The Chvátal Dual of a Pure Integer Programme

H.P.Williams

London School of Economics

Duality in LP and IPThe Value Function of an LP

Minimise x2

subject to: 2x1 + x2 >= b1

5x1 + 2x2 <= b2

-x1 + x2 >= b3

x1 , x2 >= 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 ) <= 0 giving Max( -4, -30, -20) <= 0 .

(5, -2, 0), (1/3, 0, 2/3), (0, 0, 1) are vertices of Dual Polytope .

They give marginal rates of change (shadow prices) of optimal objective with

respect to b1, b2, b3 .

(5, -2,, 0), (1/3, 0, 2/3), (0, 0, 1) are extreme rays of Dual Polytope .

What are the corresponding quantities for an IP ?

Duality in LP and IPThe Value Function of an IP

Minimise x2

subject to: 2x1 + x2 >= b1

5x1 + 2x2 <= b2

-x1 + x2 >= b3

x1 , x2 >= 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 Chvάtal Functions .

Consistency Tester same as for LP (in this example)

IP Solution

9 Optimal IP Solution (2 , 9) . Min x2

c3 st 2x1+ x2 >= 13

8 . . c1 . . 5x1 + 2x2 <= 30

Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5

7 . . . . c2 . x1 , x2 >= 0 x2

6 . . . . .

5 . . . . .

0 1 2 3 4 x1

•

IP Solution after removing constraint 1

Min x2

8 c1 . . . c3 st 5x1 + 2x2 <= 30

-x1 + x2 >= 5

. . . . x1 , x2 >= 0

x2 7 . . .

6 . . . . . c2

Optimal IP Solution (0 , 5)

5

0 1 2 3 4 x1

•

IP Solution

9 Optimal IP Solution (2 , 9) . Min x2

c3 st 2x1+ x2 >= 13

8 . . c1 . . 5x1 + 2x2 <= 30

Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5

7 . . . . c2 . x1 , x2 >= 0 x2

6 . . . . .

5 . . . . .

0 1 2 3 4 x1

•

IP Solution after removing constraint 2

9 . . . Min x2

c1 c3 st 2x1+ x2 >= 13

8 . . . . Optimal IP Solution (3, 8) -x1 + x2 >= 5

7 . . . . . x1 , x2 >= 0

x2

6 . . . . .

5 . . . . . x1

0 1 2 3 4

•

IP Solution

9 Optimal IP Solution (2 , 9) . Min x2

c3 st 2x1+ x2 >= 13

8 . . c1 . . 5x1 + 2x2 <= 30

Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5

7 . . . . c2 . x1 , x2 >= 0 x2

6 . . . . .

5 . . . . .

0 1 2 3 4 x1

•

IP Solution after removing constraint 3

9 . . . . Min x2

st 2x1+ x2 >= 13

8 . . . . . 5x1 + 2x2 <= 30

c1 c2 x1 , x2 >= 0

7 . . . . . x2

6 . . . . .

5 . . . . .Optimal IP Solution (4 , 5)

0 1 2 3 4 x1

IP Solution

9 Optimal IP Solution (2 , 9) . Min x2

c3 st 2x1+ x2 >= 13

8 . . c1 . . 5x1 + 2x2 <= 30

Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5

7 . . . . c2 . x1 , x2 >= 0 x2

6 . . . . .

5 . . . . . x1

0 1 2 3 4

•

Gomory and Chvátal Functions

Max( 5b1-2b2, ┌1/3(b1 + 2b3)

┐, b3 , b1+ 2 ┌1/5 (-b2+ 2 ┌1/3(b1 + 2b3)

┐ ) ┐ )

If b1=13, b2=30, b3=5 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 )

Pricing by optimal Chvátal FunctionIntroduce new variable X3

LP Case

Minimise X2 +1.5X3

Subject to: 2X1+X2 +X3 >=13

5X1+2X2+X3<=30

-X1+X2 +X3 >=5

X1 , X2 >= 0

IP Case

Minimise X2 +1.5X3

Subject to: 2X1+X2 +X3 >=13

5X1+2X2+X3<=30

-X1+X2 +X3 >=5

X1 , X2 >= 0 and integer

Function

bb11+ 2 + 2 ┌┌1/5 (-b1/5 (-b22+ 2 + 2 ┌┌1/3(b1/3(b11 + 2b + 2b33))┐┐) =3) =3

does not price out X3

Solution X1 = 3, X2 = 7, X3 =1

Function

⅓b1+ ⅔b3=1

prices out X3

SolutionX1 = 22

/3 , X2 = 72/3 , X3 = 0

Why are valuations on discrete resources of interest ?

Allocation of Fixed Costs

Maximise ∑j pi xi - f y

st xi - Di y <= 0 for all I

y ε {0,1} depending on whether facility built.f is fixed cost.

xi is level of service provided to i (up to level D i )pi is unit profit to i.

A ‘dual value’ vi on xi - Di y <= 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’

A Representation for Chvátal Functions

b1 b3 - b2

1 2

Multiply and add

on arcs 1

1

Divide and round

up on nodes

2

2

Giving b1+ 2 ┌1/5( -b2+ 2 ┌

1/3( b1 + 2b3) ┐ ) ┐

LP Relaxation is 19/15 b1 - 2/5 b2 +8/15 b3

3

5

1

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 ?

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.

The Triangle Inequality

• ┌

a┐

+ ┌

b┐

>= ┌

a + b┐

• Hence of value in defining Discrete Metrics

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

7

6

5

4

3

2

1 . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 --- x

Polyhedral and Non-Polyhedral Monoids

The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 0A 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┐ < = 0

Calculating the optimal Chvátal Function over a Cone

• Value Function over a Cone is a Chvátal

Function

IP Solution

9 Optimal IP Solution (2 , 9) . Min x2

c3 st 2x1+ x2 >= 13

8 . . c1 . . 5x1 + 2x2 <= 30

Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5

7 . . . . c2 . x1 , x2 >= 0 x2

6 . . . . .

5 . . . . .

0 1 2 3 4 x1

•

An Example

Minimise x2

subject to: 2x1 + x2 >= b1

-x1 + x2 >= 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’

x1 ‘ >= ┌1/3( b1 + 2b3) ┐

x2’ >= ┌1/2(b1+ ┌

1/3(b1 + 2b3) ┐ ) ┐

x1’ >= ┌ 1/2(b3 +┌1/2(b1+

┌ 1/3(b1 + 2b3)

┐ ) ┐) ┐

= ┌1/3( b1 + 2b3) ┐

Unchanged. Hence optimal Chvátal Function

Calculating a Chvátal Functionover a Cone

• ie we have sign pattern x’

n x’n-1 x’

n-2 … x’1

Min + - + b1

- - + b2

. . . >= . . . - - + bn-1

------------------------------- + . . … . bn

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

Calculating the optimal Chvátal Function

Minimise x2

subject to: 2x1 + x2 >= b1 -x1 + x2 >= 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 >= b1

5x1 + 2x2 <= b2

-x1 + x2 >= b3

x2 >= ┌1/3(b1+ 2b3)┐

x1 , x2 >= 0 and integer

Gives x2 = b1+ 2 ┌ 1/5 (-b2+ 2 ┌

1/3(b1 + 2b3) ┐ ) ┐ )

x1 = ┌1/2(b1-( b1+ 2 ┌ 1/5 (-b2+ 2

┌ 1/3(b1 + 2b3)

┐ ) ┐ ) ┐

Substituting gives feasible solution to IP implying optimal Chvátal Function.

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.