Sample Approximation Methods for Stochastic Program

Jerry Shen

Zeliha Akca

March 3, 2005

Two-Stage SP with Recourse

0,..)('min xbAxTSxQxcx

)(xQWhere : Expected recourse cost of choosing x in first stage

0,)()(..'min),( yxThWyTSyqxQy

def

)(),( dPxQ

Interior sampling methods

LShaped Method (Dantzig and Infanger)

Stochastic Decomposition (Higle and Sen)

Stochastic Quasi-gradient methods (Ermoliev)

Exterior sampling methods

Monte Carlo

Quasi-Monte Carlo

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

)(),( dPxQ

N

j

jN xQ

11 ),(

Variance Reduction Techniques

Decrease the sample variance:

-Improve statistical efficiency

-Improve time efficiency

-Decrease necessary number of random number generation

Variance Reduction Techniques

Antithetic Variables Stratified Sampling Conditional Sampling Latin Hypercube Sampling Common Random numbers Combination of these

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)

)]2,1(2)2()1([4

1)

2

21( XXCovXVarXVar

XXVar

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.

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])

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)

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

S3 S4

S2

1

w1

Need N senarios Create N/4 realizations from each

Si

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)

n

UiX

jijj

i

)(

2

1

1)(

n

XVarLHS

Application of LHS:

Divide the range of each input to N partition Take a realization from each partition with prob. 1/N

W1:a1 aNa3a2 ….

W2: b2b1 b3 bN….

Scenario1=(a4,b56)

Scenario2=(a6,bN)

ScenarioN=(a40,b8)

Scenariok=(a26,b3)

Random match

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

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

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

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

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 unbiase

d) Rule2: X1,…Xn is a QMC set with probability 1

(keeps the properties that QMC had) RQMC can be viewed as variance reduction t

echniques to MC

Randomized Quasi-Monte Carlo (Cont.)

Randomizations: Random shift ( Xi=(Ai+U)mod1 ) Digital b-ary shift Scrambling Random Linear Scrambling

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

r

j jr II1

1 ˆˆ

2

1)1(1 )ˆˆ(

r

j jrr II

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

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

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.