Corner Polyhedra and

2-Dimensional Cuttimg Planes

George Nemhauser Symposium

June 26-27 2007

Integer Programming - Notation

Some or all of (x,t) Integer

(x,t) Non-Negative

Max cx

Bx Nt b

V

L.P., I.P and Corner Polyhedron

1 1

1 1

Corner Polyhedr

Integer Programming

(Mod 1)

on at basis B

Variables x Integer

Non-negativity Relaxed on

;

at ba

x

sis B

Bx Nt b

Ix

B Nt B

B Nt B b

b

Equations

V

L.P., I.P and Corner Polyhedron

ComparingInteger 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

1 1

1 1

i

(Mod 1)

{ } and

Cutting Plane; non-negative scalar ( )

( ) 1

i g

i

i i g i ii

B Nt B b

B N v B b v

v

if t v v then t v

General Cutting Planes

i

Additive group G (ususally N space)

with elements v

Non-Negative ( ) such that

For any {t } from the origin to

the path ( ) 1.

g

i i gi

i ii

v

path v

t v v

length t v

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.

1 1

1 1

-1

Equations for Corner Polyhedr

(Mod 1)

(Mod 1)

Add and Subtract Columns of

This forms group G

on

B N

Ix B Nt B b

B Nt B b

First Type Data and Variables Integer

11 1 1

2 2 2 2

3 3 3 3

4 4 4 4

i

fc n f

c n f fv

c n f f

c n f f

Mod(1) B-1N has exactly Det(B) distinct

Columns vi

Structure Theorem

o

is a facet if and only if it is a basic feasible

solution of this list of equations and inequalities

(g)+ (g-g ) 1 (all g)

(g)+ (g') ( ') (all g, g')g g

Typical Structured Faces

Shooting Theorem

0

The Facet first hit by the random direction v

is the Facet solving the L.P.

min vg

(g)+ (g -g) 1 (all g)

(g)+ (g') ( ') (all g, g')g g

Concentration of HitsEllis Johnson and Lisa Evans

Second Type: Data non-integer , some Variables Continuous

G is n-space, elements v are n-vectors

Cutting Plane is Non-Negative ( ) such that

For any

( ) 1.

i i gi

i ii

v

path t v v

t v

Cutting Planes Must Be Created

,1

,2

i ,3 ,

,

Usually only one equation is used

From the n dimensional equation

If v ; ( ) ( )

.

.

i i gi

i

i

i i i j

i n

t v v

v

v

v v v

v

Cutting Planes Direct Construction

• Example: Gomory Mixed Integer Cut

• Variables ti Integer

• Variables t+, t- Non-Integer

( ) Gomory Mixed Integer Cut

Integer Variables

x

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

0.5

1

1.5

2

2.5

3

( ) Gomory Mixed Integer Cut

Continuous Variables

x

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

0.5

1

1.5

2

2.5

3

Integer Cuts lead to Cuts for the Continuous Variables

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

0.5

1

1.5

2

2.5

3

Two Integer Variables Examples: Both are Facets

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Integer Variables Example 2

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.

Integer versus Continuous

• Integer Theory More Developed

• But more developed cutting planes weaker than the Gomory Mixed Integer Cut for continuous variables

Comparing

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

New Direction

• Reverse the present Direction

• Create continuous facets

• Turn them into facets for the integer problem

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

0.5

1

1.5

2

2.5

3

Start With Continuous x

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

0.5

1

1.5

2

2.5

3

Create Integer Cut: Shifting and Minimizing

The Continuous Problem and A Theorem

1 1

Pure Continuous Problem: All t continuous

:The Gomory

(Mo

Mixed

d 1)

Theor Integer em Cut is the only

cutting plane that is a facet for both the pure integer and the

B Nt B b

pure continuous one dimensional problems.

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

-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 periodic figure

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

0.5

1

1.5

2

2.5

3

The 2D Periodic figure- a facet-1

0

1

2

XXX

-1

0

1

2

YYY

00.250.50.751ZZZ

-1

0

1

2

YYY

00.250.50.751ZZZ

One Periodic Unit

Creating Another Facet

-1 1 2 3

-1.5

-1

-0.5

0.5

1

1.5

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

xi Integer ti Continuous

1 1

2 2

x 0.34, 1.12 -0.11, 1.01 1.10+

-0.35, 0.44 0.70, -0.44 0.14

Bx+Nt=b

t

x t

Basis B

1 1

1 1

2 2 2

1 0 0.75, 0.15 0.6

0 1 0,35, 0.55 0.8

Ix B N B b

x t

x t

Corner Polyhedron Equations

1

2 2

1 1

0.75, 0.15 0.6

0.35, 0.55 0.8

t

t

B Nt B b

T-SpaceGomory Mixed Integer Cuts

1 2 3 4t1

1

2

3

4

t2

T- Space – some 2D Cuts Added

1 2 3 4t1

1

2

3

4

t2

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=f0

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

Shifting

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).