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

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Integer Programming Max c x Ax=b Some or All x Integer

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

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CUTS WASTE Roll of Paper at Mill

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The Effect of the Number of Industries (8)

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The Effect of the Number of Industries (3)

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The Effect of the Number of Industries (2)

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How Do You Solve I.Ps? Branch and Bound, Cutting Planes

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

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

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Equations

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T-Space

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Corner Polyhedra and Groups

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Structure of Corner Polyhedra I

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Structure of Corner Polyhedra II

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Shooting Theorem:

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Concentration of Hits Ellis Johnson and Lisa Evans

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Cutting Planes From Corner Polyhedra

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Why Does this Work?

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Equations 2

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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=f 0

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Cutting Planes are Plentiful Hierarchy: Valid, Minimal, Facet

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Hierarchy

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Example: Two Facets

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Low is Good - High is Bad

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Example 3

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Gomory-Johnson Theorem

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3-Slope Example

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Continuous Variables t

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Origin of Continuous Variables Procedure

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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

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Comparing

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The Continuous Problem and A Theorem

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Cuts Provide Two Different Functions on the Real Line

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Start with Continuous Case

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Direction Create continuous facets Turn them into facets for the integer problem

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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.

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Creating 2D facets

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The triopoly figure

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This corresponds to

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The related periodic figure

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This Corresponds To

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Results for a Very Small Problem

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Gomory Mixed Integer Cuts

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2D Cuts Added

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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

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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.

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END

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References Some Polyhedra Related to Combinatorial Problems, Journal of Linear Algebra and Its Applications, Vol. 2, No. 4, October 1969, pp.451-558 Some Continuous Functions Related to Corner Polyhedra, Part I with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 1, North-Holland, August, 1972, pp. 23-85. Some Continuous Functions Related to Corner Polyhedra, Part II with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 3, North-Holland, December 1972, pp. 359- 389. T-space and Cutting Planes Paper, with Ellis L. Johnson, Mathematical Programming, Ser. B 96: Springer-Verlag, pp 341-375 (2003).

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