Thoughts About Integer Programming

University of Montreal, January 26, 2007

Integer Programming

Max c xAx=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)

Eight Industries 8pnew

0 .2 .4 .6 .8 1

1 1

Z 1

U 1U 2

Z C

The Effect of the Number of Industries (3)

Three Industries 3pnew

0 .2 .4 .6 .8 1

1 1

Z 1

U 1U 2

Z C

The Effect of the Number of Industries (2)

Two Industries 2ptest

0 .2 .4 .6 .8 1

1 1

Z 1

U 1U 2

Z C

How Do You Solve I.P’s?

• Branch and Bound, Cutting Planes

V

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

1

1 1

1

Equa

Equations for Corner Polyhedr

tions for Integer Program

(Mod 1

ming

)

on

Bx Nt b

x B Nt B b

B Nt B b

Equations

T-Space

2 4 6 8 10t1

1

2

3

4

5

6

t2

1 1

-1

Equations for Corner Polyhedr

(Mod 1)

Add and Subtract Columns of

This forms group G

B N

on

B Nt B b

Corner Polyhedra and Groups

Structure of Corner Polyhedra I

Structure of Corner Polyhedra II

Shooting Theorem:

Concentration of HitsEllis Johnson and Lisa Evans

Cutting Planes From Corner Polyhedra

2 4 6 8 10t1

1

2

3

4

5

6

t2

1

1 1

1

Equa

Equations for Corner Polyhedr

tions for Integer Program

(Mod 1

ming

)

on

Bx Nt b

x B Nt B b

B Nt B b

Why Does this Work?

Equations 2

1 1

0i

0i

(Mod 1)

(Mod1). ake fractional parts

(Mod1

Select a single row of

Any t of the Corner Polyhedron must satisfy

this relatio

)

n

T

i i

i i

B Nt B b

c t c

f t f

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=f0

0i

0i

Since any t of the Corner Polyhedron must satisfy

(Mod1), we have

( ) ( ) 1

i i

i i

f t f

f t f

Cutting Planes are PlentifulHierarchy: Valid, Minimal, Facet

2 4 6 8 10t1

1

2

3

4

5

6

t2

Hierarchy

Subadditive π(u1)+ π(u2) π(u1+u2)

Minimal π(u1)+π(u0-u1 ) = 1 (Normalization π(u0) =1)

Facet π(u1)+π(u2) = π(u1+u2)if whenever π (u1)+π (u2) = π (u1+u2)implies π(u1)=π(u1)

Example: Two Facets

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Low is Good - High is Bad

Example 3

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Gomory-Johnson Theorem

If (x) has only two slopes and satisfies

the minimality condition (x)+ (1-x)=1

then it is a facet.

3-Slope Example

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Continuous Variables t

Origin of Continuous Variables Procedure

0 0i

i

i

If for some t then ( / )( )

For large apply ; the result is (( / )) ( ) 1

( ) ) 1

( ) 0 ( ) 0.

i i i i i ii

i i i i i

i i

i i

c t c c k k t c

k c k k t

s c t

where s c s c for x and s x s x for x

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

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

The Continuous Problem and A Theorem

1 1

Pure Continuous Problem: All t continuous

T - Space looks different - Lines and Sheets as well as dots

:The Gomory Mixed Integer Cut is the only

(Mod 1)

Theore m(?)

cut that is a facet for both

B Nt B b

the pure integer and the

pure continuous one dimensional problems.

Cuts Provide Two Different Functions on the Real Line

-2 -1.5 -1 -0.5 0.5 1 1.5 2

0.5

1

1.5

2

2.5

3

Start with Continuous Case

-2 -1.5 -1 -0.5 0.5 1 1.5 2

0.5

1

1.5

2

2.5

3

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

-1.5 -1 -0.5 0.5 1 1.5 2

-1.5

-1

-0.5

0.5

1

1.5

The triopoly figure

0 1 2

-0.5

0

0.5

00.250.50.751

-0.5

0

0.5

This corresponds to

-2 -1.5 -1 -0.5 0.5 1 1.5 2

0.5

1

1.5

2

2.5

3

The related periodic figure

-1

0

1

2

XXX

-1

0

1

2

YYY

00.250.50.751ZZZ

-1

0

1

2

YYY

00.250.50.751ZZZ

This Corresponds To

-2 -1.5 -1 -0.5 0.5 1 1.5 2

0.5

1

1.5

2

2.5

3

Results for a Very Small Problem

Gomory Mixed Integer Cuts

1 2 3 4t1

1

2

3

4

t2

2D Cuts Added

1 2 3 4t1

1

2

3

4

t2

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.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).