On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts

Abstract

We examine the connections between the classes of cuts in the title. We show that lift-and-project (L&P) cuts from a given disjunction are equivalent to generalized intersection cuts (GICs) from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction. While L&P cuts from split disjunctions are known to be equivalent to standard intersection cuts (SICs) from the strip obtained by complementing the terms of the split, we show that L&P cuts from more general disjunctions may not be equivalent to any SIC. In particular, we give easily verifiable necessary and sufficient conditions for a L&P cut from a given disjunction D to be equivalent to a SIC from the polyhedral counterpart of D. Irregular L&P cuts, i.e. those that violate these conditions, have interesting properties. For instance, unlike the regular ones, they may cut off part of the corner polyhedron associated with the LP solution from which they are derived. Furthermore, they are not exceptional: their frequency exceeds that of regular cuts. A numerical example illustrates some of the above properties.

When Lift-and-Project Cuts Are Different

“…Case 1 : |N | = n. Thenà N is an n × n nonsigular submatrix ofà such that u t j = 0 for all j / ∈ N and all t ∈ T , i.e.,w satisfies the condition of Theorem 10 of Balas and Kis (2016).…”

“…From Theorem 10 of Balas and Kis (2016), a CGLP solutionw such thatv t e > 0 is regular ifà has an n × n nonsingular submatrixà J such thatū t j is nonbasic for all j / ∈ J, t ∈ T , in which caseᾱx ≥β is equivalent to the intersection cut from the LP cone associated with the cobasis indexed by J and the P I -free convex set defined by {x :…”

“…More specifically, a positive CGLP multiplierū t i for some row i of Ax ≥ b for any t ∈ T maps the corresponding surplus variable s i as nonbasic in the LP, a positive multiplier u t m+j for the bound x j ≥ 0 maps x j as nonbasic at the lower bound in the LP, and a positive CGLPū t m+p+j for the bound x j ≤ 1 maps x j as nonbasic at the upper bound in the LP. From Theorem 12 of Balas and Kis (2016), the sufficient condition for the regularity ofw given in Theorem 10 is also necessary, i.e., if the condition is not met, thenw is irregular.…”

“…More recently, Balas and Kis (2016) have shown that the correspondence between liftand-project cuts and intersection cuts may not necessarily hold for lift-and-project cuts from arbitrary disjunctions. More specifically, they have proven that it holds if, and only if, there is a basic CGLP solution associated with the cut that maps to an LP basis that corresponds to a standard intersection cut.…”

“…In this work, we focus on the equivalence with respect to lift-and-project cuts without strengthening, and we present the following contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions from Balas and Kis (2016) and show how it extends to CGLP solutions that are not basic in Section 3. Second, we introduce and prove the validity of an MILP that checks whether there is a regular CGLP solution for a given cut in Section 4.…”