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 x 2 subject to: 2x 1 + x 2 >= b 1 5x 1 + 2x 2 <= b 2 -x 1 + x 2 >= b 3 x 1, x 2 >= 0 Value Function of LP is Max( 5b 1 - 2b 2, 1/3( b 1 + 2b 3 ), b 3 ) If b 1 = 13, b 2 = 30, b 3 = 5 we have Max( 5, 7 2 /3, 5 ) = 7 2 /3, Consistency Tester is Max( 2b 1 – b 2, -b 2, -b 2 + 2b 3 ) <= 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 b 1, b 2, b 3. (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 ?

3. Duality in LP and IP The Value Function of an IP Minimise x 2 subject to: 2x 1 + x 2 >= b 1 5x 1 + 2x 2 <= b 2 -x 1 + x 2 >= b 3 x 1, x 2 >= 0 and integer Value Function of IP is Max( 5b 1 - 2b 2, ┌ 1/3( b 1 + 2b 3 ) ┐, b 3, b 1 + 2 ┌ 1/5 (-b 2 + 2 ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) This is known as a Gomory Function. The component expressions are known as Chvάtal Functions. Consistency Tester same as for LP (in this example)

4. IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= 13 8.. c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= 5 7.... c 2. x 1, x 2 >= 0 x 2 6..... 5..... 0 1 2 3 4 x 1

5. IP Solution after removing constraint 1 Min x 2 8 c1... c 3 st 5x 1 + 2x 2 <= 30 -x 1 + x 2 >= 5.... x 1, x 2 >= 0 x 2 7... 6..... c 2 Optimal IP Solution (0, 5) 5 0 1 2 3 4 x 1

6. IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= 13 8.. c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= 5 7.... c 2. x 1, x 2 >= 0 x 2 6..... 5..... 0 1 2 3 4 x 1

7. IP Solution after removing constraint 2 9... Min x 2 c1 c3 st 2x 1 + x 2 >= 13 8.... Optimal IP Solution (3, 8) -x 1 + x 2 >= 5 7..... x 1, x 2 >= 0 x 2 6..... 5..... x 1 0 1 2 3 4

8. IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= 13 8.. c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= 5 7.... c 2. x 1, x 2 >= 0 x 2 6..... 5..... 0 1 2 3 4 x 1

9. IP Solution after removing constraint 3 9.... Min x 2 st 2x 1 + x 2 >= 13 8..... 5x 1 + 2x 2 <= 30 c1 c2 x 1, x 2 >= 0 7..... x 2 6..... 5..... Optimal IP Solution (4, 5) 0 1 2 3 4 x 1

10. IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= 13 8.. c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= 5 7.... c 2. x 1, x 2 >= 0 x 2 6..... 5..... x 1 0 1 2 3 4

11. Gomory and Chvátal Functions Max( 5b 1 -2b 2, ┌ 1/3(b 1 + 2b 3 ) ┐, b 3, b 1 + 2 ┌ 1/5 (-b 2 + 2 ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) If b 1 =13, b 2 =30, b 3 =5 we have Max(5,8,5,9)=9 Chvátal Function b 1 + 2 ┌ 1/5 (-b 2 + 2 ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ determines the optimum. LP Relaxation is 19/15 b 1 - 2/5 b 2 +8/15 b 2 (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 b i )

12. Pricing by optimal Chvátal Function Introduce new variable X 3 LP Case Minimise X 2 +1.5X 3 Subject to: 2X 1 +X 2 +X 3 >=13 5X 1 +2X 2 +X 3 <=30 -X 1 +X 2 +X 3 >=5 X 1, X 2 >= 0 IP Case Minimise X 2 +1.5X 3 Subject to: 2X 1 +X 2 +X 3 >=13 5X 1 +2X 2 +X 3 <=30 -X 1 +X 2 +X 3 >=5 X 1, X 2 >= 0 and integer Function b 1 + 2 ┌ 1/5 (-b 2 + 2 ┌ 1/3(b 1 + 2b 3 ) ┐ ) =3 does not price out X3 Solution X1 = 3, X2 = 7, X3 =1 Function ⅓b 1 + ⅔b 3 =1 prices out X 3 Solution X 1 = 2 2 /3, X 2 = 7 2 /3, X 3 = 0

13. Why are valuations on discrete resources of interest ? Allocation of Fixed Costs Maximise ∑ j p i x i - f y st x i - D i y <= 0 for all I y ε {0,1} depending on whether facility built. f is fixed cost. x i is level of service provided to i (up to level D i ) p i is unit profit to i. A ‘dual value’ v i on x i - D i y <= 0 would result in Maximise ∑ j (p i – v i ) x i - (f – (∑ i D i v i ) y Ie an allocation of the fixed cost back to the ‘consumers’

14. A Representation for Chvátal Functions b 1 b 3 - b 2 1 2 Multiply and add on arcs 1 1 Divide and round up on nodes 2 2 Giving b 1 + 2 ┌ 1/5( -b 2 + 2 ┌ 1/3( b 1 + 2b 3 ) ┐ ) ┐ LP Relaxation is 19/15 b 1 - 2/5 b 2 +8/15 b 3 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 ┐ >= ┌ 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 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

19. Polyhedral and Non-Polyhedral Monoids The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 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 ┐ < = 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 x 2 c 3 st 2x 1 + x 2 >= 13 8.. c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= 5 7.... c 2. x 1, x 2 >= 0 x 2 6..... 5..... 0 1 2 3 4 x 1

22. An Example Minimise x 2 subject to: 2x 1 + x 2 >= b 1 -x 1 + x 2 >= b 3 x 1, x 2 integer These are constraints which are binding at LP Optimum. Convert 1 st 2 rows to Hermite Normal Form by (integer) elementary column operations 0 1 1 0 x 1 x 1 ’ - 1 1 2 1 E = -1 2 E -1 = where E = 1 0 -1 1 2 -1 x 2 x 2 ’

23. x 1 ‘ >= ┌ 1/3( b 1 + 2b 3 ) ┐ x 2 ’ >= ┌ 1/2(b 1 + ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ x 1 ’ >= ┌ 1/2(b 3 + ┌ 1/2(b 1 + ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) ┐ = ┌ 1/3( b 1 + 2b 3 ) ┐ Unchanged. Hence optimal Chvátal Function

24. Calculating a Chvátal Function over a Cone ie we have sign pattern x ’ n x ’ n-1 x ’ n-2 … x ’ 1 Min + - + b 1 - - + b 2... >=... - - + b n-1 ------------------------------- +.. …. b n

25. Calculating the optimal Chvátal Function over a Cone e Take ‘first estimate’ for x n ’ (Optimal LP Chvátal Function) Substitute to give new rhs for problem with variables x n-1 ’,,, x n-2 ’,, …, x 1 ’ Repeat for x n-2 ’,, …, x 1 ’.. Repeat to give new estimate for x n ’.. Continue until Chvátal Function unchanged between successive iterations

26. Calculating the optimal Chvátal Function Minimise x 2 subject to: 2x 1 + x 2 >= b 1 -x 1 + x 2 >= b 3 (ie over cone) gives x 1 = ┌ 1/2(b 1 + ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ - ┌ 1/3(b 1 + 2b 3 ) ┐ x 2 = ┌ 1/3(b 1 + 2b 3 ) ┐ (NB values of variables not Chvátal Functions) Substitute values for b i. If feasible for IP gives optimal Chvátal Function. Otherwise repeat procedure for IP Minimise x 2 subject to: 2x 1 + x 2 >= b 1 5x 1 + 2x 2 <= b 2 -x 1 + x 2 >= b 3 x 2 >= ┌ 1/3(b 1 + 2b 3 ) ┐ x 1, x 2 >= 0 and integer Gives x 2 = b 1 + 2 ┌ 1/5 (-b 2 + 2 ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) x 1 = ┌ 1/2( b 1 -( b 1 + 2 ┌ 1/5 (-b 2 + 2 ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) ┐ Substituting gives feasible solution to IP implying optimal Chvátal Function.

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.