Forty Years of Corner Polyhedra

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

Corner Polyhedra Origins Stock Cutting

• Computing Lots of Knapsacks

• Periodicity observed

• Gomory-Gilmore 1966 "The Theory and Computation of Knapsack Functions“

1 1

1 1

Integer Programming Equations

Corner Polyhedr

(Mod 1)

at basis B

on Relaxation

Variables x Integer

Non-negativity Relaxed on x

Bx Nt b

Ix B Nt B b

B Nt B b

Equations

V

L.P., I.P and Corner Polyhedron

Another View - T-Space

2 4 6 8 10t1

1

2

3

4

5

6

t2

Cutting Planes for Corner Polyhedra are Cutting Planes for

General I.P.

Valid, Minimal, Facet

T-Space View

2 4 6 8 10t1

1

2

3

4

5

6

t2

FMV

Cutting Planes for Corner Polyhedra

1 1

1 1

i

(Mod 1)

{ } and

a solution

Valid Cutting Plane; non-negative scalar ( )

( ) 1

subaadditive, normalized

i g

i i gi

i

i i g i ii

B Nt B b

B N v B b v

t v v

v

if t v v then t v

Structure Theorem- 1969

o

( ) / ( )

is a facet of the corner polyhedron

produced by G 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 M I M B

g g

, g')

Typical Structured Facescomputed using Balinski program

Size Problem :Shooting Geometry

2 4 6 8 10t1

1

2

3

4

5

6

t2

Size Problem -Shooting Theorem

0

the Facet solving the L.P.

min v

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

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

Is the Facet first hit by the random direction v

g g

Concentration of HitsEllis Johnson and Lisa Evans

Much More to be Learned

ComparingInteger 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

Cutting Planes for Type Two

• Example: Gomory Mixed Integer Cut

• Variables ti Integer

• Variables t+, t- Non-Integer

• Valid subadditive function

Typical Structured Faces

Interpolating to get cutting plane function on the real line

2 4 6 8 100

0.2

0.4

0.6

0.8

1

Interpolating

2 4 6 8 100

0.5

1

1.5

2

2.5

3

Interpolating

2 4 6 8 100

1

2

3

4

Gomory-Johnson Theorem

If (x) has only two slopes and satisfies

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

then it is a facet.

Integer Variables Example 2

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

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

Comparing

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

( ) 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

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

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

0.5

1

1.5

2

2.5

3

( )x

Gomory Mixed Integer CutContinuous Variables

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

-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 Intersecting

Shifting and Intersecting

i

i i

i

Cutting Plane; non-negative scalar ( )

( ) 1

If a t is integer, v can be changed by

an integer . So (v ) min ( )

shifting + intersecting

i

i i g i ii

i

v

if t v v then t v

v v

One Dimension Continuous Problem

1 1

All t continuous

Theorem: The Gomory Mixed Integer cut is the only

(Mod 1)

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

pure continuous one di

B Nt B b

mensional 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

Two Dimensional 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

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

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.

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

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

Thoughts About Integer Programming

University of Montreal, January 26, 2007 40th Birthday Celebration of the

Department of Computer Science and Operations Research

Corner Polyhedra and

2-Dimensional Cuttimg Planes

George Nemhauser Symposium

June 26-27 2007

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

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).