### 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 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.

Original language | English |
---|---|

Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Mathematical Programming, Series B |

DOIs | |

Publication status | Accepted/In press - Jan 14 2016 |

### Fingerprint

### Keywords

- Corner polyhedra
- Generalized intersection cuts
- Integer programming
- Intersection cuts
- Lift-and-project cuts

### ASJC Scopus subject areas

- Mathematics(all)
- Software

### Cite this

**On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts.** / Balas, Egon; Kis, T.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts

AU - Balas, Egon

AU - Kis, T.

PY - 2016/1/14

Y1 - 2016/1/14

N2 - 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 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.

AB - 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 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.

KW - Corner polyhedra

KW - Generalized intersection cuts

KW - Integer programming

KW - Intersection cuts

KW - Lift-and-project cuts

UR - http://www.scopus.com/inward/record.url?scp=84954319624&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84954319624&partnerID=8YFLogxK

U2 - 10.1007/s10107-015-0975-1

DO - 10.1007/s10107-015-0975-1

M3 - Article

AN - SCOPUS:84954319624

SP - 1

EP - 30

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -