1. 1 Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten

2. 2 Outline Introduction Lifting Procedures: Review Generalization of the Lifting Procedures Summary

3. 3 Example: Vending Machine A Swiss Roll costs 40 cents No change light blinks We have 3 quarters, 5 dimes, and 10 cents We prefer to use as few coins as possible How many of each type of coins should we use?

4. 4 Decision variables: Vending Machine (2) Payment equation: Objective function:

5. 5 LP Relaxation P coin

6. 6 Projection Q coin

7. 7 Cutting Planes We will use at most one quarter We will use at least one dime

8. 8 Convex Hull H coin

9. 9 Definitions: Polyhedra

10. 10 Definitions: Faces

11. 11 Definitions: Cones

12. 12 Outline Introduction Lifting Procedures: Review Generalization of the Lifting Procedures Summary

13. 13

14. 14 Consider where B n is the space of n-dimensional binary vectors Define Define S1 as S\S0 Let P be the convex hull of S Let P 0 be the convex hull of S 0 Let P 1 be the convex hull of S 1 Definitions

15. 15 Let be a valid inequality for P 0 Then for some is called a lifting from P 0 to P of the inequality if it is valid for P It is valid if and only if the coefficient  satisfies: Traditional Lifting

16. 16 Literature Review Wolsey, 1976 Zemel, 1978 Balas & Zemel, 1984 Nemhauser & Wolsey, 1988 Boyd & Pulleyblank, 1991 Gu et al, 1995 No guarantee that a facet defining inequality of P 0 lifts to a facet defining inequality of P if the dimension gap is larger than 1

17. 17 Example Let S = {(0,0,0), (1,1,0), (1,0,1), (0,1,1)}

18. 18 Example Polytope

19. 19 Example Traditional Lifting is a facet defining inequality for P 0 Then for some is a lifting from P 0 to P of the inequality if it is valid for P It is valid if and only if the coefficient  satisfies: Strongest lifted inequality is

20. 20 Example Traditional Lifting (2) Due to the symmetry of the polytope, no matter in which order the variables are lifted, the resulting lifted inequalities are always trivial inequalities

21. 21 Outline Introduction Lifting Procedures: Review Generalization of the Lifting Procedures Summary

22. 22 Take into account all equalities that hold for P 0 but not for P  and  should satisfy the solutions of S 1 : for (x,y,z) = (0,1,1) for (x,y,z) = (1,0,1) Example Extended lifting Extreme point:  =  = ½ Corresponding inequality:

23. 23 Extended Lifting For every facet defining inequality of P 0, we can construct at least one facet defining inequality of P. We do need a minimal representation of all equations that hold for P 0 but not for P. We do need to find the extreme points of the lifting polyhedron of the inequality ‘extended lifting of the inequality a T x  a 0 ’

24. 24 Lift all equalities that hold for P 0 but not for P  and  should satisfy the solutions of S 1 : for (x,y,z) = (0,1,1) for (x,y,z) = (1,0,1) Example Equality lifting Two extreme rays: ( ,  ) = (-1,1) and ( ,  ) = (-1,-1) Corresponding inequalities:

25. 25 Equality Lifting With a minimal representation of all equations (‘equality set of P 0’ ) that hold for P 0 but not for P, we can construct at least one facet of P. We do need to find the extreme rays of the lifting cone of the equality set of P 0. ‘extended lifting of the equality system’

26. 26 Complete Lifting The other way around: for every facet of P is the lifting of at least one face of P 0. We do need to find the extreme rays of the complete lifting cone of the polytope P 0. ‘complete lifting of the minimal facial description of P 0 ’

27. 27 Outline Introduction Lifting Procedures: Review Generalization of the Lifting Procedures Summary

28. 28 Summary Every facet can be lifted to a facet Equalities can be lifted to a facet There are complete descriptions of the set of solutions that are partly a facial description, partly a listing of solutions. Lifting procedures describe the relations between these descriptions.

29. 29 Polarity context Suppose P 0 is the empty set. We do need a minimal representation of all equations that hold for P 0 but not for P, for example: x 1 =0, x 2 =0, … x n =0, and 0=1. The lifting cone of the equality set of P 0 reduces to the polar cone of P:

30. 30 Duality context  If  is an extreme ray of this cone, your inequality defines a facet of P 0 orand If ( ,  ) is an extreme ray of this cone, your inequality defines a facet of P

31. 31 Outline Introduction Lifting Procedures: Review Generalization of the Lifting Procedures Summary