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SEARCHING FOR THE DUAL OF AN INTEGER PROGRAMME

SEARCHING FOR THE DUAL OF AN INTEGER PROGRAMME. H.P.Williams London School of Economics. Why is Duality important ? Desirable properties for a Dual. Possible Duals for an Integer Programme (IP). Projecting out variables of an IP to give a ‘Dual’.

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SEARCHING FOR THE DUAL OF AN INTEGER PROGRAMME

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  1. SEARCHING FOR THE DUAL OF AN INTEGER PROGRAMME H.P.Williams London School of Economics

  2. Why is Duality important ? Desirable properties for a Dual. Possible Duals for an Integer Programme (IP). Projecting out variables of an IP to give a ‘Dual’.

  3. Why is Duality Important in Mathematical Programming? • COMPUTATIONAL Provides a Bound on Optimal Objective Value (Proof of Optimality). • ECONOMIC Gives effects of changes in Resource Levels (marginal values). • MATHEMATICAL Reflexive structure (Dual of Dual = Primal).

  4. LINEAR PROGRAMMING DUALITY All these properties are met in Linear Programming (LP). 1. If PRIMAL LP solvable optimal solution to DUAL provides tight bound on objective. ie No duality gap. Proves if a feasible solution optimal. 2. Dual values give marginal (shadow prices) of RHS changes. (If alternate dual values (primal degeneracy) we get two-valued shadow prices). Dual values ‘price out’ uneconomic activities and split profits from economic ones between cost centres. (many applications, eg Transfer prices, Decomposition etc. ) 3. Dual of LP is itself an LP. Dual of Dual = Primal. NOT ALL THESE PROPERTIES CAN BE RECAPTURED IN IP DUALITY

  5. POSSIBLE INTEGER PROGRAMMING DUALS 1. SURROGATE DUAL Add together constraints in different multiples to create Knapsack Problem which minimises Duality Gap. Cannot, generally, close gap. Of computational use in providing bound on optimal objective value. Multipliers (dual values) of constraints can be given economic interpretation as relative values of different resources (Forgó). Not a Reflexive structure.

  6. 2. LAGRANGIAN DUAL (Structure dependent) Add together some constraints in different multiples and add/subtract from objective (giving model with special structure) so as to minimise duality gap. Cannot, generaly, close gap. Of computational value in providing bound on optimal objective value. For particular structures multipliers could be given economic interpretation. Not a reflexive structure. Easier to calculate optimal multipliers than for Surrogate Dual but wider duality gap.

  7. 3. Duals for Mixed 0-1 IPs Can regard as a Disjunction of LPs. Take dual values for each LP in disjunction and choose optimal sets. (Fix variables at optimal values and apply duality to model these decisions).

  8. 4. THE CHVÁTAL DUAL. Use a Chvátal Function in place of dual values. Primal IP Minimise Σj cj xj s.t. Σj a ij xj >= bi for all i xj >=0, xjεZ for all j Chvátal Dual Maximise C(bi) s.t. C(aij)>= cj for all j C ε Class of Chvátal Functions The class of Chvátal Functions on (b1, b2, …, bn) is the set of functions obtained by repeated (and nested) (i) Non-negative linear combinations. (ii) Integer round-up. Closes duality gap (tight bound on optimal objective value). Possible economic interpretation. Can be regarded as reflexive.

  9. An Example • An IP Minimise X2 Subject to: 2X1 + X2 >=13 5X1 + 2X2<=30 -X1 + X2 >=5 X1 , X2 >= 0 and integer Optimal Solution X1 = 2, X2 = 9 Optimal Chvátal Function b1+ 2 ┌1/5 (-b2+ 2┌1/3(b1 + 2b3) ┐ ) ┐ (Optimal LP Chvátal Function is 1/3(b1 + 2b3) since (1/3,0,2/3) is optimal dual vertex)

  10. b1+ 2 ┌1/5 (-b2+ 2┌1/3(b1 + 2b3) ┐ )┐ shows how optimal solution varies with changed in RHSs (analogous to LP shadow prices) Closes Duality Gap Prices out uneconomic activities

  11. Proof of Optimality b1+ 2 ┌1/5 (-b2+ 2┌1/3(b1 + 2b3) ┐ ) ┐ 2X1 + X2 >=13 x1 -X1 + X2 >=5 x2 ---------------------------- 3X2 >=23  X2 >= 8 x 2 -5X1 - 2X2 >= -30 x 1 ----------------------------- -5X1 >= -14  -X1 >= -2 x 2 2X1 + X2 >= 13 x1 --------------------------- X2 >= 9 The Optimal Objective Value

  12. 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 is19/15 b1 - 2/5 b2 +8/15 b3 3 5 1

  13. A Chvátal Function of one argument ┌ ½ ( x + 3 ┌ x /9 ┐ ) ┐ is (9, 6) Shift Periodic. 2/3is ‘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

  14. References • CE Blair and RG Jeroslow, The value function of an integerprogramme, Mathematical Programming 23(1982) 237-273. • V Chvátal, Edmonds polytopes and a hierarchy of combinatorialproblems, Discrete Mathematics 4(1973) 305-307. • D.Kirby and HP Williams, Representing integral monoids by inequalities Journal of Combinatorial Mathematicsand Combinatorial Computing 23 (1997) 87-95. • F Rhodes and HP Williams Discrete subadditive functions as Gomory functions, Mathematical Proceedings of the CambridgePhilosophicalSociety 117 (1995) 559-574. • HP Williams, Constructing the value function for an integer linear programme over a cone, Computational Optimisation andApplications 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.

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