Sample Approximation Methods for Stochastic Program

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 • Need N senarios • Create N/4 realizations from each Si 1 S1 S2 S3 S4 w1 1/3

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 {ui1,ui2,..uiN} • 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: a1 a2 a3 …. aN Scenario1=(a4,b56) Random match Scenario2=(a6,bN) W2: b1 b2 b3 …. bN Scenariok=(a26,b3) ScenarioN=(a40,b8)

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 A1,…Ai be a QMC point set • RQMC: Xi is a randomized version of Ai. • Rule1: Xi ~ U[0,1]d. (makes estimator unbiased) • Rule2: X1,…Xn 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 ( Xi=(Ai+U)mod1 ) • Digital b-ary shift • Scrambling • Random Linear Scrambling

21. Replicating Quasi-Monte Carlo • 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

22. Padding • 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

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

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.