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Forty Years of Corner Polyhedra

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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. Corner Polyhedra relevant to both

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Corner Polyhedra Origins Stock Cutting Computing Lots of Knapsacks Periodicity observed Gomory-Gilmore 1966 "The Theory and Computation of Knapsack Functions

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Equations

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L.P., I.P and Corner Polyhedron

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Another View - T-Space

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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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T-Space View

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Cutting Planes for Corner Polyhedra

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Structure Theorem- 1969

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Typical Structured Faces computed using Balinski program

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Size Problem : Shooting Geometry

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Size Problem -Shooting Theorem

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Concentration of Hits Ellis Johnson and Lisa Evans

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Much More to be Learned

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Comparing Integer Programs and Corner Polyhedron General Integer Programs – Complex, no obvious structure Corner Polyhedra – Highly structured, but complexity increases rapidly with group size. Next Step: Making this supply of cutting planes available for non-integer data and continuous variables. Gomory-Johnson 1970

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Cutting Planes for Type Two Example: Gomory Mixed Integer Cut Variables t i Integer Variables t +, t - Non-Integer Valid subadditive function

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Typical Structured Faces

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Interpolating to get cutting plane function on the real line

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Interpolating

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Gomory-Johnson Theorem

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Integer Variables Example 2

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Integer Based Cuts A great variety of cutting planes generated from Integer Theory But more developed cutting planes weaker than the Gomory Mixed Integer Cut for their continuous variables

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Comparing

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Integer Cuts lead to Cuts for the Continuous Variables

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Gomory Mixed Integer Cut Continuous Variables

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New Direction Reverse the present Direction Create facets for continous variables Turn them into facets for the integer problem Montreal January 2007, Georgia Tech August 2007

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Start With Continuous x

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Create Integer Cut: Shifting and Intersecting

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Shifting and Intersecting

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One Dimension Continuous Problem

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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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Two Dimensional Periodic Figure

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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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But there are four sided figures too Corneujois and Margot have given a complete characterization of the two dimensional cutting planes for the pure continuous problem.

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All of the three sided polygons create Facets For the continuous problem 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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Thoughts About Integer Programming University of Montreal, January 26, th Birthday Celebration of the Department of Computer Science and Operations Research

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

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

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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 Some Continuous Functions Related to Corner Polyhedra, Part I with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 1, North-Holland, August, 1972, pp Some Continuous Functions Related to Corner Polyhedra, Part II with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 3, North-Holland, December 1972, pp T-space and Cutting Planes Paper, with Ellis L. Johnson, Mathematical Programming, Ser. B 96: Springer-Verlag, pp (2003).

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