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 Trade Theory – Conflicting National Interests

CUTS WASTE Roll of Paper at Mill

The Effect of the Number of Industries (8)

The Effect of the Number of Industries (3)

The Effect of the Number of Industries (2)

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

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

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

Equations

T-Space

Corner Polyhedra and Groups

Structure of Corner Polyhedra I

Structure of Corner Polyhedra II

Shooting Theorem:

Concentration of Hits Ellis Johnson and Lisa Evans

Cutting Planes From Corner Polyhedra

Why Does this Work?

Equations 2

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

Cutting Planes are Plentiful Hierarchy: Valid, Minimal, Facet

Hierarchy

Example: Two Facets

Low is Good - High is Bad

Example 3

Gomory-Johnson Theorem

3-Slope Example

Continuous Variables t

Origin of Continuous Variables Procedure

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

Comparing

The Continuous Problem and A Theorem

Cuts Provide Two Different Functions on the Real Line

Start with Continuous Case

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

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.

Creating 2D facets

The triopoly figure

This corresponds to

The related periodic figure

This Corresponds To

Results for a Very Small Problem

Gomory Mixed Integer Cuts

2D Cuts Added

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

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

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).