corner polyhedra and 2-dimensional cuttimg planes george nemhauser symposium june 26-27 2007
TRANSCRIPT
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).