The Chv

1
The Chvátal Dual of a Pure Integer Programme

• H.P.Williams
• London School of Economics

2
Duality in LP and IP

• The Value Function of an LP
• Minimise
  x2
• 
  subject to 2x1 x2 gt b1
• 
  
  5x1 2x2 lt b2
• 
  
  -x1 x2 gt b3
• 
  
  x1 , x2 gt 0
• Value Function of LP is Max( 5b1 - 2b2 , 1/3(
  b1 2b3) , b3)
• If b1 13, b2 30, b3 5 we have Max( 5,
  72/3 , 5 ) 72/3
• ,
• Consistency Tester is Max( 2b1 b2 , -b2 , -b2
  2b3 ) lt 0 giving Max( -4, -30, -20) lt 0 .
• (5, -2, 0), (1/3, 0, 2/3), (0, 0, 1) are
  vertices of Dual Polytope .

3
Duality in LP and IP

• The Value Function of an IP
• 
  Minimise x2
• subject to
  2x1 x2 gt b1
• 
  5x1 2x2 lt b2
• 
  -x1 x2 gt
  b3
• 
  x1 , x2
  gt 0 and integer
• Value Function of IP is
• Max( 5b1 - 2b2 ,1/3( b1 2b3) , b3 , b1
  2 1/5 (-b2 2 1/3(b1 2b3) ) )
• This is known as a Gomory Function.
• The component expressions are known as Chv?tal
  Functions .
• Consistency Tester same as for LP (in this
  example)

4
IP Solution

• 9 Optimal IP Solution (2 , 9) .
  Min x2
• c3
  st 2x1 x2 gt 13
• 8 . . c1 . .
  5x1 2x2 lt 30
• Optimal LP Solution (2 2/3 , 7 2/3)
  
  -x1 x2 gt 5
• 7 . . . . c2 .
  x1 , x2 gt 0
• x2
• 6 . . . .
  .
• 5 . . . . .
  
• 0 1 2 3 4
  x1

5
IP Solution after removing constraint 1

• 
  Min x2
• 
  
• 8 c1 . . . c3
  st 5x1 2x2 lt 30
• 
  
  
  -x1 x2 gt 5
• . . .
  . x1 , x2 gt 0
• x2 7 . . .
• 6 . . . .
  . c2
• Optimal IP Solution (0 , 5)
  
• 5
  
• 0 1 2 3
  4 x1

6
IP Solution

• 9 Optimal IP Solution (2 , 9) .
  Min x2
• c3
  st 2x1 x2 gt 13
• 8 . . c1 . .
  5x1 2x2 lt 30
• Optimal LP Solution (2 2/3 , 7 2/3)
  
  -x1 x2 gt 5
• 7 . . . . c2 .
  x1 , x2 gt 0
• x2
• 6 . . . .
  .
• 5 . . . . .
  
• 0 1 2 3 4
  x1

7
IP Solution after removing constraint 2

• 9 . . .
  Min x2
• c1 c3
  st 2x1 x2 gt 13
• 8 . . . . Optimal IP
  Solution (3, 8)
• 
  -x1 x2 gt 5
• 7 . . . .
  . x1 , x2 gt 0
• x2
• 6 . . . .
  .
• 5 . . . . .
  x1
• 0 1 2 3 4

8
IP Solution

• 9 Optimal IP Solution (2 , 9) .
  Min x2
• c3
  st 2x1 x2 gt 13
• 8 . . c1 . .
  5x1 2x2 lt 30
• Optimal LP Solution (2 2/3 , 7 2/3)
  
  -x1 x2 gt 5
• 7 . . . . c2 .
  x1 , x2 gt 0
• x2
• 6 . . . .
  .
• 5 . . . . .
  
• 0 1 2 3 4
  x1

9
IP Solution after removing constraint 3

• 9 . . . .
  Min x2
• 
  st 2x1 x2 gt 13
• 8 . . . . .
  5x1 2x2 lt 30
• c1 c2
  x1 ,
  x2 gt 0
• 7 . . . .
  .
• x2
• 6 . . . .
  .
• 5 . . . .
  .Optimal IP Solution (4 , 5)
• 0 1 2 3 4
  x1

10
IP Solution

• 9 Optimal IP Solution (2 , 9) .
  Min x2
• c3
  st 2x1 x2 gt 13
• 8 . . c1 . .
  5x1 2x2 lt 30
• Optimal LP Solution (2 2/3 , 7 2/3)
  
  -x1 x2 gt 5
• 7 . . . . c2 .
  x1 , x2 gt 0
• x2
• 6 . . . .
  .
• 5 . . . . .
  x1
• 0 1 2 3 4

11
Gomory and Chvátal Functions

• Max( 5b1-2b2, 1/3(b1 2b3), b3 , b1 2 1/5
  (-b2 2 1/3(b1 2b3) ) )
• If b113, b230, b35 we have Max(5,8,5,9)9
• Chvátal Function b1 2 1/5 (-b2 2 1/3(b1
  2b3) ) determines
• the optimum.
• LP Relaxation is 19/15 b1 - 2/5 b2 8/15 b2
• (19/15, -2/5, 8/15) is an interior point of dual
  polytope but
• (5, -2, 0), (1/3, 0, 2/3) and (0,0,1) are
  vertices corresponding to possible
• LP optima (for different bi )

12
Pricing by optimal Chvátal FunctionIntroduce new
variable X3

• Function
• ?b1 ?b31
• prices out X3
• Solution
• X1 22/3 , X2 72/3 , X3 0

• LP Case
• Minimise X2 1.5X3
• Subject to 2X1X2 X3 gt13
• 5X12X2X3lt30
• -X1X2 X3 gt5
• X1 , X2 gt 0

IP Case Minimise X2 1.5X3 Subject to 2X1X2 X3
gt13 5X12X2X3lt30
-X1X2 X3 gt5 X1 , X2
gt 0 and integer
Function b1 2 1/5 (-b2 2 1/3(b1 2b3))
3 does not price out X3 Solution X1 3,
X2 7, X3 1
13
Why are valuations on discrete resources of
interest ?

• Allocation of Fixed Costs
• Maximise ?j pi xi - f y
• st xi - Di y lt 0
  for all I
• y e 0,1 depending on whether facility built.
• f is fixed cost.
• xi is level of service provided to i (up to level
  Di )
• pi is unit profit to i.
• A dual value vi on xi - Di y lt 0 would
  result in
• Maximise ?j (pi vi ) xi - (f
  (?i D i v i) y
• Ie an allocation of the fixed cost back to the
  consumers

14
A Representation for Chvátal Functions

• 
  
  b1 b3
  - b2
• 
  
  
  1
  2
• Multiply and add
• on arcs
  
  1
• 
  1
  
• Divide and round
  
• up on nodes
• 
  
  
• 
  
  
  
• 
  2
  

3
5
1
15
Simplifications sometimes possible

• 2/7 7/3 n 2/3 n
  
• But 7/3 2/7 n ? 2/3 n
  eg n 1
• 1/3 5/6 n
  5/18 n
• But 2/3 5/6 n ? 5/9 n
  eg n 5
• Is there a Normal Form ?

16
Properties of Chvátal Functions

• They involve non-negative linear combinations
  (with possibly negative coefficients on the
  arguments) and nested integer round-up.
• They obey the triangle inequality.
• They are shift-periodic ie value is increased in
  cyclic pattern with increases in value of
  arguments.
• They take the place of inequalities to define
  non-polyhedral integer monoids.

17
The Triangle Inequality

• a b gt a b
• Hence of value in defining Discrete Metrics

18
A Shift Periodic Chvátal Function of one argument

• ½ ( x 3 x /9 ) is (9, 6) Shift
  Periodic. 2/3 is long-run marginal value
• 14
• 13
• 12
• 11
• 10
• 9
• 8

19
Polyhedral and Non-Polyhedral Monoids

• The integer lattice within the polytope -2x
  7y gt 0
• 
  x 3y gt 0
• A Polyhedral Monoid
• 4 . . . . . . . . . .
  . . . . .
• 3 . . . . . . . . . .
  . . . . .
• 2 . . . . . . . . . .
  . . . . .
• 1 . . . . . . . . .
  . . . . . . .
• 0 . . . . . . . . . .
  . . . . .
• 0 1 2 3 4 5
  6 7 8 9 10 11 12
  13 14
• Projection A Non-Polyhedral Monoid (Generators
  3 and 7)
• x . . x . . x x . x
  x . x x x .
• Defined by -x /3 2x /7 lt 0

20
Calculating the optimal Chvátal Function over a
Cone

• Value Function over a Cone is a Chvátal
• Function

21
IP Solution

• 9 Optimal IP Solution (2 , 9) .
  Min x2
• c3
  st 2x1 x2 gt 13
• 8 . . c1 . .
  5x1 2x2 lt 30
• Optimal LP Solution (2 2/3 , 7 2/3)
  
  -x1 x2 gt 5
• 7 . . . . c2 .
  x1 , x2 gt 0
• x2
• 6 . . . .
  .
• 5 . . . . .
  
• 0 1 2 3 4
  x1

22
An Example

• Minimise x2
• subject to 2x1 x2 gt b1
• -x1 x2 gt
  b3 x1, x2 integer
• These are constraints which are binding at LP
  Optimum.
• Convert 1st 2 rows to Hermite Normal Form by
  (integer) elementary column operations
• 0 1 1 0
  x1 x1
  -1 1
• 2 1 E -1 2 E-1
  where E 1
  0
• -1 1 2 -1
  x2 x2

23

• x1 gt 1/3( b1 2b3)
• x2 gt 1/2(b1 1/3(b1 2b3) )
• x1 gt 1/2(b3 1/2(b1 1/3(b1 2b3)
  ) )
• 1/3( b1 2b3)
• Unchanged. Hence optimal Chvátal Function

24
Calculating a Chvátal Functionover a Cone

• ie we have sign pattern
• xn xn-1 xn-2
  x1
• Min
• -
  b1
• - -
  b2
• .
  .
• .
  gt .
• .
  .
• - -
  bn-1
• -------------------------------
• . . .
  bn

25
Calculating the optimal Chvátal Function over a
Cone

• e
• Take first estimate for xn (Optimal LP Chvátal
  Function)
• Substitute to give new rhs for problem with
  variables xn-1,,, xn-2 ,, , x1
• Repeat for xn-2 ,, , x1 ..
• Repeat to give new estimate for xn ..
• Continue until Chvátal Function unchanged between
  successive iterations

26
Calculating the optimal Chvátal Function

• Minimise x2
• subject to 2x1 x2 gt b1
• -x1 x2 gt b3 (ie over
  cone) gives
• x 1 1/2(b1 1/3(b1 2b3) ) - 1/3(b1
  2b3) x2 1/3(b1 2b3)
• (NB values of variables not Chvátal Functions)
• Substitute values for bi . If feasible for IP
  gives optimal Chvátal Function. Otherwise repeat
  procedure for IP
• Minimise x2
• subject to 2x1 x2 gt b1
• 5x1 2x2
  lt b2
• -x1
  x2 gt b3
• x2 gt
  1/3(b1 2b3)
• x1 , x2 gt 0
  and integer

27
References

• CE Blair and RG Jeroslow, The value function of
  an integer programme, Mathematical Programming
  23(1982) 237-273.
• V Chvátal, Edmonds polytopes and a hierarchy of
  combinatorial problems, Discrete Mathematics
  4(1973) 305-307.
• D.Kirby and HP Williams, Representing integral
  monoids by inequalities Journal of Combinatorial
  Mathematics and Combinatorial Computing 23 (1997)
  87-95.
• F Rhodes and HP Williams Discrete subadditive
  functions as Gomory functions, Mathematical
  Proceedings of the Cambridge Philosophical
  Society 117 (1995) 559-574.
• HP Williams, Constructing the value function for
  an integer linear programme over a cone,
  Computational Optimisation and Applications 6
  (1996) 15-26.
• HP Williams, Integer Programming and Pricing
  Revisited, Journal of Mathematics Applied in
  Business and Industry 8(1997) 203-214..
• LA Wolsey, The b-hull of an integer programme,
  Discrete Applied Mathematics 3(1981) 193-201.