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

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Monte Carlo Sampling

• Sample independently from U0,1d
• 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

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Variance Reduction Techniques

• Decrease the sample variance
• -Improve statistical efficiency
• -Improve time efficiency
• -Decrease necessary number of random number
  generation

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Variance Reduction Techniques

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

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Antithetic variables

• X1, X2 be r.v. and estimator is (X1X2)/2
• Need negatively correlation
• X1h(U1,U2,..Um) X2h(1-U1,1-U2,..1-Um)

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Application of Antithetic in Sampling

• Need N scenarios,
• 1. Create N/2 uniform (0,1) r.v,
• 2. Use yiUniform(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.

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Conditional Sampling

• Use EXY to estimate X.
• EXYEX,
• Var(X)EVar(XY)Var(EXY)gtVar(EXY)

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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 EL1/REL1EL2..ELR
• Var((L1L2..LR)/R)ltVar(L)

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Application of Stratified Sampling

• Need N senarios
• Create N/4 realizations from each Si

1
S1
S2
S3
S4
w1
1/3

• S1 w1Uniform(1,5/2) and w2Uniform(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

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Latin Hyper Cube Sampling

• Create independent random points
• uiU(i-1)/N,i/N for i1,2,..N
• Create i1,i2,..iN as a random permutation of
  1..N
• Take sample ui1,ui2,..uiN
• Conover (1979)
• Owen(1998)

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Application of LHS

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

W1
a1
aN
a3
a2
.
Scenario1(a4,b56)
Random match
Scenario2(a6,bN)
W2
b2
b1
b3
bN
.
Scenariok(a26,b3)
ScenarioN(a40,b8)
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Common Random Numbers

• Estimate a1-a2EX1-EX2
• 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

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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

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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

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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

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Randomized Quasi-Monte Carlo

• Let A1,Ai be a QMC point set
• RQMC Xi is a randomized version of Ai.
• Rule1 Xi U0,1d. (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

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Randomized Quasi-Monte Carlo (Cont.)

• Randomizations
• Random shift ( Xi(AiU)mod1 )
• Digital b-ary shift
• Scrambling
• Random Linear Scrambling

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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

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Padding

• Partitioning the set of d-dimensions to two
  subsets 1,,s, s1,,d
• Use QMC or RQMC rule on the first subset
• Use MC or LHS rule on the second subset

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Latin Supercube Sampling

• Partitioning the set of d-dimensions to groups of
  s-dimension subsets. (dks)
• Find QMC or RQMC point set on each group

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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. LEcuyer and C.Lemieux. 2002. Recent Advances
  in Randomized Quasi-Monte Carlo Methods. Book
  Modeling UncertaintyAn 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.