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Computational methods for processing two stage stochastic programming problems

Computational methods for processing two stage stochastic programming problems. Frank Ellison Csaba Fabian Gautam Mitra. Department of Mathematical Sciences Brunel University Uxbridge Middlesex UB8 3PH. OUTLINE. Overview of 2-stage SP methods Diagrammatic Presentation of Level Method

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Computational methods for processing two stage stochastic programming problems

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  1. Computational methods for processing two stage stochastic programming problems Frank Ellison Csaba Fabian Gautam Mitra Department of Mathematical Sciences Brunel University Uxbridge Middlesex UB8 3PH

  2. OUTLINE • Overview of 2-stage SP methods • Diagrammatic Presentation of Level Method • Theoretical Comparisons between Methods • Mathematical Description • Test Models • Parameter Variation • Solver Comparisons • Conclusions

  3. 1) Overview of 2-stage SP methods • L-shaped Method (LSM) • Van Slyke and Wets (1969) • Regularised Decomposition (RDC) • Ruszczynski (1986) • Stochastic Decomposition (SD) • Higle and Sen (1991)

  4. 1) Overview of 2-stage SP methods • Level Method • Lemarechal, Nemirovskii, Nesterov (1995) • Level Decomposition (LDC) • Fabian, Szoke (2007)

  5. Cutting-plane method

  6. φ(x) X

  7. φ(x) X x1

  8. φ(x) X x1

  9. φ(x) X x1

  10. φ(x) X x1 x2

  11. φ(x) X x1 x2

  12. φ(x) X x1 x2

  13. φ(x) X x1 x2

  14. φ(x) X x1 x2

  15. Level method

  16. φ(x) x1 x2

  17. φ(x) x1 x2

  18. φ(x) x1 x2

  19. φ(x) x1 x2

  20. φ(x) Δ x1 x2

  21. φ(x) Δ λΔ x1 x2

  22. φ(x) Δ λΔ x1 x2

  23. φ(x) Δ λΔ x1 x2

  24. 3) Theoretical Comparisons between Methods • Convergence:- • LSM/RDC:- Finite • LDC:- Bounded • Confidence:- • SD:- Not high, But may be good enough • LSM/RDC/LDC:- Epsilon-gap (accurate) • Performance:- • SD:- Can tackle unlimited no of scenarios • LSM:- Patchy, best for small-to-medium no of scenarios • RDC:- Better than LSM on the whole • LDC:- Not so good for small, otherwise compares well with LSM

  25. 4.1) Mathematical Description • First Stage Problem is:- Min f(x)  cTx + E[Q(x,)] s.t.:- x X (polyhedral region) • Model Problem (MP) is:-Min cTx +  s.t.:- x XDx +  ≥ d (cuts)

  26. 4.2) Mathematical Description Let x0 be current iterate (from last pass) Let UB, LB be current upper, lower bounds (as in LSM) LB is the minimum of the current MP UB is the best value of f known so far Let  be a parameter:- 0 ≤  < 1

  27. 4.3) Mathematical Description Then the new iterate x1 is the optimal solution to:- Min ½(x-x0)T(x-x0)s.t.:- x XDx +  ≥ d (cuts) cTx + ≤ .UB + (1-)LB

  28. 5) Test Models

  29. 6.1) Parameter Variation

  30. 6.2) Parameter Variation

  31. 6.3) Parameter Variation

  32. 7) Solver Comparisons

  33. Conclusions • LDC shows better times than than LSM – especially for larger/harder problems • LDC offers extensions to:- • Regularise infeasibility cuts • Progressive approximation of the distribution • LDC offers better solution confidence than SD – but SD is faster & can tackle indefinite nos of scenarios

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