1. OSiL: An XML-based schema for stochastic programs H.I. Gassmann, R. Fourer, J. Ma, R.K. Martin SP XI Vienna, August 2007

2. © 2007 H.I. Gassmann Outline Motivation and review A four-stage investment problem OSiL format Conclusions and future work

3. © 2007 H.I. Gassmann Why a standard? Benchmarking Archiving Algorithm development Distributed computing Sharing of problem instances

4. © 2007 H.I. Gassmann Why XML? Existing parsers to check syntax Easy to generate automatically Tree structure naturally mirrors expression trees for nonlinear functions Arbitrary precision and name space Automatic attribute checking (e.g., nonnegativity) Querying capabilities via XQuery Encryption standards being developed Easy integration into broader IT infrastructure

5. © 2007 H.I. Gassmann Stochastic programs Any data item with nonzero subscript may be random (including dimensions where mathematically sensible) ~ stands for arbitrary relations ( , =,  ) “ ”  means ~ with probability 1 or with probability at least  or with expected violation at most v or …

6. © 2007 H.I. Gassmann Problem classes and time domain Single-stage problems –Mean-variance problems (Markowitz) –Robust optimization –Chance-constrained problems –Reformulated and solved as deterministic nonlinear problems

7. © 2007 H.I. Gassmann Problem classes and time domain (cont’d) Two-stage problems with recourse –Solved by Deterministic equivalent methods Benders decomposition Stochastic quasigradient methods Stochastic decomposition (Higle and Sen) Monte Carlo sampling (Shapiro and Homem-de-Mello) Regression approximation (Deák) –Distributions Known Approximated (by scenario generation) Partially known (moments, distribution type, support, etc.)

8. © 2007 H.I. Gassmann Problem classes and time domain (cont’d) Multi-stage recourse models –Deterministic equivalent –Nested Benders decomposition –Progressive hedging –Multistage stochastic decomposition –Probabilistic constraints and risk measures can be added as “linking constraints”

9. © 2007 H.I. Gassmann Problem classes and time domain (cont’d) Horizon problems (Grinold, Sethi) Markov reward processes Continuous time problems

10. © 2007 H.I. Gassmann Example (Birge) … S0S0 S2S2 S1S1 I = 2, T = 3, B = 55, R = 80,  t1 = {1.25, 1.06},  t2 = {1.14, 1.12}

11. © 2007 H.I. Gassmann Markup languages Intersperse text (data) and information about it (formatting, etc.) Examples –TeX (extensible through user \def) –HTML –VRML –XML

12. © 2007 H.I. Gassmann OSiL Schema Written in XML Very flexible Intended to handle as many types of mathematical programs as possible –Linear and integer –Nonlinear –Stochastic –…

13. © 2007 H.I. Gassmann OSiL Schema – Header information

14. © 2007 H.I. Gassmann Header information – Example <osil xmlns="os.optimizationservices.org“ xmlns:xsi=http://www.w3.org/2001/XMLSchemainstancehttp://www.w3.org/2001/XMLSchemainstance xsi:schemaLocation="OSiL.xsd"> FinancialPlan_JohnBirge Birge and Louveaux, Stochastic Programming Three-stage stochastic investment problem...

15. © 2007 H.I. Gassmann OSiL Schema – Deterministic data …

16. © 2007 H.I. Gassmann Instance data – Variables, objectives, constraints 1. -4.

17. © 2007 H.I. Gassmann Instance data – Core matrix (sparse matrix form) <linearConstraintCoefficients numberOfValues=“14"> 0 2 4 6 8 10 12 13 14 1 1.25 1 1.14 1 1.25 1 1.14 1 1.25 1 1.14 1 0 1 0 1 2 1 2 3 2 3

18. © 2007 H.I. Gassmann OSiL Schema – Dynamic structure

19. © 2007 H.I. Gassmann Dynamic information – Example

20. © 2007 H.I. Gassmann Explicit and implicit event trees

21. © 2007 H.I. Gassmann OSiL Schema – Scenario trees

22. © 2007 H.I. Gassmann Scenario tree – Example 1.06 1.12 …

23. © 2007 H.I. Gassmann Node-by-node representation for stochastic problem dimensions

24. © 2007 H.I. Gassmann Node-by-node – Example 1.06 1.12 …

25. © 2007 H.I. Gassmann Distributions (implicit tree)

26. © 2007 H.I. Gassmann Discrete random vector 0.5 1.25 1.14 0.5 1.06 1.12 …

27. © 2007 H.I. Gassmann General distribution (incomplete information)

28. © 2007 H.I. Gassmann Transformations Random variables separated from model entities Linked to stochastic problem elements by transformations (linear or nonlinear) Useful for factor models and other stochastic processes

29. © 2007 H.I. Gassmann OSiL schema – Linear transformations

30. © 2007 H.I. Gassmann Linear transformation – Example <stochasticElements numberOfElements="6"> <matrixCoefficients numberOfElements="6"> 0 1 2 3 4 5 6 0 1 2 3 4 1 1.0

31. © 2007 H.I. Gassmann Penalties and probabilistic constraints

32. © 2007 H.I. Gassmann Nonlinear expression – (2x 0 – x 1 2 ) 2 + (1 – x 0 ) 2

33. © 2007 H.I. Gassmann Capabilities Arbitrary nonlinear expressions Arbitrary distributions Scenario trees Stochastic problem dimensions Simple recourse Soft constraints with arbitrary penalties Probabilistic constraints Arbitrary moment constraints

34. © 2007 H.I. Gassmann Further work Internal data structures (OSInstance) OSiLWriter: SMPS  OSiL Library of problems (netlib, POSTS, Ariyawansa and Felt, Watson, SIPLib,…) Readers Solver interfaces Buy-in

35. © 2007 H.I. Gassmann QUESTIONS? http://www.optimizationservices.org http://myweb.dal.ca/gassmann