projection and inverse projection as a method of reformulating linear and integer programmes
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Projection and inverse projection as a method of reformulating linear and integer programmes. H.P. Williams London School of Economics. h.p.williams @ lse.ac.uk www.lse.ac.uk/depts/op-research/personal/Williams. Example (Maximise z ). Subject to:. Project out x 1. - PowerPoint PPT PresentationTRANSCRIPT
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Projection and inverse Projection and inverse projection as a method projection as a method of reformulating linear of reformulating linear
and integer and integer programmesprogrammesH.P. WilliamsH.P. Williams
London School of EconomicsLondon School of Economics
h.p.williams @ lse.ac.uk
www.lse.ac.uk/depts/op-research/personal/Williams
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ExampleExample(Maximise (Maximise zz))
Subject to:
Project out x1
1 2 3
1 2 3
1 2 3
1
2
3
4 5 3 0 0
2 1
2 3 2
0 3
0 4
0 5
x x x z C
x x x C
x x x C
x C
x C
x C
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3
2 3
2 3
2 3
2 3
2 3
7 8
5 3 0
2 5
2 3
, 0
x x z
x x z
x x
x x
x x
3
3
3
3
3
13 2 21
5
7 15
2 3
0
x z
x
x z
x
x
383
43
15
z
z
z
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How to carry out ProjectionHow to carry out Projection
The following statements are equivalent:
jigf
xjigxfx
ji
ji
, all
:, all ][
Proof Immediate
Take )( j
j
ii
gMinx
fMaxx
or
(Decision Procedure of Langford for Theory of Dense Linear Order)
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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
With every inequality of form
"" jgx
"" ifx
Can lead to combinatorial explosion in number of inequalities.
Equations and associated variables can be eliminated prior to this by Gaussian Elimination.
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ExampleExample(Maximise (Maximise zz))
Subject to:1 2 3
1 2 3
1 2 3
1
2
3
4 5 3 0 0
2 1
2 3 2
0 3
0 4
0 5
x x x z C
x x x C
x x x C
x C
x C
x C
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Write in formWrite in form
32
321
32
23
354
1
0
2
xx
zxxxxx
0
0
3
2
x
x
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Eliminate Eliminate xx11
0
0
230
354
10
232
354
12
3
2
32
32
3232
3232
x
x
xx
zxx
xxxx
zxxxx
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Have added (in suitable multiples) every inequality in which x1 has a positive coefficient to every inequality in which x1 has a negative coefficient
i.e. 2 3
2 3
2 3
2 3
2
3
7 8 0 4 1
5 3 0 0 4 3
2 5 1 2
2 3 2 3
0 4
0 5
x x z C C
x x z C C
x x C C
x x C C
x C
x C
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C0 C1 C2 C3 C4 C5
1 1 11
1
1 1
4
Can continue process to eliminate x2 and x3
This is Fourier-Motzkin.
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C0 C1 C2 C3 C4 C5
-- + + - 0
+ +- - 0 0
x2
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.
- + + + -
3/38Z 50 15Z 30 43Z
- -- - + - + -x3+
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Can choose which variables to project out and order in which to do so.
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EXAMPLE OF PARTIAL EXAMPLE OF PARTIAL PROJECTIONPROJECTIONBenders’ 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.
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Application of Projection to Non-Application of Projection to Non-Exponential Formulations of the Exponential Formulations of the
Travelling Salesman ProblemTravelling Salesman Problem
Example
Sequential Formulation (Miller, Tucker and Zemlin)
1,1
1
1
),...,3,2(
visitediscity in which number sequence
tourofpart 1
jinxnuu
ix
jx
ni
u
jiiffx
ijji
jij
iij
i
ij
Constraints and Variables
)(0 2n
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Project out iu variables
Gives n
SSx
Sjiij
,
all directed cycles
A relaxation of Conventional Formulation
all proper subsets
1,
Sxsji
ij
nS ...,2,1
nS ...,2,1
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Other non-exponential formulations give Other non-exponential formulations give modifications of subtour elimination modifications of subtour elimination constraints of (exponential) constraints of (exponential) conventional formulationconventional formulation
Single Commodity Flow Formulation gives
nSn
SSx
Sjiij ,...,2,1 all
1,
Modified Single Commodity Flow Formulation
11
1
11
n
SSxx
nSjSi
SjSi
ijij
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1,
SxSji
ij
Multi Commodity Flow Formulation
i.e. of equal strength to Conventional Flow Formulation
Time Staged Formulation
11
1
1
1
,11
n
SSxx
nx
n Sjiij
SjSi
SjSi
ijij
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Minimise 2y1 + 3y2
subject to: -y1 + y2 4
y1 + y2 5
-y1 + 2y2 3
y1, y2 0
INVERSE PROJECTION Apply the dual procedures to eliminate constraints (as opposed to variables).
-
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Write in form
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
Minimise 2y1 + 3y2
Eliminate homogeneous constraints by adding columns (in suitable multiples) where coefficients have opposite sign.
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Implemented by a transformation of variables.
4y0
y2y3
y14u1
4u2
u3
u4
First homogeneous constraint vanishes.
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Substitute 4y0 = 4u1 + 4u2
y1 = 4u1 + u3
y2 = u3 + u4
y3 = 4u2 + u4
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Model becomes
Minimise 8u1 + 5u
3
+ 3u
4
subject to - u1
+ 5u
2
+ 2u
3
+ u4 - y
4
= 0
- 7u1
- 3u
2
+ u3 + 2u
4
- y
5
= 0
u1 + u2 = 1
u1, u2, u3, u4, y4, y5 0
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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
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Applying corresponding transformations to identity matrix gives relation between final and original variables
y0 1 0
1 1 0
y1 11 1 7/3
0 0
y2 7 18/
35 1
y3 0 013/
3 9 1
y4 13 20 0 1
y5 0 1 0 7 2
w1 w2 w3 w4 w5
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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)
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NB: Transformations mirror those of (Primal) Projection.
Redundant transformations if variable depends on more than n+1
of original variables after n constraints eliminated.
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Example of Inverse Projection Example of Inverse Projection (On some constraints)
Dantzig-Wolfe Decomposition
.
.
.
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Project out subproblems into single (convexity) constraints
1111….1
1111….
.
.
.
1111….1
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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).
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INTEGER PROJECTIONINTEGER PROJECTION
0..
0
0
agbfeiagbf
agabxbf
gbx
fax
a, b positive integers
x an integer variable
f and g linear expressions
Need to state condition “a multiple of ab lies between bf and –ag”
The Elimination of Integer Variables
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Can be done in a finite way by
]1-t, . . . 0,1,2,t
b) (mod 0,0
1-a, . . . 0,1,2,s
a) (mod 0,0
i.e.
1-a, . . . 0,1,2,s
a)(mod0,
tgatagbf
sfbsagbf
sfagbsbf
Presburger Arithmetic I.e. Arithmetic “without multiplication”)
Alternatively
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Projection of an IP may not produce an IP in lower dimension
e.g. Project out x2
1
1
7
(mod3)
0,1,2
x u
x u
u
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But if coefficient of x is unity in one of inequalities can apply F-M elimination
gaf
Z
gaxafx
gaxZ
gfxfx
:
sexpressioninteger
,:
NB The projection of an IP generally does not result in an IP in a lower dimension
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INTEGER PROJECTION
Example Minimize z
i.e.
2 1 2
2 1 2
2 1 2
2 1 2
2 1 2
2 1 2
5 4 5 10 2 2
5 2 5 30 6 2
5 3 10 45 9 2
3 4 5 6 2 15
2 5 6 2 15
2 3 10 18 3(2 15)
x x z x
x x z x
x x z x
x x x
x x x
x x x
1 2
1 2
1 2
1 2
1 2
5 2 0
2 4 5
6 2 5
6 2 15
9 3 10
z x x
x x
x x
x x
x x
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Eliminate 1x
2
2
2
2
2
2
2
2 16 25 2
6 2 25 6
9 33 50 9
14 0
4 10
0 65
2 (mod5) 0,..., 4
2 3 (mod 6) 0,...,5
z x u
z x u
z x u
x v
x v
v
z x u u
x v v
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Eliminate x2 (taking into account congruence relations)
50 225 50 8
210 25 210 16
126 700 126 33 14
12 40 12 2
36 530 36 33 4
42 175 42 7
0 -65+v
z u w
z u s
z u v s
z u v w
z u v s
z u v w
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Optimal Solution5,5,3,0
3,0,6 21
swvu
xxz
Optimal Solution a Lattice Point within Polytope. Reduces problem from an infinite number to a finite number of solutions.
1(mod2)
v 0(mod2)
v 2(mod3) 0,..., 4
0(mod5) 0,...,5
2(mod3) 0,..., 29
0(mod5) 0,...,989
6 5 7 (mod9)
4 7 2 1(mod11)
v
w
w u
w v
s w
s s
v s
z u s
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C1
C2
C3
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Inverse Integer Projection Inverse Integer Projection
0integer,,
1
01039
01526
0526
0542
654321
621
521
421
321
xxxxxx
y
yxxx
yxxx
yxxx
yxxx
Example Minimize z 21 25 xx
Subject to:
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24x
3x
1u
2u
3u
12x
5y
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Eliminate constraint 1 by a transformation of variables
1 2 3
1 1 2
2 2
3
1
2
3 1 3
1 2 3
1
3
1
21
(NB: , , not sign constrained)4
0 ( not sign constrained)
1
2 0 (mod 2)
0 (mod 4)
0 (mod 5)
x u u u
x u x x y
x u u u u
y u
u u u
u
u
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Leads to:
Minimise 321 5542
1uuu
Subject to:
Eliminate constraints 2,3,4
1 2 3 4
1 2 5
1 2 3 6
3
1 2 3
1
4 5 6 1 3
5 6 4 2 0
7 6 2 0
21 18 26 4 0
5
0 (mod2)
0 (mod4)
, , , 0 ( , not sign constrained)
u u u x
u u x
u u u x
u
u u u
u
x x x u u
=
=
=
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Leads to:
Minimise
Subject to:
0,,,
(0264923
(03969723
209300652342
4321
421
4321
432
wwww
www
wwww
www
83720) mod
251160) mod
Optimal LP Solution. w4 = 3220 Objective = 4.5
N.B. Could replace congruence conditions by Mixed Integer Constraints.
Feasible Solutions defined by Convexity Constraint + Congruence Conditions
Illustrates Hilbert Basis result
1 2 3 4
1253 259 345 351
251160w w w w
1 2 3
1 2
Optimal IP Solution. 630, 4830, 280 objective 6
giving 0, 3 (at P)
w w w
x x
1 2giving 0, 3 (At P)x x
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Alternatively we could scale variables and write in form
14
13
12
11 1624501853036
9041760
1wwww Minimise
Subject to:
corresponds to the extreme ray
1 1 1 11 2 3 4
1 11 2
1 1 1 11 2 3 4
11
1 1 12 3 4
11
12
13
14
4550 16380 16380 0 (mod 32760)
15925 0 (mod 10920)
, , , 0
and , , to vertices
Optimal Solution: 1890
0.9692
(Objective 6)
0.0307
0
w w w w
w w
w w w w
w
w w w
w
w
w
w
1 1 12 3 4 1w w w
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Projection and
Sign Patterns
+ -
+ -
: :
+ or -
0 0
0 0
: :
0 0
Can remove columns and all constraints in which non-negative entry.
+ + + 0 0 0 -
- - - 0 0 0 +
Can remove constraints and all variables in which non-negative entry
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0
0
0
Can add constraints in suitable multiple and remove variable
Eg Network Flow
can remove node
Integer Variable (and other variables integer)
1 -1
1 -1
1 -1
+
or +
: :
+
0 0
0 0
: :
0 0
Can regard variable as continuous
And generalisations of above
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REFERENCESREFERENCES
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