Projection and inverse projection as a method of reformulating

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```					Projection and inverse
projection as a method
of reformulating linear
and integer
H.P. Williams
programmes
London School of Economics

h.p.williams @ lse.ac.uk
www.lse.ac.uk/depts/op-research/personal/Williams   1
Example
(Maximise z)

Subject to:     4 x1  5 x2  3x3  z  0   C0
 x1  x2  x3      2      C1
x1  x2  2 x3     3     C 2
 x1                 0     C3
 x2          0     C 4
 x3  0      C5
Project out x1

2


 x2  7 x3  z  8     13x3  2 z  21       z  43
5 x2  3x3  z  0
    x3
7 x3  z
5
 15
   z 15
2 x2  x3      5
z  38
3
x2  2 x3        3      2 x3          3
x2 , x3  0                   x3  0                 3
How to carry out Projection
The following statements are equivalent:

x[ f i  x  g j ]                       all i, j : x  
fi  g j                               all i, j

Proof                               Immediate

              Take           x  Max f i 
i
 
(or x  Min g j )
j

(Decision Procedure of Langford for Theory of Dense Linear Order)   4
We can take fi and g i as linear expressions in the other
variables (apart from x)
The constraints of any linear programme can be put in
this form and x eliminated (projected out)
But need to combine every inequality of form " x  g j "
With every inequality of form " x  f i "

Can lead to combinatorial explosion in number of
inequalities.
Equations and associated variables can be eliminated
prior to this by Gaussian Elimination.                     5
Example
(Maximise z)

Subject to:   4 x1  5 x2  3x3  z   0   C0
 x1  x2  x3          2   C1
x1  x2  2 x3        3   C 2
 x1                    0   C3
 x2             0   C 4
 x3      0   C5

6
Write in form

 1
 2  x2  x3    5 x2  3x3  z 
 x1  4
0    3  x  2 x
    2     3

 x2  0
 x3  0

7
Eliminate x1

 2  x 2  x3 
1
5 x2  3 x3  z 
4
 2  x 2  x3    3  x 2  2 x3

0 
1
5 x2  3 x3  z 
4
0       3  x 2  2 x3
 x2      0
 x3      0

8
i.e.     x2  7 x3  z        8        C 0  4 C1
5 x2  3 x3  z  0             C 0  4 C3
2 x2  x3           5        C1  C 2
x2  2 x3             3        C 2  C3
 x2                   0        C4
 x3        0        C5

Have added (in suitable multiples) every inequality in which x1
has a positive coefficient to every inequality in which x1 has a
negative coefficient
9
C0        C1          C2           C3            C4        C5
4
1        1
4   1
1    1        1

1         1

Can continue process to eliminate x2 and x3
This is Fourier-Motzkin.
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+           -            +           -              0                 0
C0             C1           C2          C3             C4                C5

x2     -         -            +                +               -                0

x3     -          -            +           -
-         +              +             -

-                      +                     +               +          -
Z  43                Z  38 / 3              05             Z  15     03
Can be shown (Kohler) that if, after n variables have been eliminated, an inequality
depends on more than n+1 of original inequalities it is redundant.
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Can choose which variables to project out and order in
which to do so.

12
EXAMPLE OF PARTIAL
PROJECTION
Benders’ Decomposition

Partially project out some variables (for MIP, continuous
variables) to give bound on objective (Benders Cut)


Potentially very large increase in
constraints.
Performed partially and iteratively.
13
Application of Projection to Non-
Exponential Formulations of the
Travelling Salesman Problem

Example
Sequential Formulation (Miller, Tucker and Zemlin)
xij  1 iff i  j part of tour
ui  sequencenumber in which city is visited
(i  2,3,...,n)
x
i
ij   1          j

x
j
ij    1         i

ui  u j  n xij  n  1     i, j  1

0(n2 )                             Constraints and Variables   14
Project out         ui        variables
S
Gives         x
i , jS
ij    S 
n
all directed cycles

S   ,2..., n
1

A relaxation of Conventional Formulation

x
i , js
ij
 S 1                all proper subsets

S   ,2..., n
1
15
Other non-exponential formulations give
modifications of subtour elimination
constraints of (exponential)
conventional formulation
Single Commodity Flow Formulation gives

S
x            S          all S   ,2,...,n
1
n 1
ij
i , jS

Modified Single Commodity Flow Formulation

1                                 S
1 xij  i  xij  S  n  1
n  1 iS            S
jS        jS 1
16
Multi Commodity Flow Formulation

x
i , jS
ij
 S 1           i.e. of equal strength to
Conventional Flow
Formulation

Time Staged Formulation

1                  1                             S
1 xij  n  1 i xij  iS xij  S  n  1
n  1 iS                 S        , j
jS              jS 1

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INVERSE PROJECTION

Apply the dual procedures to eliminate constraints (as opposed to variables).

Minimise                2y1    +      3y2

subject to:              -y1   +       y2        -4

y1   +       y2           5

-y1   +      2y2           3

y1, y2  0

18
Write in form

Minimise       2y1 +        3y2

subject to:     4y0 -       y1 +      y2 -    y3 =   0

-5y0    + y1       +   y2 -     y4 =   0

-3y0 -     y1 +       2y2 -   y5 =   0

y0                              =    1

y1, y2, y3, y4, y5  0

Eliminate homogeneous constraints by adding columns (in suitable multiples) where
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coefficients have opposite sign.
Implemented by a transformation of variables.

4u1                y1
4y0

4u2

u3

y2                            y3
u4

First homogeneous constraint vanishes.

20
4y0   =   4u1   +   4u2
Substitute

y1   =   4u1   +    u3

y2   =    u3   +    u4

y3   =   4u2   +    u4

21
Model becomes

Minimise                     8u1   +   5u3   +   3u4

subject to               - u1      +   5u2   +   2u3   +    u4   -   y4   =   0

- 7u1     -   3u2   +    u3   +   2u4   -   y5   =   0

u1   +    u2                                =   1

u1, u2, u3, u4, y4, y5  0

22
Repeat with transformations to eliminate second and third homogeneous constraints.

Results in

Minimise      43w1   +   5w2   +   38/
3w3     +      15w4    +   3w5

w1              +         w3      +        w4              =   1

w1, w2, w3, w4, w5  0

23
Applying corresponding transformations to identity matrix gives relation between final
and original variables

w1       w2        w3        w4      w5

y0    1        0        1         1        0

y1   11        1        7/
3       0        0

y2    7        1
8/
3
5        1

13/
y3    0        0              3   9        1

y4   13                  0        0        1
2

y5    0        1         0        7        2

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Gives all vertices and extreme rays of model

y1   y2

C: w1 = 1 gives            (11     7) Objective = 43

w2 gives extreme ray ( 1      1)

B: w3 = 1 gives            (7/3   8/
3)   Objective = 38/3

A: w4 = 1 gives            ( 0     5) Objective = 15

w5 gives extreme ray ( 0      1)

25
NB:   Transformations mirror those of (Primal) Projection.
Redundant transformations if variable depends on more than n+1
of original variables after n constraints eliminated.     26
Example of Inverse Projection
(On some constraints)

Dantzig-Wolfe Decomposition

.
.
.
27
Project out subproblems into single (convexity) constraints

1111….1

1111….

.
.
.
1111….1

28
Also Modal Formulations

i.e. Variables represent extreme modes of processes rather than
quantities

NB Inverse Projection is not the same as Reverse Projection
(not well defined).

29
INTEGER PROJECTION
The Elimination of Integer Variables
 ax  f  0
a, b positive integers
bx  g  0
x an integer variable
f and g linear expressions


bf  abx   ag

bf   ag            i.e.      bf  ag  0

Need to state condition “a multiple of ab lies between bf and –ag”

30
Can be done in a finite way by

bf  bs  ag,               f  s  0 (mod a)
s 0,1,2,. . . , a - 1
i.e.
bf  ag  bs  0, f  s  0 (mod a)
s 0,1,2,. . . , a - 1
Alternatively

bf  ag  at  0, g  t  0 (mod b)
t                     
0,1,2,. . . , t - 1 ]

Presburger Arithmetic I.e. Arithmetic “without multiplication”)
31
Projection of an IP may not produce an IP in lower dimension
e.g. Project out x2

x1  7u

x1  u (mod3)
u  0,1, 2

32
But if coefficient of x is unity in one of inequalities can apply F-M
elimination x : f  x                       f ,g
Z               ax  g         integer expressions



x : af        ax        g
Z


af  g
NB The projection of an IP generally does not result in an IP in a lower
dimension
33
INTEGER PROJECTION
Example         Minimize z
z  5 x1  2 x2  0
2 x1  4 x2  5
6 x1  2 x2  5
 6 x1  2 x2  15
9 x1  3x2  10
i.e.     5  4 x2  5   10 x1  2  z  2 x2 
5  2 x2  5   30 x1  6  z  2 x2 
5  3 x2  10   45 x1  9  z  2 x2 
3  4 x2  5   6 x1  2 x2  15
2 x2  5  6 x1  2 x2  15
2  3 x2  10   18 x1  3(2 x2  15)
34
Eliminate x1
2 z  16 x2  25  2u
6 z  2 x2    25  6u
9 z  33x2   50  9u
14 x2      0v
4 x2   10  v
0      65  v

z  2 x2        u (mod5)   u  0,..., 4
2 x2  3  v (mod 6)    v  0,...,5

35
Eliminate x2 (taking into account congruence relations)

50 z  225  50u  8w
210 z       25  210u  16 s
126 z   700  126u  33v  14s
12 z       40  12u  v  2w
36 z   530  36u  33v  4s
42 z  175  42u  v  7 w
0  -65+v

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v  1(mod2)
v  w  0 (mod2)
v  w  2 (mod3)            u  0,..., 4
w  0 (mod5)            v  0,...,5
s  2 (mod3)             w  0,..., 29 
s     0 (mod5)          s  0,...,989     
6v  5 s  7 (mod9)
4 z  7u  2 s  1(mod11)

Optimal Solution        z  6, x1  0, x 2  3
u  0, v  3, w  5, s  5

Optimal Solution a Lattice Point within Polytope. Reduces problem
37
from an infinite number to a finite number of solutions.
C2

C3

C1
38
Inverse Integer Projection
Example       Minimize z         5 x1  2 x2

Subject to:        2 x1  4 x2  x3  5 y  0
6 x1  2 x2  x4  5 y  0
 6 x1  2 x2  x5  15 y  0
9 x1  3 x2  x6  10 y  0
y 1

x1 , x2 integer, x3 x4 x5 x6  0
39
4x2
u1

2x1               x3
u2

u3        5y

40
Eliminate constraint 1 by a transformation of variables

1
2

x1  u1  u 2  u 3   
1
x2  u1         (NB: x1 , x 2 , y not sign constrained)
4
x3  u 2     u 2  0 (u1u3 not sign constrained)
1
y  u3
2
u1  u2  u3  0 (mod 2)
u1           0 (mod 4)
u3  0 (mod 5)
41

Minimise
1
4u1  5u2  5u3 
2

Subject to:   5u1  6 u2  4u3  2 x4  0
7 u1  6 u2                2 x5    =0
21u1  18u2  26u3  4 x6           =0
u3 = 5

u1  u2  u3  0 (mod2)
u1                  0 (mod4)
x4 , x5 , x6 ,  0     (u1 , u3 not sign constrained)
Eliminate constraints 2,3,4
42

1
Minimise               253w1  259w2  345w3  351w4 
251160

Subject to:                     42w 2  23w3  65w 4  209300
23w1  7 w 2  69w3  39w 4  0 (mod 251160)
23w1  49w 2             26w 4  0 (mod 83720)

w1, w 2 , w3 , w 4  0
Optimal LP Solution. w4 = 3220 Objective = 4.5

Optimal IP Solution. w1  630, w2  4830, w3  280 objective 6
giving x1  0, x2  3 (at P) giving x1  0, x2  3     (At P)

N.B. Could replace congruence conditions by Mixed Integer Constraints.
Feasible Solutions defined by Convexity Constraint + Congruence Conditions
43
Illustrates Hilbert Basis result
Alternatively we could scale variables and write in form

Minimise
1
9041760

3036w11  185w21  450w31  162w41                   
Subject to: w2  w3  w4  1
1    1    1

w11  4550 w21  16380 w31  16380 w41                 0 (mod 32760)
w11  15925w21                                          0 (mod 10920)

w11 , w21 , w31 , w41  0

w11         corresponds to the extreme ray
and w21 , w31 , w41 to vertices

Optimal Solution:          w11  1890
w21  0.9692
(Objective  6)
w31  0.0307                             44
w 4
1
0
Projection and
Sign Patterns
+            -        Can remove columns and all constraints in
which non-negative entry.
+            -
:            :
+   or       -
0            0
0            0
:            :
0            0

+ +  + 0 0   0  -   Can remove constraints and all
variables in which non-negative
- -  - 0 0   0  +   entry

45

                                            Eg Network Flow
0
suitable multiple and remove           can remove node
0     variable


0
Integer Variable (and other variables integer)
1        -1
1        -1
Can regard variable as continuous
1        -1
        +
   or   +
:         :
         +
0         0
0         0         And generalisations of above
:         :
46
0         0
REFERENCES

1.   K.P. Martin Large Scale Linear and Integer Optimization : A Unified
Approach, Kluwer 1999
2.   Porta Version 2, Free Software Foundation, Boston 1991
3.   H.P. Williams (1986) Fourier’s Method of Linear Programming and its
Dual, Am. Math, Monthly, 93, 681-94
4.   H.P. Williams (1976) Fourier-Motzkin Extension to Integer Programming
Problems, Journal of Combinatorial Theory, 21, 118-123
5.   H.P. Williams (1983) A Characterisation of all Feasible Solutions to an
Integer Programme, Discrete Applied Mathematics, 5, 147-155

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