Lifting Procedures

Lifting Procedures Houston Chapter of INFORMS 30 May 2002 Maarten Oosten

Outline • Introduction • Lifting Procedures: Review • Generalization of the Lifting Procedures • Summary

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?

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

LP Relaxation Pcoin

Projection Qcoin

Cutting Planes • We will use at most one quarter • We will use at least one dime

Convex Hull Hcoin

Definitions: Polyhedra

Definitions: Faces

Definitions: Cones

Definitions • Consider where Bnis the space of n-dimensional binary vectors • Define • Define S1 as S\S0 • Let P be the convex hull of S • Let P0 be the convex hull of S0 • Let P1 be the convex hull of S1

Traditional Lifting • Let be a valid inequality for P0 • Then for some is called a lifting from P0 to P of the inequality if it is valid for P • It is valid if and only if the coefficient  satisfies:

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 P0 lifts to a facet defining inequality of P if the dimension gap is larger than 1

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

Example Polytope

Example Traditional Lifting • is a facet defining inequality for P0 • Then for some is a lifting from P0 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

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

Corresponding inequality: Example Extended lifting • Take into account all equalities that hold for P0 but not for P • and  should satisfy the solutions of S1: for (x,y,z) = (0,1,1) for (x,y,z) = (1,0,1) Extreme point:  =  = ½

‘extended lifting of the inequality aTx a0’ Extended Lifting • For every facet defining inequality of P0, we can construct at least one facet defining inequality of P. • We do need a minimal representation of all equations that hold for P0 but not for P. • We do need to find the extreme points of the lifting polyhedron of the inequality

Corresponding inequalities: Example Equality lifting Lift all equalities that hold for P0 but not for P • and  should satisfy the solutions of S1: for (x,y,z) = (0,1,1) for (x,y,z) = (1,0,1) Two extreme rays: (,) = (-1,1) and (,) = (-1,-1)

‘extended lifting of the equality system’ Equality Lifting • With a minimal representation of all equations (‘equality set of P0’) that hold for P0 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 P0.

‘complete lifting of the minimal facial description of P0’ Complete Lifting • The other way around: for every facet of P is the lifting of at least one face of P0. • We do need to find the extreme rays of the complete lifting cone of the polytope P0.

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.

Polarity context • Suppose P0 is the empty set. • We do need a minimal representation of all equations that hold for P0 but not for P, for example: x1=0, x2=0, … xn=0, and 0=1. • The lifting cone of the equality set of P0 reduces to the polar cone of P:

 If  is an extreme ray of this cone, your inequality defines a facet of P0 If (,) is an extreme ray of this cone, your inequality defines a facet of P or and Duality context

Lifting Procedures