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Thoughts About Integer Programming

Thoughts About Integer Programming. University of Montreal, January 26, 2007. Integer Programming Max c x Ax=b Some or All x Integer. Why Does Integer Prrogramming Matter?. Navy Task Force Patterns in Stock Cutting Economies of Scale in Industries

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Thoughts About Integer Programming

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  1. Thoughts About Integer Programming University of Montreal, January 26, 2007

  2. IntegerProgrammingMax c xAx=bSome or All x Integer

  3. Why Does Integer Prrogramming Matter? • Navy Task Force • Patterns in Stock Cutting • Economies of Scale in Industries • Trade Theory – Conflicting National Interests

  4. WASTE CUTS Roll of Paper at Mill

  5. The Effect of the Number of Industries (8)

  6. The Effect of the Number of Industries (3)

  7. The Effect of the Number of Industries (2)

  8. How Do You Solve I.P’s? • Branch and Bound, Cutting Planes

  9. L.P.,I.P.and Corner Polyhedron

  10. I.P. and Corner Polyhedron • Integer Programs – Complex, no obvious structure • Corner Polyhedra – Highly Structured • We use Corner Polyhedra to generate cutting planes

  11. Equations

  12. T-Space

  13. Corner Polyhedra and Groups

  14. Structure of Corner Polyhedra I

  15. Structure of Corner Polyhedra II

  16. Shooting Theorem:

  17. Concentration of HitsEllis Johnson and Lisa Evans

  18. Cutting Planes From Corner Polyhedra

  19. Why Does this Work?

  20. Equations 2

  21. Why π(x) Produces the Equality • It is subadditive: π(x) + π(y)  π(x+y) • It has π(x) =1 at the goal point x=f0

  22. Cutting Planes are PlentifulHierarchy: Valid, Minimal, Facet

  23. Hierarchy

  24. Example: Two Facets

  25. Low is Good - High is Bad

  26. Example 3

  27. Gomory-Johnson Theorem

  28. 3-Slope Example

  29. Continuous Variables t

  30. Origin of Continuous Variables Procedure

  31. Integer versus Continuous • Integer Variables Case More Developed • But all of the more developed cutting planes are weaker than the Gomory Mixed Integer Cut with respect to continuous variables

  32. Comparing

  33. The Continuous Problem and A Theorem

  34. Cuts Provide Two Different Functions on the Real Line

  35. Start with Continuous Case

  36. Direction • Create continuous facets • Turn them into facets for the integer problem

  37. Helpful Theorem Theorem(?) If  is a facet of the continous problem, then (kv)=k (v). This will enable us to create 2-dimensional facets for the continuous problem.

  38. Creating 2D facets

  39. The triopoly figure

  40. This corresponds to

  41. The related periodic figure

  42. This Corresponds To

  43. Results for a Very Small Problem

  44. Gomory Mixed Integer Cuts

  45. 2D Cuts Added

  46. Summary • Corner Polyhedra are very structured • There is much to learn about them • It seems likely that that structure can can be exploited to produce better computations

  47. Challenges • Generalize cuts from 2D to n dimensions • Work with families of cutting planes (like stock cutting) • Introduce data fuzziness to exploit large facets and ignore small ones • Clarify issues about functions that are not piecewise linear.

  48. END

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