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Corner Polyhedra and 2-Dimensional Cuttimg Planes George Nemhauser Symposium June 26-27 2007

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Integer Programming - Notation

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L.P., I.P and Corner Polyhedron

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Equations

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L.P., I.P and Corner Polyhedron

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Comparing Integer Programs and Corner Polyhedron General Integer Programs – Complex, no obvious structure Corner Polyhedra – Highly structured

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Cutting Planes for Corner Polyhedra are Cutting Planes for General I.P.

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Valid, Minimal, Facet

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Cutting Planes

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General Cutting Planes

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Two Types of I.P. All Variables (x,t) and data (B,N) integer. Example: Traveling Salesman Some Variables (x,t) Integer, some continuous, data continuous. Example: Scheduling,Economies of scale.

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First Type Data and Variables Integer

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Mod(1) B -1 N has exactly Det(B) distinct Columns v i

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Structure Theorem

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Typical Structured Faces

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Shooting Theorem

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Concentration of Hits Ellis Johnson and Lisa Evans

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Second Type: Data non-integer, some Variables Continuous

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Cutting Planes Must Be Created

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Cutting Planes Direct Construction Example: Gomory Mixed Integer Cut Variables t i Integer Variables t +, t - Non-Integer

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Integer Cuts lead to Cuts for the Continuous Variables

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Two Integer Variables Examples: Both are Facets

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Integer Variables Example 2

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Gomory-Johnson Theorem

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Integer versus Continuous Integer Theory More Developed But more developed cutting planes weaker than the Gomory Mixed Integer Cut for continuous variables

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Comparing

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New Direction Reverse the present Direction Create continuous facets Turn them into facets for the integer problem

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Start With Continuous x

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Create Integer Cut: Shifting and Minimizing

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The Continuous Problem and A Theorem

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Direction Move on to More Dimensions

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Helper 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 periodic figure

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The 2D Periodic figure- a facet

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One Periodic Unit

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Creating Another Facet

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The Periodic Figure - Another Facet

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More

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These are all Facets For the continuous problem (all the facets) For the Integer Problem For the General problem Two Dimensional analog of Gomory Mixed Integer Cut

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x i Integer t i Continuous

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Basis B

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Corner Polyhedron Equations

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T-Space Gomory Mixed Integer Cuts

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T- Space – some 2D Cuts Added

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Summary Corner Polyhedra are very structured The structure can be exploited to create the 2D facets analogous to the Gomory Mixed Integer Cut There is much more to learn about Corner Polyhedra and it is learnable

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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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Backup Slides

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One Periodic Unit

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Why π(x) Produces the Inequality It is subadditive π(x) + π(y) π(x+y) on the unit interval (Mod 1) It has π(x) =1 at the goal point x=f 0

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Origin of Continuous Variables Procedure

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Shifting

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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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