Sequence Independent Lifting in Mixed Integer Programming

We investigate lifting i e the process of taking a valid inequality for a polyhe dron and extending it to a valid inequality in a higher dimensional space Lifting is usually applied sequentially that is variables in a set are lifted one after the other This may be computationally unattractive since it involves the solution of an optimization problem to compute a lifting coe cient for each variable To relieve this computational burden we study sequence independent lifting which only in volves the solution of one optimization problem We show that if a certain lifting function is superadditive then the lifting coe cients are independent of the lifting sequence We introduce the idea of valid superadditive lifting functions to obtain good aproximations to maximum lifting We apply these results to strengthen Balas lifting theorem for cover inequalities and to produce lifted ow cover inequalities for a single node ow problem Introduction This paper investigates a general principle called lifting which is the process of con structing from a given valid inequality for a low dimensional polydedron a valid inequal ity for a higher dimensional polyhedron When the lifting coe cients are maximum low dimensional facets are turned into higher dimensional facets The idea of lifting was introduced by Gomory in the context of the group problem Its computational possibilities were emphasized in Padberg and the approach was generalized by Wolsey Zemel and Balas and Zemel Lifting is usually applied sequentially variables in a set are lifted one after the other and a separate optimization problem has to be solved to determine each lifting coe cient The resulting inequality depends on the order in which the variables are lifted A better lifting coe cient for a given variable is obtained if the variable is lifted earlier in the This research was supported by NSF Grant No DDM sequence In sequence independent lifting all of the coe cients can be obtained by solving a single optimization problem and the resulting inequality is independent of the order in which the variables are lifted Sequential lifting has been instrumental to the success of branch and cut algorithms for integer programs BIPs based on cover inequalities see e g Crowder Johnson and Padberg and Gu Nemhauser and Savelsbergh The idea is to nd simple valid inequalities cover inequalities from individual rows knapsack inequalities of the problem that are violated by LP optimal solutions and then to strengthen these cuts by lifting Balas gave a speci c sequence independent formula for the lifting coe cients for a cover inequality The coe cients obtained from the formula do not always yield maximum lifting coe cients but they can be computed very fast and yield valid inequal ities For BIPs Wolsey proved that if a lifting function is superadditive then the maximum lifting coe cients are independent of the lifting order Gu Gu gen eralized Wolsey s result to mixed integer programs MBIPs and to lifting several variables simultaneously For MBIPs the computation of lifting coe cients is typically much more complex than for BIPs Therefore the role of sequence independent lifting as a way to relieve the computational burden is especially important Gu Nemhauser and Savelsbergh show that lifting can be done e ectively for MBIPs To bene t from the computational advantages of superadditive lifting functions they relax the true lifting function to an approximate lifting function that is superadditive and use this approximate lifting function to compute lifting coe cients Marchand and Wolsey take a slightly di erent approach They group variables to be lifted in such a way that the lifting function associated with each group is superadditive Both of these papers have shown that sequence independent lifting is an important tool in the solutions of MBIPs In this paper we present the fundamental concepts and we prove the basic theorems related to sequence independent lifting Some of these theorems have been stated in Gu Nemhauser and Savelsbergh but no proofs were presented In addition we use the theory of superadditive lifting to strengthen Balas result on sequence independent lifting of cover inequalities and we give a new derivation of a result by Pochet which gives a complete characterization of all facet inducing lifted ow cover inequalities for single node ow models with only out ow arcs In Section we discuss the essential ideas of lifting In Section we present the fundamental result for MBIPs that superadditive lifting functions lead to sequence inde pendent lifting We also show how this result can be exploited by working with relaxed lifting Applications of sequence independent lifting results are given in Sections and Lifting Consider the set of feasible points for a mixed integer program given by X fx R jN j X j N ajxj d X j Ck wjxj rk k t xj f g j I Ng Here fCk k tg is a partition ofN aj j N and d arem and wj j N and rk are mk We assume that aj d and rk but not necessarily wj are nonnegative Initially we consider the subset of X with xj for j N n C given by X fx R jC j X