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

Integer Programming - Notation

L.P., I.P and Corner Polyhedron

Equations

L.P., I.P and Corner Polyhedron

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

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

Valid, Minimal, Facet

Cutting Planes

General Cutting Planes

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.

First Type Data and Variables Integer

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

Structure Theorem

Typical Structured Faces

Shooting Theorem

Concentration of Hits Ellis Johnson and Lisa Evans

Second Type: Data non-integer, some Variables Continuous

Cutting Planes Must Be Created

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

Integer Cuts lead to Cuts for the Continuous Variables

Two Integer Variables Examples: Both are Facets

Integer Variables Example 2

Gomory-Johnson Theorem

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

Comparing

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

Start With Continuous x

Create Integer Cut: Shifting and Minimizing

The Continuous Problem and A Theorem

Direction Move on to More Dimensions

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.

Creating 2D facets

The triopoly figure

This corresponds to

The periodic figure

The 2D Periodic figure- a facet

One Periodic Unit

Creating Another Facet

The Periodic Figure - Another Facet

More

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

x i Integer t i Continuous

Basis B

Corner Polyhedron Equations

T-Space Gomory Mixed Integer Cuts

T- Space – some 2D Cuts Added

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

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.

END

Backup Slides

One Periodic Unit

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

Origin of Continuous Variables Procedure

Shifting

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).