample average approximation (replacing P by P n ( )): { } 1 n min f 0 (ξ i, x) : x X, 1 n f j (ξ i, x) 0, j = 1,..., d n n i=1 i=1 Then results on the convergence in probability of and, hence, of d F (P, P n ( )) v(p ) v(p n ( )) may be obtained using the general stability results, empirical process theory and covering numbers for F as subsets of L p (, P ). Page 11 of 35

12 3. Two-stage stochastic programming models We consider the two-stage stochastic program { } min c, x + ˆΦ(ξ, x)p (dξ) : x X, where ˆΦ(ξ, x) := inf{ q(ξ), y : y Y, W (ξ)y = h(ξ) T (ξ)x} P := Pξ 1 P() is the probability distribution of the random vector ξ, c R m, X R m is a bounded polyhedron, q(ξ) R m, Y R m is a polyhedral cone, W (ξ) a r m-matrix, h(ξ) R r and T (ξ) a r m-matrix. Page 12 of 35 We assume that q(ξ), h(ξ), W (ξ) and T (ξ) are affine functions of ξ (e.g., some of their components or elements are random).

13 Example:(two-stage model with simple recourse) m = s = 1, d = 0, f 0 (ξ, x) := max{0, ξ x}, := R, X := [ 1, 1], P := δ 0 (unit mass at 0), P n := (1 1 n )δ n δ n 2, n N. { x, x [ 1, 0) f 0(ξ, x)p (dξ) = 0, x [0, 1] v(p ) = 0, S(P ) = [0, 1], f 0(ξ, x)p n (dξ) = (1 1 n ) max{0, x} + 1 n max{0, n2 x} v(p n ) = n 1 n, S(P n) = {1} (n N). Note: P n w P, but first order moments do not converge! f(ξ)p n(dξ) = (1 1 n )f(0) + 1 n f(n2 ) f(0), f C b (). But, ξ P n(dξ) = 1 n n2 = n (n N) and ξ P (dξ) = 0 Page 13 of 35

14 Structural properties of two-stage models We consider the infimum function of the parametrized linear (secondstage) program and the dual feasible set of the second-stage program, namely, Φ(ξ, u, t):=inf{ u, y :W (ξ)y = t, y Y } ((ξ, u, t) R m R r ) D(ξ) := {z R r :W (ξ) z q(ξ) Y } (ξ ), where W (ξ) is the transposed of W (ξ) and Y the polar cone of Y (i.e., Y = {y : y, y 0, y Y }). Then we have ˆΦ(ξ, x)=φ(ξ, q(ξ), h(ξ) T (ξ)x)=sup{ h(ξ) T (ξ)x, z :z D(ξ)}. Page 14 of 35 Theorem: (Walkup/Wets 69) For any ξ, the function Φ(ξ,, ) is finite and continuous on the polyhedral set D(ξ) W (ξ)y. Furthermore, the function Φ(ξ, u, ) is piecewise linear convex on the polyhedral set W (ξ)y for fixed u D(ξ), and Φ(ξ,, t) is piecewise linear concave on D(ξ) for fixed t W (ξ)y.

15 Assumptions: (A1) relatively complete recourse: for any (ξ, x) X, h(ξ) T (ξ)x W (ξ)y ; (A2) dual feasibility: D(ξ) holds for all ξ. Note that (A1) is satisfied if W (ξ)y = R r (complete recourse). In general, (A1) and (A2) impose a condition on the support of P. Proposition: Then the deterministic equivalent of the two-stage model represents a finite convex program (with polyhedral constraints) if the integrals Φ(ξ, q(ξ), h(ξ) T (ξ)x)p (dξ) are finite for all x X. For fixed recourse (W (ξ) W ), it suffices to assume ξ 2 P (dξ) <. Convex subdifferentials, optimality conditions, conditions for differentiability, duality results are well known. (Ruszczyński/Shapiro, Handbook, 2003) Page 15 of 35

16 Towards stability We define the integrand f 0 : R m R by c, x +Φ(ξ, q(ξ), h(ξ) T (ξ)x) if h(ξ) T (ξ)x f 0 (ξ, x)= W (ξ)y, D(ξ), + otherwise, and note that f 0 is a convex random lsc function with X dom f 0 if (A1) and (A2) are satisfied. The two-stage stochastic program can thus be expressed as { } min f 0 (ξ, x)p (dξ) : x X. Then the general stability theory applies! Simple examples of two-stage stochastic programs show that, in general, the set-valued mapping S(.) is not inner semicontinuous at P. Furthermore, explicit descriptions of conditioning functions ψ P of stochastic programs (like linear or quadratic growth at solution sets) are only known in some specific cases. Page 16 of 35

17 Stability w.r.t. the Fortet-Mourier metric ζ H Proposition: Suppose the stochastic program satisfies (A1) and (A2). Assume that the mapping ξ D(ξ) is bounded-valued and there exists a constant L > 0, and a nondecreasing function h : R + R + with h(0) = 0 such that dl (D(ξ), D( ξ)) L max{1, h( ξ ), h( ξ )} ξ ξ holds for all ξ, ξ. Then there exist ˆL > 0 such that f 0 (ξ, x) f 0 ( ξ, x) ˆL max{1, H( ξ ), H( ξ )} ξ ξ f 0 (ξ, x) f 0 (ξ, x) ˆL max{1, H( ξ ) ξ } x x for all ξ, ξ, x, x X, where H is defined by H(t) := h(t)t, t R +. Page 17 of 35 Note that h(t) = { 1, fixed recourse t k, lower diagonal randomness with k blocks.

18 Discrete approximations of two-stage stochastic programs Replace the (original) probability measure P by measures P n having (finite) discrete support {ξ 1,..., ξ n } (n N), i.e., n P n = p i δ ξi, and insert it into the infinite-dimensional stochastic program: n min{ c, x + p i q(ξ i ), y i : x X, y i Y, i = 1,..., n, i=1 i=1 W (ξ 1 )y 1 +T (ξ 1 )x = h(ξ 1 ) W (ξ 2 )y 2 +T (ξ 2 )x =.... = h(ξ 2 ). W (ξ n )y n +T (ξ n )x = h(ξ n )} Hence, we arrive at a (finite-dimensional) large scale block-structured linear program which allows for specific decomposition methods. (Ruszczyński/Shapiro, Handbook, 2003) Page 18 of 35

19 How to choose the discrete approximation? The quantitative stability results suggest to determine P n such that it forms the best approximation of P with respect to the semidistance d F or the probability metric ζ p, i.e., given n N solve ( ) } min {ζ p P, 1 n δ ξi : ξ i, i = 1,..., n n i=1 Such best approximations Pn = 1 n n i=1 δ ξi are known as optimal quantizations of the probability distribution P (Graf/Luschgy, Lecture Notes Math ). Page 19 of 35 Convergence properties of optimal quantizations in case of the l p - minimal metrics (or Wasserstein metrics) ( { p, l p (P, Q):= inf ξ ξ p η(dξ, d ξ) π 1 η = P, π 2 η = Q})1 are already known. Here, π i is the projection onto the i-th component. Note that ζ p (P, Q) (1 + ξ p (P + Q)(dξ)) p 1 p l p (P, Q).

20 Scenario reduction We consider discrete distributions P with scenarios ξ i and probabilities p i, i = 1,..., N, and Q being supported by a given subset of scenarios ξ j, j J {1,..., N}, of P. Optimal reduction of a given scenario set J: The best approximation of P with respect to ζ r by such a distribution Q exists and is denoted by Q. It has the distance D J := ζ r (P, Q ) = min ζ r(p, Q) = p i min ĉr(ξ i, ξ j ) Q j J i J = n 1 p i min{ c r (ξ lk, ξ lk+1 ) : n N, l k {1,..., N}, i J k=1 l 1 = i, l n = j J} and the probabilities qj = p j + p i, j J, where i J j J j := {i J : j = j(i)} and j(i) arg min ĉr(ξ i, ξ j ), i J. j J (Dupačová-Gröwe-Kuska-Römisch 03, Heitsch-Römisch 07) Page 20 of 35

21 We needed the following notation: c r (ξ, ξ) := max{1, ξ r 1, ξ r 1 } ξ ξ (ξ, ξ ). Proposition: (Rachev/Rüschendorf 98) { } ζ r (P, Q) = inf ĉ r (ξ, ξ)η(dξ, d ξ):π 1 η =P, π 2 η =Q where ĉ r c r and ĉ r is the metric (reduced cost) { k 1 } ĉ r (ξ, ξ) := inf c r (ξ li, ξ li+1 ) : k N, ξ li, ξ l1 = ξ, ξ lk = ξ. i=1 Determining the optimal scenario index set with prescribed cardinality n is, however, a combinatorial optimization problem of set covering type: Page 21 of 35 min{d J = i J p i min j J ĉr(ξ i, ξ j ) : J {1,..., N}, #J = N n} Hence, the problem of finding the optimal set J to delete is N P- hard and polynomial time solution algorithms do not exist.

22 Fast reduction heuristic Starting point (n = 1): min u {1,...,N} N p k ĉ r (ξ k, ξ u ) k=1 Algorithm: (Forward selection) Step [0]: Step [i]: Step [n+1]: J [0] := {1,..., N}. u i arg min u J [i 1] J [i] := J [i 1] \ {u i }. k J [i 1] \{u} Optimal redistribution. p k min ĉ r (ξ k, ξ j ), j J [i 1] \{u} Page 22 of 35

23 Example: (Electrical load scenario tree) 1000 (Mean shifted ternary) Load scenario tree (729 scenarios) Page 23 of <Start Animation>

24 Reduced load scenario tree obtained by the forward selection method (15 scenarios) Page 24 of

25 4. Chance constrained stochastic programs We consider the chance constrained model min{ c, x : x X, P ({ξ : T (ξ)x h(ξ)}) p}, where c R m, X and are polyhedra in R m and R s, respectively, p (0, 1), P P(), and the right-hand side h(ξ) R d and the (d, m)-matrix T (ξ) are affine functions of ξ. By specifying the general (semi-) distance we obtain d F (P, Q) := sup max x X j=0,1 f j (x, ξ)(p Q)(dξ) = sup P (H(x)) Q(H(x)), x X where f 0 (ξ, x) = c, x, f 1 (ξ, x) = p χ H(x) (ξ) and H(x) = {ξ : T (ξ)x h(ξ)} (polyhedral subsets of ). The relevant probability metrics are polyhedral discrepancies: α ph (P, Q) = sup P (B) Q(B) B B ph () Page 25 of 35

26 5. Two-stage mixed-integer stochastic programs { } min c, x + Φ(q(ξ), h(ξ) T (ξ)x)p (dξ) : x X, where Φ is given by Φ(u, t) := inf { u 1, y + u 2, ȳ : W y + W ȳ t, y Z ˆm, ȳ R m} for all pairs (u, t) R ˆm+ m R r, and c R m, X is a closed subset of R m, a polyhedron in R s, W and W are (r, ˆm)- and (r, m)-matrices, respectively, q(ξ) R ˆm+ m, h(ξ) R r, and the (r, m)-matrix T (ξ) are affine functions of ξ, and P P 2 (). Probability metric on P 2 (): { ζ 2,ph (P, Q) := sup f(ξ)(p Q)(dξ) f F } 2() B B ph () B Cα ph (P, Q) s+1 1 (if is bounded) Here, the set F 2 () contains all functions f : R such that f(ξ) max{1, ξ } ξ, f(ξ) f( ξ) max{1, ξ, ξ } ξ ξ. Page 26 of 35

27 6. Multistage stochastic programs Let {ξ t } T t=1 be a discrete-time stochastic data process defined on some probability space (Ω, A, P) and with ξ 1 deterministic. The stochastic decision x t at period t is assumed to be measurable with respect to A t (ξ) := σ(ξ 1,..., ξ t ) (nonanticipativity). Multistage stochastic optimization model: [ ] T x t X t, t = 1,..., T, min E b t (ξ t ), x t x t is A 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 1 is bounded polyhedral and X t, t = 2,..., T, are polyhedral cones, the vectors b t ( ), h t ( ) and A t,1 ( ) are affine functions of ξ t, where ξ varies in a polyhedral set. Page 27 of 35 If the process {ξ t } T t=1 has a finite number of scenarios, they exhibit a scenario tree structure.

28 To have the model well defined, we assume x L r (Ω, A, P; R m ) and ξ L r (Ω, A, P; R s ), where r 1 and r := r r 1, if only costs are random r, if only right-hand sides are random 2, if costs and 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, A, P; R m t ) : x t = E[x t A t (ξ)], t}, i.e., as a subspace constraint, by using the conditional expectation E[ A t (ξ)] with respect to the σ-algebra A t (ξ). Page 28 of 35 For T = 2 we have N r (ξ) = R m 1 L r (Ω, A, P; R m 2). infinite-dimensional optimization problem

29 Example: (Optimal purchase under uncertainty) The decisions x t correspond to the amounts to be purchased at each time period with uncertain prices are ξ t, t = 1,..., T, and such that a prescribed amount a is achieved at the end of a given time horizon. The problem is of the form [ T ] min E ξ t x t t=1 (x t, s t ) X t = R 2 +, (x t, s t ) is (ξ 1,..., ξ t )-measurable, s t s t 1 = x t, t = 2,..., T, s 1 = 0, s T = a., where the state variable s t corresponds to the amount at time t. Page 29 of 35 Let T := 3 and ξ ε denote the stochastic price process having the two scenarios ξ 1 ε = (3, 2 + ε, 3) (ε (0, 1)) and ξ 2 ε = (3, 2, 1) each endowed with probability 1 2. Let ξ denote the approximation of ξ ε given by the two scenarios ξ 1 = (3, 2, 3) and ξ 2 = (3, 2, 1) with the same probabilities 1 2.

30 2+ε We obtain Scenario trees for ξ ε (left) and ξ v(ξ ε ) = ε ((2 + ε)a + a) = 2 2 a v( ξ) = 2a, but ξ ε ξ (0 + ε + 0) ( ) = ε 2. Hence, the multistage stochastic purchasing model is not stable with respect to 1. Page 30 of 35

31 Quantitative Stability Let F denote the objective function defined on L r (Ω, A, P; R s ) L r (Ω, A, P; R m ) R by F (ξ, x) := E[ 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, P; R 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, S α (ξ) := {x X (ξ) N r (ξ) : F (ξ, x) v(ξ) + α} S(ξ) := S 0 (ξ) denote the α-approximate solution set and the solution set of the stochastic program with input ξ. Page 31 of 35

32 Assumptions: (A1) ξ L r (Ω, A, P; R s ) for some r 1. (A2) There exists a δ > 0 such that for any ξ L r (Ω, A, P; R 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) Assume that the optimal values v( ξ) are finite if ξ ξ r δ and that the objective function F is level-bounded locally uniformly at ξ, i.e., for some α > 0 there exists a bounded subset B of L r (Ω, A, P; R m ) such that S α ( ξ) is contained in B if ξ ξ r δ. Page 32 of 35

33 Theorem: (Heitsch-Römisch-Strugarek 06) Let (A1) (A3) be satisfied 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,t 1 (ξ, ξ)) holds for all ξ L r (Ω, A, P; R s ) with ξ ξ r δ. If 1 < r < and (ξ (n) ) converges to ξ in L r and with respect to d f,t, then any sequence x n S(ξ (n) ), n N, contains a subsequence converging weakly in L r to some element of S(ξ). Here, d f,τ (ξ, ξ) denotes the filtration distance of ξ and ξ defined by d f,τ (ξ, ξ) := sup x r 1 τ E[x t A t (ξ)] E[x t A t ( ξ)] r. t=2 Remark: For T = 2 we obtain the same result for the optimal values as in the two-stage case! However, we obtain weak convergence of subsequences of (random) second-stage solutions, too! Page 33 of 35

34 Consequences for designing scenario trees If ξ tr is a scenario tree process approximating ξ, one has to take care that ξ ξ tr r and d f,t (ξ, ξ tr ) are small. This is achieved for the generation of scenario trees by recursive scenario reduction (Heitsch-Römisch 05). Page 34 of 35 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 <Start Animation> Are there specific approximations ξ of ξ such that an estimate of the form v(ξ) v( ξ) L ξ ξ r is valid? Recently, such approximations ξ were characterized by Küchler 07! The conditions on ξ and approximation schemes developed by Kuhn 05, Pennanen 05, Mirkov-Pflug 07 also avoid filtration distances.

35 References: Dupačová, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming: An approach using probability metrics, Mathematical Programming 95 (2003), Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs, SIAM Journal Optimization 17 (2006), Heitsch, H., Römisch, W.: Scenario tree modelling for multistage stochastic programs, Preprint 296, DFG Research Center Matheon, Berlin 2005, and submitted. Heitsch, H., Römisch, W.: Stability and scenario trees for multistage stochastic programs, Preprint 324, DFG Research Center Matheon, Berlin 2006, and submitted to Stochastic Programming - The State of the Art (G. Dantzig, G. Infanger eds.). Heitsch, H., Römisch, W.: A note on scenario reduction for two-stage stochastic programs, Operations Research Letters (2007) (to appear). Page 35 of 35 Henrion, R., Römisch, W.: Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints, Mathematical Programming 100 (2004), Rachev, S. T., Römisch, W.: Quantitative stability in stochastic programming: The method of probability metrics, Mathematics of Operations Research 27 (2002), Römisch, W., Wets, R. J-B: Stability of ε-approximate solutions to convex stochastic programs, SIAM Journal Optimization (to appear) Römisch, W., Vigerske, S.: Quantitative stability of fully random mixed-integer two-stage stochastic programs, Preprint 367, DFG Research Center Matheon, Berlin 2007, and submitted.