The Chvátal Dual of a Pure Integer Programme H.P.Williams London School of Economics
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 ?
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 ┌ 1/5 (-b ┌ 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)
IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= c 2. x 1, x 2 >= 0 x x 1
IP Solution after removing constraint 1 Min x 2 8 c1... c 3 st 5x 1 + 2x 2 <= 30 -x 1 + x 2 >= x 1, x 2 >= 0 x c 2 Optimal IP Solution (0, 5) x 1
IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= c 2. x 1, x 2 >= 0 x x 1
IP Solution after removing constraint Min x 2 c1 c3 st 2x 1 + x 2 >= Optimal IP Solution (3, 8) -x 1 + x 2 >= x 1, x 2 >= 0 x x
IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= c 2. x 1, x 2 >= 0 x x 1
IP Solution after removing constraint Min x 2 st 2x 1 + x 2 >= x 1 + 2x 2 <= 30 c1 c2 x 1, x 2 >= x Optimal IP Solution (4, 5) x 1
IP Solution 9 Optimal IP Solution (2, 9). Min x 2 c 3 st 2x 1 + x 2 >= c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= c 2. x 1, x 2 >= 0 x x
Gomory and Chvátal Functions Max( 5b 1 -2b 2, ┌ 1/3(b 1 + 2b 3 ) ┐, b 3, b ┌ 1/5 (-b ┌ 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 ┌ 1/5 (-b ┌ 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 )
Pricing by optimal Chvátal Function Introduce new variable X 3 LP Case Minimise X X 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 X 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 ┌ 1/5 (-b ┌ 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
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’
A Representation for Chvátal Functions b 1 b 3 - b Multiply and add on arcs 1 1 Divide and round up on nodes 2 2 Giving b ┌ 1/5( -b ┌ 1/3( b 1 + 2b 3 ) ┐ ) ┐ LP Relaxation is 19/15 b 1 - 2/5 b 2 +8/15 b
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’ x
Polyhedral and Non-Polyhedral Monoids The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 0 A Polyhedral Monoid …… 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 x 2 c 3 st 2x 1 + x 2 >= c1.. 5x 1 + 2x 2 <= 30 Optimal LP Solution (2 2 / 3, 7 2 / 3 ) -x 1 + x 2 >= c 2. x 1, x 2 >= 0 x x 1
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 x 1 x 1 ’ E = -1 2 E -1 = where E = x 2 x 2 ’
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
Calculating a Chvátal Function over a Cone ie we have sign pattern x ’ n x ’ n-1 x ’ n-2 … x ’ 1 Min b b 2... >= b n …. b n
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
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 ┌ 1/5 (-b ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) x 1 = ┌ 1/2( b 1 -( b ┌ 1/5 (-b ┌ 1/3(b 1 + 2b 3 ) ┐ ) ┐ ) ┐ 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) V Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Mathematics 4(1973) D.Kirby and HP Williams, Representing integral monoids by inequalities Journal of Combinatorial Mathematics and Combinatorial Computing 23 (1997) F Rhodes and HP Williams Discrete subadditive functions as Gomory functions, Mathematical Proceedings of the Cambridge Philosophical Society 117 (1995) HP Williams, Constructing the value function for an integer linear programme over a cone, Computational Optimisation and Applications 6 (1996) HP Williams, Integer Programming and Pricing Revisited, Journal of Mathematics Applied in Business and Industry 8(1997) LA Wolsey, The b-hull of an integer programme, Discrete Applied Mathematics 3(1981)