Stochastic Integer Programming
          An Algorithmic Perspective
                                 Shabbir Ahmed
       ...
Shabbir Ahmed                2
SPX Tucson AZ Oct. 9, 2004
Outline
                   Two-stage SIP
                   • Formulation
                   • Challenges
                ...
Two-Stage SIPs
  •     Decisions in two stages
        Stage 1 decision → Observe uncertainties → Stage 2 decision
       ...
Examples

  •     Resource acquisition (Dempster et al.,1981,1983)
        Acquire machines → Observe processing times → S...
General Formulation




Shabbir Ahmed                                      6
SPX Tucson AZ Oct. 9, 2004
Difficulty 1

• Evaluating the second-stage cost          for a fixed first-stage
  decision and a particular realization ...
Difficulty 2
 • Evaluating the expected second-stage cost for a fixed first-stage
   decision.
    – If the uncertain para...
Difficulty 3
 •          is the value function of an integer program
 • Is non-convex and discontinuous (lower-semicontinu...
Difficulty 3
 • Optimizing                                  , with respect to     .



 Theorem (Stougie 1985; Schultz 199...
Non-convexity and Discontinuity




Shabbir Ahmed                                  11
SPX Tucson AZ Oct. 9, 2004
Simple Integer Recourse
                                (Stochastic RHS)




Shabbir Ahmed                                ...
Dealing with the difficulties (SIR)

• No Difficulty 1



     where                    .

• Let                        an...
Dealing with Difficulty 2 (SIR)

Theorem (Louveaux and van der Vlerk, 1993)

                               and




• In m...
Dealing with Difficulty 3 (SIR)
Let                                           then




In general,                  is not...
Example




Shabbir Ahmed                          16
SPX Tucson AZ Oct. 9, 2004
Convexification
• Klein Haneveld et al (1995) give an algorithm for constructing the
  convex hull in case of discrete dis...
Example




Shabbir Ahmed                Klein Haneveld et al., 1995   18
SPX Tucson AZ Oct. 9, 2004
Approximations

• Similar results: by perturbing the distribution, convex (continuous
  simple recourse type) approximatio...
General Mixed-Integer Recourse

 • Assume that the recourse function is well-defined.

 • Dealing with Difficulty 1
    – ...
(One way of) Dealing with Difficulty 2
The Sample Average Approximation Method
  • Need to solve

  • Let     be the set o...
Convergence of the SAA Method

  Theorem (Kleywegt et al. 2001)

  If         is finite then        and                   ...
Convergence of the SAA Method

  Theorem (Ahmed and Shapiro, 2002)

  If            is bounded (not necessarily finite) th...
Practical SAA Method
• Select a sample size              , solve   independent SAA problems.

• Let   and                 ...
Sample SAA Computation




                             Two-stage SIP with 108 scenarios.

Shabbir Ahmed                  ...
Solving Similar IPs
• Still need to solve many “similar” MIPs.

• In case of stochastic linear programming, this problem i...
Dealing with Difficulty 3
        • Assume a finite (“manageable”) number of scenarios.




                             w...
Deterministic Equivalent




              • Large-scale MIP.
              • If not too many scenarios, use, e.g., CPLEX....
Two-Stage Decomposition

      Solve
                                               is computationally
                   ...
Binary First-stage
Integer L-Shaped Method (Laporte & Louveaux, 1993)

Here                                    Let
Denote
...
Integer L-Shaped Method
                                             Master
                                             P...
Integer L-Shaped (Remarks)
       • Master problem is a 0-1 MIP … solve by branch & bound.
       • (Decomposed) MIP subpr...
Disjunctive Decomposition
       Sen and Higle (2000)
       • Binary first-stage, Mixed-binary second-stage and fixed
   ...
The C3 Theorem
        Sen and Higle (2000)
        • There exists a function such that the cut can be translated
        ...
Convexification
        • Since only binary first-stage solutions are relevant, convexify
          (linearize) the concav...
The D2 Algorithm (Remarks)

        • As long as we have a separation scheme for finding “proper”
          valid inequali...
Mixed-Integer First Stage

• Binary first-stage ⇒ Need to evaluate/approximate value function
  only at binary solutions (...
Pure Integer Second Stage

                                       • Inherent finiteness.

                                ...
Continuous Branch & Bound
                             Objective            P        Objective              P


          ...
Finiteness Issue

                                         • The most common branching scheme is
                         ...
Pure Integer Second Stage
 B&B in the tender space (Ahmed et al., 2004)
 • Fixed technology matrix.
 • Solve the problem i...
The B&B Algorithm

• Lower bounding: Second-stage value
  function is non-decreasing and lsc.


• Upper bounding: Function...
Sample Computation




              • Test set: Capacity acquisition-assignment problems.
              • Ahmed and Garci...
More on Pure Integer Second Stage
    • Hemmecke and Schultz, 2003
       – Rhs uncertainty. Pure integer first stage.
   ...
Multi-Stage SIPs

 •     Decisions in multiple (but, a finite number of) stages

     Stage 1 decision → Observe uncertain...
The Scenario Tree
•   Assuming finite support, the evolution of the uncertain parameters
    can be modeled as a scenario ...
A Formulation

 •     Deterministic equivalent formulation.
 •     Solution at a node depends on the solutions in ancestor...
Scenario Decomposition




Shabbir Ahmed                                         48
SPX Tucson AZ Oct. 9, 2004
The Scenario Formulation




                                  Non-anticipativity constraints



Shabbir Ahmed            ...
Lagrangian Relaxation

 •     Any feasible solution to the Lagrangian dual provides a lower bound
       to the true optim...
Dual Decomposition
 Caroe and Schultz (1999)

 • Use Lagrangian dual as the lower bounding scheme within a branch
   and b...
Polyhedral Results

    • Given valid inequalities for a
      deterministic MIP, can we find a valid
      inequality for...
Example: Uncapacitated Lot-sizing


                              • Common substructure in
                               ...
Stochastic Uncapacitated Lot-sizing

 Guan et al. (2004)

 • Given any subset of the
   nodes, the corresponding
   (l,S) ...
Concluding Remarks

 •        Would have liked to talk about Approximation Algorithms for SIP.

 •        Survey articles ...
References
 S. Ahmed and A. Shapiro. The sample average approximation method for stochastic programs with integer recourse...
References
 G. Laporte and F.V. Louveaux. The integer L-shaped method for stochastic integer programs with complete recour...
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Stochastic Integer Programming. An Algorithmic Perspective

  1. 1. Stochastic Integer Programming An Algorithmic Perspective Shabbir Ahmed sahmed@isye.gatech.edu www.isye.gatech.edu/~sahmed School of Industrial & Systems Engineering
  2. 2. Shabbir Ahmed 2 SPX Tucson AZ Oct. 9, 2004
  3. 3. Outline Two-stage SIP • Formulation • Challenges • Simple Integer Recourse • Structural results • General integer recourse • A few Decomposition algorithms Multi-stage SIP • Formulation • Algorithms • Scenario decomposition • Polyhedral results Shabbir Ahmed 3 SPX Tucson AZ Oct. 9, 2004
  4. 4. Two-Stage SIPs • Decisions in two stages Stage 1 decision → Observe uncertainties → Stage 2 decision (“here and now”) (“recourse”) • Known distribution The probability distribution of the uncertainties is known • Exogenous uncertainties Stage 1 decisions do not affect the distribution • Discrete/Combinatorial decisions • Goal: Minimize cost of stage 1 decisions + Expected cost of stage 2 decisions Shabbir Ahmed 4 SPX Tucson AZ Oct. 9, 2004
  5. 5. Examples • Resource acquisition (Dempster et al.,1981,1983) Acquire machines → Observe processing times → Schedule jobs. • Location-Routing (Laporte et al., 1994) Locate depots → Observe demand → Route vehicles. • Ground Holding in Airline Operations (Ball et al., 2003) Schedule arrival/departure → Observe delays → Decide optimal holding pattern. Shabbir Ahmed 5 SPX Tucson AZ Oct. 9, 2004
  6. 6. General Formulation Shabbir Ahmed 6 SPX Tucson AZ Oct. 9, 2004
  7. 7. Difficulty 1 • Evaluating the second-stage cost for a fixed first-stage decision and a particular realization of the uncertain parameters. • Involves solving (possibly) NP-hard integer program • E.g. Second stage: schedule jobs after observing processing requirements. • Most SIP research assumes away this difficulty. Shabbir Ahmed 7 SPX Tucson AZ Oct. 9, 2004
  8. 8. Difficulty 2 • Evaluating the expected second-stage cost for a fixed first-stage decision. – If the uncertain parameters have a continuous distribution: involves integrating the value function of an integer program and is in general impossible. – If the uncertain parameters have a discrete distribution: if and each component has 3 independent realizations, then !! involves solving a huge number of similar integer programs. Shabbir Ahmed 8 SPX Tucson AZ Oct. 9, 2004
  9. 9. Difficulty 3 • is the value function of an integer program • Is non-convex and discontinuous (lower-semicontinuous) Shabbir Ahmed 9 SPX Tucson AZ Oct. 9, 2004
  10. 10. Difficulty 3 • Optimizing , with respect to . Theorem (Stougie 1985; Schultz 1993,1995) If for all and , and then is real-valued and lower-semicontinuous on . If , in addition, has an absolutely continuous density, then is continuous on . • In general, is non-convex and often discontinuous, and therefore, so is . Shabbir Ahmed 10 SPX Tucson AZ Oct. 9, 2004
  11. 11. Non-convexity and Discontinuity Shabbir Ahmed 11 SPX Tucson AZ Oct. 9, 2004
  12. 12. Simple Integer Recourse (Stochastic RHS) Shabbir Ahmed 12 SPX Tucson AZ Oct. 9, 2004
  13. 13. Dealing with the difficulties (SIR) • No Difficulty 1 where . • Let and then • Dealing with Difficulty 2: if we know how to evaluate the univariate functions and , we are done. Shabbir Ahmed 13 SPX Tucson AZ Oct. 9, 2004
  14. 14. Dealing with Difficulty 2 (SIR) Theorem (Louveaux and van der Vlerk, 1993) and • In many cases, the above sums are easy to evaluate. • Separability allows for the easy evaluation of . • Difficulty 2 resolved. Shabbir Ahmed 14 SPX Tucson AZ Oct. 9, 2004
  15. 15. Dealing with Difficulty 3 (SIR) Let then In general, is not convex. Theorem (Klein Haneveld et al., 1995) There exists a random variable such that Here denotes the convex hull of a function over its entire domain. Shabbir Ahmed 15 SPX Tucson AZ Oct. 9, 2004
  16. 16. Example Shabbir Ahmed 16 SPX Tucson AZ Oct. 9, 2004
  17. 17. Convexification • Klein Haneveld et al (1995) give an algorithm for constructing the convex hull in case of discrete distributions. Theorem (Klein Haneveld, 1995) If is of full row rank then • In some cases • Then, we only need to solve a problem with continuous simple recourse Shabbir Ahmed 17 SPX Tucson AZ Oct. 9, 2004
  18. 18. Example Shabbir Ahmed Klein Haneveld et al., 1995 18 SPX Tucson AZ Oct. 9, 2004
  19. 19. Approximations • Similar results: by perturbing the distribution, convex (continuous simple recourse type) approximations/lower bounding functions for the SIR function can be obtained. – Uniform error bound on the approximation • Solve SIPs with SIR approximately or obtain lower bounds via solving continuous recourse models. – Can we use these within a branch and bound scheme? – Can we get convex hull/lower bound restricted to a subset of the domain? Shabbir Ahmed 19 SPX Tucson AZ Oct. 9, 2004
  20. 20. General Mixed-Integer Recourse • Assume that the recourse function is well-defined. • Dealing with Difficulty 1 – Assume that the second stage MIPs are “easily” solvable exactly. – Some literature on using approximations (e.g. Dempster 1983). • Dealing with Difficulty 2 – Approximate the distribution by a discrete distribution with a “manageable” number of realizations, e.g., by sampling. – Still need to solve several similar MIPs. Shabbir Ahmed 20 SPX Tucson AZ Oct. 9, 2004
  21. 21. (One way of) Dealing with Difficulty 2 The Sample Average Approximation Method • Need to solve • Let be the set of -optimal solutions and be the optimal objective value. • Generate i.i.d samples and solve the Sample Average Approximating (SAA) Problem: • Let be the set of -optimal solutions (with ) and be the optimal objective value. Shabbir Ahmed 21 SPX Tucson AZ Oct. 9, 2004
  22. 22. Convergence of the SAA Method Theorem (Kleywegt et al. 2001) If is finite then and . Moreover the convergence is exponentially fast. The sample size needed to obtain an -optimal solution to the true problem with probability is Shabbir Ahmed 22 SPX Tucson AZ Oct. 9, 2004
  23. 23. Convergence of the SAA Method Theorem (Ahmed and Shapiro, 2002) If is bounded (not necessarily finite) then the sample size needed to obtain an -optimal solution to the true problem with probability is For pure integer recourse, only right hand-side uncertainty, and discrete distribution with scenarios, the sample size needed to obtain an -optimal solution to the true problem with probability is Shabbir Ahmed 23 SPX Tucson AZ Oct. 9, 2004
  24. 24. Practical SAA Method • Select a sample size , solve independent SAA problems. • Let and be the optimal value and optimal solution of the I-th SAA problem. • A point estimate of a lower bound on is given by (Mak et al, 1999) • A point estimate of the objective value of a candidate solution is where is a large sample. • Then a point estimate of the optimality gap of the candidate solution is obtained. • Variability in the point estimates can be used to obtain confidence intervals. Shabbir Ahmed 24 SPX Tucson AZ Oct. 9, 2004
  25. 25. Sample SAA Computation Two-stage SIP with 108 scenarios. Shabbir Ahmed 25 SPX Tucson AZ Oct. 9, 2004
  26. 26. Solving Similar IPs • Still need to solve many “similar” MIPs. • In case of stochastic linear programming, this problem is tackled using various warm-start strategies arising from exploiting LP duality. • Unfortunately, a computationally useful IP duality theory is not yet mature. • Two of the approaches in the SIP literature for pure integer second- stage: – Gröbner Basis (Schultz et al., 1998) – Value function construction (Kong et al., 2004) Shabbir Ahmed 26 SPX Tucson AZ Oct. 9, 2004
  27. 27. Dealing with Difficulty 3 • Assume a finite (“manageable”) number of scenarios. where and Shabbir Ahmed 27 SPX Tucson AZ Oct. 9, 2004
  28. 28. Deterministic Equivalent • Large-scale MIP. • If not too many scenarios, use, e.g., CPLEX. • Otherwise decompose Shabbir Ahmed 28 SPX Tucson AZ Oct. 9, 2004
  29. 29. Two-Stage Decomposition Solve is computationally tractable lower bounding approximation of Evaluation is via Evaluate decomposition Refinement No Yes Shabbir Ahmed 29 SPX Tucson AZ Oct. 9, 2004 STOP
  30. 30. Binary First-stage Integer L-Shaped Method (Laporte & Louveaux, 1993) Here Let Denote Let Note Then and Shabbir Ahmed 30 SPX Tucson AZ Oct. 9, 2004
  31. 31. Integer L-Shaped Method Master Problem Cuts Evaluation is via Evaluate decomposition Refinement No Yes Shabbir Ahmed 31 SPX Tucson AZ Oct. 9, 2004 STOP
  32. 32. Integer L-Shaped (Remarks) • Master problem is a 0-1 MIP … solve by branch & bound. • (Decomposed) MIP subproblems in the evaluation steps. • Implementation: Do not B&B to optimality … branch-and-cut. – Add cuts whenever a binary solution is encountered in the B&B search. • Cut quality depends on the quality of the lower bound. – Can be improved if more information is available on the value function. • Other types of valid cuts can also be added. – Standard LP Benders cuts are valid, but weak. • Finite termination guaranteed. • Application: Stochastic vehicle routing (Laporte et al. 2002) Shabbir Ahmed 32 SPX Tucson AZ Oct. 9, 2004
  33. 33. Disjunctive Decomposition Sen and Higle (2000) • Binary first-stage, Mixed-binary second-stage and fixed recourse. • Goal – Avoid solving MIP subproblems during evaluation. – Exploit similarity of subproblems. • Given , for each solve the LP-relaxation. If solution is fractional, find a valid inequality for the LP. Only valid for current subproblem Shabbir Ahmed 33 SPX Tucson AZ Oct. 9, 2004
  34. 34. The C3 Theorem Sen and Higle (2000) • There exists a function such that the cut can be translated to be valid for the subproblem corresponding to any scenario and any first-stage solution. • The cut-coefficients do not change (“common”). • However, is piece-wise linear and concave in . Shabbir Ahmed 34 SPX Tucson AZ Oct. 9, 2004
  35. 35. Convexification • Since only binary first-stage solutions are relevant, convexify (linearize) the concave function . • RHS linear in first-stage variables. Pass Benders cuts to master. Shabbir Ahmed 35 SPX Tucson AZ Oct. 9, 2004
  36. 36. The D2 Algorithm (Remarks) • As long as we have a separation scheme for finding “proper” valid inequalities for the subproblems, the algorithm terminates in a finite number of steps with the optimum. • Application: Server location (Ntaimo and Sen, 2003) • Sen and Sherali (2004) extend the approach to when the second-stage problems are partially solved by Branch-and- cut. Shabbir Ahmed 36 SPX Tucson AZ Oct. 9, 2004
  37. 37. Mixed-Integer First Stage • Binary first-stage ⇒ Need to evaluate/approximate value function only at binary solutions (finiteness inherent). • Mixed-integer first-stage ⇒ Optimize a non-convex discontinuous objective over a (semi)-continuous domain. • If Pure integer Second-stage: Theorem (Schultz et al.1998) The expected second-stage value function is piece-wise constant (over polyhedral regions), and an optimal solution to the problem lies at an extreme point of one of these polyhedra. Shabbir Ahmed 37 SPX Tucson AZ Oct. 9, 2004
  38. 38. Pure Integer Second Stage • Inherent finiteness. • Algorithm: Check all these extreme points. • Difficulties – Polyhedra not easy to characterize. – May be too many. • Alternative: Use continuous branch and bound. Shabbir Ahmed 38 SPX Tucson AZ Oct. 9, 2004
  39. 39. Continuous Branch & Bound Objective P Objective P R R U L L Variable Variable (a) Lower bounding (b) Upper bounding Objective P R U R R2 L R1 R1 R2 fathom subdivide Variable Shabbir Ahmed 39 SPX Tucson AZ Oct. 9, 2004 (c) Domain subdivision (d) Search tree
  40. 40. Finiteness Issue • The most common branching scheme is rectangular partitioning - branching along a variable axis. • The resulting partitions are rectangular. • The polyhedral shaped discontinuous pieces cannot be isolated by a finite number of rectangular partitions. Finite sequences • There will be some partitions containing discontinuities. A potentially infinite sequence • These will require infinite partitioning for bounds to converge. Shabbir Ahmed 40 SPX Tucson AZ Oct. 9, 2004
  41. 41. Pure Integer Second Stage B&B in the tender space (Ahmed et al., 2004) • Fixed technology matrix. • Solve the problem in the space of the tender variables. • Discontinuous are orthogonal to the tender axes. Shabbir Ahmed 41 SPX Tucson AZ Oct. 9, 2004
  42. 42. The B&B Algorithm • Lower bounding: Second-stage value function is non-decreasing and lsc. • Upper bounding: Function evaluation. χL χB χU • Branching: Partition along the discontinuities. Maintains rectangular partitions. χL χB • Finite convergence to global optima. χB χU • Improved lower bounding methods. Shabbir Ahmed 42 SPX Tucson AZ Oct. 9, 2004
  43. 43. Sample Computation • Test set: Capacity acquisition-assignment problems. • Ahmed and Garcia, 2003. Shabbir Ahmed 43 SPX Tucson AZ Oct. 9, 2004
  44. 44. More on Pure Integer Second Stage • Hemmecke and Schultz, 2003 – Rhs uncertainty. Pure integer first stage. – IP Test Sets (Computational Algebra). • van der Vlerk, 2004 – Rhs uncertainty. – Convex lower bounds by changing the distribution. • Kong et al., 2004 – Rhs uncertainty. Pure integer first stage. – Construct and optimize value function. • Kong et al. 2004 – Conditions for total unimodularity. – Benders: MIP master may be better than LP master. Shabbir Ahmed 44 SPX Tucson AZ Oct. 9, 2004
  45. 45. Multi-Stage SIPs • Decisions in multiple (but, a finite number of) stages Stage 1 decision → Observe uncertainties → Stage 2 decision → Observe uncertainties → Stage 3 decision → ….. • Example Applications: – Unit commitment (Takriti et al. 1996, Caroe & Schultz, 1999). – Capacity/Production planning (Ahmed et al. 2001, Lulli and Sen, 2002 ). – Asset liability management (Drijver et al., 2000). • Challenges: Same as before … but now in multiple-folds! Shabbir Ahmed 45 SPX Tucson AZ Oct. 9, 2004
  46. 46. The Scenario Tree • Assuming finite support, the evolution of the uncertain parameters can be modeled as a scenario tree. Shabbir Ahmed 46 SPX Tucson AZ Oct. 9, 2004
  47. 47. A Formulation • Deterministic equivalent formulation. • Solution at a node depends on the solutions in ancestor nodes. • Index according to node – a tree formulation. Shabbir Ahmed 47 SPX Tucson AZ Oct. 9, 2004
  48. 48. Scenario Decomposition Shabbir Ahmed 48 SPX Tucson AZ Oct. 9, 2004
  49. 49. The Scenario Formulation Non-anticipativity constraints Shabbir Ahmed 49 SPX Tucson AZ Oct. 9, 2004
  50. 50. Lagrangian Relaxation • Any feasible solution to the Lagrangian dual provides a lower bound to the true optimal value. • Evaluating the dual function requires solving one deterministic problem per scenario (Decomposition). • Dual involves maximizing a concave non-smooth function … can be solved using non-smooth optimization techniques. • Difficulty: Many dual multipliers. • Difficulty: Duality Gap Shabbir Ahmed 50 SPX Tucson AZ Oct. 9, 2004
  51. 51. Dual Decomposition Caroe and Schultz (1999) • Use Lagrangian dual as the lower bounding scheme within a branch and bound algorithm. • Branch to enforce non-anticipativity. • Finite termination in case of pure integer solutions. • Applicable to two-stage stochastic integer programs. • Application: Unit commitment problem. Shabbir Ahmed 51 SPX Tucson AZ Oct. 9, 2004
  52. 52. Polyhedral Results • Given valid inequalities for a deterministic MIP, can we find a valid inequality for the stochastic counterpart? • Generating “tree” inequalities from “path” inequalities. • Branch and cut schemes for deterministic equivalent. • Tighten subproblems within decomposition based branch and cut schemes. Shabbir Ahmed 52 SPX Tucson AZ Oct. 9, 2004
  53. 53. Example: Uncapacitated Lot-sizing • Common substructure in many production planning problems • (l,S) inequalities sufficient to describe convex hull. • Exponential family … polynomially separable. Shabbir Ahmed 53 SPX Tucson AZ Oct. 9, 2004
  54. 54. Stochastic Uncapacitated Lot-sizing Guan et al. (2004) • Given any subset of the nodes, the corresponding (l,S) inequalities are valid. • These inequalities can be “combined” to generate a new family of inequalities. • Necessary and sufficient conditions for the inequalities to be facet- defining. • Excellent performance within branch & cut. • The combining idea is quite general and can be applied to other SIPs. Shabbir Ahmed 54 SPX Tucson AZ Oct. 9, 2004
  55. 55. Concluding Remarks • Would have liked to talk about Approximation Algorithms for SIP. • Survey articles on SIP: – Klein Haneveld and van der Vlerk (1998) – Louveaux and Schultz (2003) – Schultz et al. (1995) – Sen (2004) • WWW Resources: – SP Community Page: http://www.stoprog.org – Bibliography (2003): http://mally.eco.rug.nl/biblio/SIP.HTML – Test Problems: http://www.isye.gatech.edu/~sahmed/siplib/ Shabbir Ahmed 55 SPX Tucson AZ Oct. 9, 2004
  56. 56. References S. Ahmed and A. Shapiro. The sample average approximation method for stochastic programs with integer recourse. Optimization Online, http://www.optimization-online.org, 2002. S. Ahmed and R. Garcia. Dynamic Capacity Acquisition and Assignment under Uncertainty. Annals of Operations Research, 124:267-283, 2003 S. Ahmed, M. Tawarmalani, and N. V. Sahinidis. A finite branch and bound algorithm for two-stage stochastic integer programs. Mathematical Programming, 100:355-377, 2004. M. O. Ball, R. Hoffman, A. R. Odoni, and R. Rifkin. A stochastic integer program with dual network structure and its application to the ground-holding problem. Operations Research, 51(1):167-171, 2003. C. C. Carøe and R. Schultz. Dual decomposition in stochastic integer programming. Operations Research Letters, 24(1-2):37-45, 1999. M.A.H. Dempster, M.L. Fisher, L. Jansen, B.J. Lageweg, J.K. Lenstra, and A.H.G. Rinnooy Kan. Analytical evaluation of hierarchical planning systems. Operations Research, 29:707-716, 1981. M.A.H. Dempster, M.L. Fisher, L. Jansen, B.J. Lageweg, J.K. Lenstra, and A.H.G. Rinnooy Kan. Analysis of heuristics for stochastic programming: results for hierarchical scheduling problems. Mathematics of Operations Research, 8:525-537, 1983. R. Hemmecke and R. Schultz. Decomposition of test sets in stochastic integer programming. Mathematical Programming, 94:323-341, 2003. Y. Guan, S. Ahmed and G.L. Nemhauser. A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem. Stochastic Programming E-Print Series, 2004. W.K. Klein Haneveld, L. Stougie and M.H. van der Vlerk. On the convex hull of the simple integer recourse objective function , Annals of Operations Research, 56:209-224, 1995. W.K. Klein Haneveld and M. H. van der Vlerk. Stochastic integer programming: general models and algorithms. Ann. Oper. Res., 85:39-57, 1999. A.J. Kleywegt, A. Shapiro, and T. Homem-de-Mello. The Sample Average Approximation Method for Stochastic Discrete Optimization. SIAM Journal on Optimization, 12:479-502, 2001. N. Kong, A.J. Schaefer and B. Hunsaker. Two-Stage Integer Programs with Stochastic Right-Hand Sides: A Superadditive Dual Approach, Technical Report, University of Pittsburgh, 2004. F.V. Louveaux and R. Schultz. Stochastic Integer Programming. Chapter 4 in Handbooks in Operations Research and Management Science, Vol. 10, Stochastic Programming (Ruszczynski and Shapiro etd.), Elsevier, 2003. Shabbir Ahmed 56 SPX Tucson AZ Oct. 9, 2004
  57. 57. References G. Laporte and F.V. Louveaux. The integer L-shaped method for stochastic integer programs with complete recourse. Operations Research Letters, 13:133-142, 1993. G. Laporte, F.V. Louveaux, and L. van Hamme. Exact solution of a stochastic location problem by an integer L-shaped algorithm. Transportation Science, 28(2):95-103, 1994. G. Laporte, F.V. Louveaux, and L. van Hamme. An integer L-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Operations Research, 50:415-423, 2002. F. Louveaux and M.H. van der Vlerk. Stochastic programming with simple integer recourse, Mathematical Programming, 61(3):301-325, 1993. W.K. Mak, D.P. Morton and R.K. Wood. Monte Carlo Bounding Techniques for Determining Solution Quality in Stochastic Programs. Operations Research Letters, 24:47-56, 1999. L. Ntaimo and S. Sen. The Million-Variable “March” for Stochastic Combinatorial Optimization. To appear in the Journal of Global Optimization, 2004. R. Schultz. Continuity properties of expectation functions in stochastic integer programming. Mathematics of Operations Research, 18(3):578-589, 1993. R. Schultz. On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Mathematical Programming, 70:73-89, 1995. R. Schultz, L. Stougie, and M.H. van der Vlerk. Two-stage stochastic integer programming: a survey. Statistica Neerlandica, 50(3):404-416, 1996. R. Schultz, L. Stougie and M.H. van der Vlerk. Solving stochastic programs with complete integer recourse: a framework using Groebner Bases , Mathematical Programming, 83(2):229-252, 1998. S. Sen and J. Higle. The C3 Theorem and a D2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification. Technical Report, University of AZ. 2003. S. Sen and H. D. Sherali. Decomposition with Branch-and-Cut Approaches for Two Stage Stochastic Mixed-Integer Programming. Technical Report, University of AZ. 2004. S. Sen. Algorithms for Stochastic Mixed-Integer Programming Models. Technical Report, University of AZ. 2004. L. Stougie. Design and analysis of algorithms for stochastic integer programming, PhD Thesis, Center for Mathematics and Computer Science, Amsterdam, 1987. M. H. van der Vlerk. Convex approximations for complete integer recourse models. Mathematical Programming, 99(2):297-310, 2004. Shabbir Ahmed 57 SPX Tucson AZ Oct. 9, 2004

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