1. Sample Approximation Methods for Stochastic Program Jerry Shen Zeliha Akca March 3, 2005

2. Two-Stage SP with Recourse Where : Expected recourse cost of choosing x in first stage

3. Interior sampling methods LShaped Method (Dantzig and Infanger) Stochastic Decomposition (Higle and Sen) Stochastic Quasi-gradient methods (Ermoliev)

4. Exterior sampling methods Monte Carlo Quasi-Monte Carlo

5. Monte Carlo Sampling Sample independently from U[0,1] d Error estimation is comparatively easy Monte Carlo errors are of O(n -1/2 ) Error does depend on dimension d Can be combined with variance reduction techniques

6. Variance Reduction Techniques Decrease the sample variance: -Improve statistical efficiency -Improve time efficiency -Decrease necessary number of random number generation

7. Variance Reduction Techniques Antithetic Variables Stratified Sampling Conditional Sampling Latin Hypercube Sampling Common Random numbers Combination of these

8. Antithetic variables: X1, X2 be r.v. and estimator is (X1+X2)/2 Need negatively correlation X1=h(U1,U2,..Um) X2=h(1-U1,1-U2,..1- Um)

9. Application of Antithetic in Sampling: Need N scenarios, 1. Create N/2 uniform (0,1) r.v, 2. Use yi~Uniform(0,1) for N/2 realizations and use (1-yi) for the other N/2. 3. Solve the model with these N scenarios. 4. Find the objective function. 5. Repeat M times with different N/2 uniform realizations. 6. Measure sample mean and variance.

10. Conditional Sampling Use E[X|Y] to estimate X. E[X|Y]=E[X], Var(X)=E[Var(X|Y)]+Var(E[X|Y])>=Var(E[X|Y])

11. Stratified Sampling Need N realizations from probability region, Suppose R conditions, Take N/R realization from each condition L is the estimate from all region L1,L2,..LR are estimates from each condition Idea is: E[L]=1/R{E[L1]+E[L2]+..+E[LR]} Var((L1+L2+..+LR)/R)<=Var(L)

12. Application of Stratified Sampling S1={ w1~Uniform(1,5/2) and w2~Uniform(1/3,2/3)} Solve the model for each region Take the average of these four objective functions Repeat M times Measure sample mean and variance of M samples 1/3 S1 S3S4 S2 1 w1 Need N senarios Create N/4 realizations from each Si

13. Latin Hyper Cube Sampling Create independent random points ui~U[(i-1)/N,i/N] for i=1,2,..N Create {i1,i2,..iN} as a random permutation of {1..N} Take sample {u i1,u i2,..u iN } Conover (1979): Owen(1998)

14. Application of LHS: Divide the range of each input to N partition Take a realization from each partition with prob. 1/N W1: a1aNa3a2…. W2: b2b1b3bN…. Scenario1=(a4,b56) Scenario2=(a6,bN) ScenarioN=(a40,b8) Scenario k =(a26,b3) Random match

15. Common Random Numbers: Estimate α1-α2=E[X1]-E[X2] X1 is from system 1 and X2 is from system 2 Use same seed to create random numbers in both systems Idea is: Var(X1-X2)=Var(X1)+Var(X2)- 2Cov(X1,X2) Need X1 and X2 are positively correlated

16. Quasi-Monte Carlo Sampling A deterministic counterpart to the MC. Find more regularly distributed point sets from d- dimensional unit hypercube instead of random point set in MC Implementation is as easy as MC but has faster convergence of the approximations Smaller sample size, cheaper computations compare to MC Quasi-Monte Carlo errors are of O(n -1 (log n) d ) which is asymptotically superior to MC

17. Quasi-Monte Carlo Sampling (Cont.) No practical way to estimate the size of Error Unpromising high dimension behavior Morokoff and Caflisch (1995) Paskov and Traub (1995) Caflisch Morokoff and Owen (1997) Hard to construct QMC point sets with meaningful QMC properties and reasonably small values of n under high dimension

18. Quasi-Monte Carlo Sampling (Cont.) Constructors: Lattice Rules Sobol’ Sequences Generalized Faure Sequences Niederreiter Sequences Polynomial Lattice Rules Small PRNGs Halton sequence Sequences of Korobov rules

19. Randomized Quasi-Monte Carlo Let A 1,…A i be a QMC point set RQMC: X i is a randomized version of A i. Rule1: X i ~ U[0,1] d. (makes estimator unbiased) Rule2: X 1,…X n is a QMC set with probability 1 (keeps the properties that QMC had) RQMC can be viewed as variance reduction techniques to MC

20. Randomized Quasi-Monte Carlo (Cont.) Randomizations: Random shift ( X i =(A i +U)mod1 ) Digital b-ary shift Scrambling Random Linear Scrambling

21. Take a small number r of independent replicates of QMC points. Unbiased estimate of error is Unbiased estimate of variance is Making r large increase the accuracy of variance estimate Replicating Quasi-Monte Carlo

22. Partitioning the set of d-dimensions to two subsets {1,…,s}, {s+1,…,d} Use QMC or RQMC rule on the first subset Use MC or LHS rule on the second subset Padding

23. Partitioning the set of d-dimensions to groups of s- dimension subsets. (d=ks) Find QMC or RQMC point set on each group Latin Supercube Sampling

24. Reference A.Oven 1998. Monte Carlo Extension of Quasi-Monte Carlo. 1998 Winter Simulation Conference. M.Koivu 2004. Variance Reduction in Sample Approximations of Stochastic Programs. Mathematical Programming. J.Linderoth A.Shapiro and S.Wright. 2002. The Empirical Behavior of Sampling Methods for Stochastic Programming. Optimization technical report 02-01. P. L’Ecuyer and C.Lemieux. 2002. Recent Advances in Randomized Quasi-Monte Carlo Methods. Book: Modeling Uncertainty:An Examination of Stochastic Theory, Methods, and Applications, pg 419-474. H.Niederreiter. 1992. Book: Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of CBMS-NSF Reginal Conference Series in Applied Mathematics.