## Safe Bounds in Linear and Mixed-Integer Programming (0)

Venue: | Math. Prog |

Citations: | 21 - 2 self |

### BibTeX

@ARTICLE{Neumaier_safebounds,

author = {Arnold Neumaier and Oleg Shcherbina},

title = {Safe Bounds in Linear and Mixed-Integer Programming},

journal = {Math. Prog},

year = {},

volume = {99},

pages = {283--296}

}

### Years of Citing Articles

### OpenURL

### Abstract

Current mixed-integer linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size.

### Citations

529 |
Methods and Applications of Interval Analysis
- Moore
- 1979
(Show Context)
Citation Context ... of seven solvers tried. We therefore discuss the use of directed rounding and interval arithmetic to guarantee correct decisions in branch-and-cut procedures. (For basics on interval arithmetic, see =-=[13, 15, 16-=-].) It turns out that safety in solving MIPs can be rigorously guaranteed with limited additional computational eort by careful preparation of the LP subproblems and careful postprocessing of the LP s... |

506 |
Interval Methods for Systems of Equations
- Neumaier
- 1990
(Show Context)
Citation Context ... of seven solvers tried. We therefore discuss the use of directed rounding and interval arithmetic to guarantee correct decisions in branch-and-cut procedures. (For basics on interval arithmetic, see =-=[13, 15, 16-=-].) It turns out that safety in solving MIPs can be rigorously guaranteed with limited additional computational eort by careful preparation of the LP subproblems and careful postprocessing of the LP s... |

64 |
Verification methods for dense and sparse systems of equations
- Rump
- 1994
(Show Context)
Citation Context ...osure for the solution of a linear system for the active part (and then check the validity of the nonactive inequalities). See Neumaier [15, 17] for appropriate techniques in the dense case, and Rump =-=[19-=-] for the sparse case. Very recent work by Jansson [9] improves on this in certain cases. Fortunately, rigorous upper bounds are not required in the logic of branch-and-cut algorithms. 4 Certicates of... |

59 |
Introduction to Numerical Analysis
- Neumaier
- 2001
(Show Context)
Citation Context ... of seven solvers tried. We therefore discuss the use of directed rounding and interval arithmetic to guarantee correct decisions in branch-and-cut procedures. (For basics on interval arithmetic, see =-=[13, 15, 16-=-].) It turns out that safety in solving MIPs can be rigorously guaranteed with limited additional computational eort by careful preparation of the LP subproblems and careful postprocessing of the LP s... |

33 |
Analysis of mathematical programming problems prior to applying the simplex algorithm
- Brearley, Mitra, et al.
- 1975
(Show Context)
Citation Context ...f solutions in linear and mixed integer programming. The only study we are aware of, by Fourer & Gay [5], is restricted to the presolve phase of solving a linear program (LP); cf. also Brearly et al. =-=[1]-=-. It contains the observation that rounding errors in presolve may change the status of an LP from feasible to infeasible, or vice versa. The solution they presented was a straightforward application ... |

18 |
Rigorous lower and upper bounds in linear programming
- Jansson
- 2004
(Show Context)
Citation Context ...used. Since, typically, most of the CPU time is spent in the LP solver, this also implies that the addition of safety should increase the run time by a small amount only. A recent preprint of Jansson =-=[9]-=- contains new results on rigorously solving linear programs with uncertain coecients, using interval arithmetic. For the case of exact coecients, some of his results overlap with those of Section 3 of... |

15 | Computational experience and the explanatory value of condition measures for linear optimization
- Ordóñez, Freund
- 2003
(Show Context)
Citation Context ...o longer produce problems of the same class. Moreover, the in uence of rounding errors on the solution may be large if the problem has a large condition number. A recent study by Ord o~ nez & Freund [=-=18]-=- revealed that 72% of the (real life!) problems of the Netlib LP collection [14] are ill-conditioned; after preprocessing (with CPLEX 7.1 presolve), 19% still remained ill-conditioned. Thus ill-condit... |

13 | Experience with a primal presolve algorithm
- Fourer, Gay
- 1993
(Show Context)
Citation Context ...wever, very little work seems to have been done about the eect of rounding errors on the quality of solutions in linear and mixed integer programming. The only study we are aware of, by Fourer & Gay [=-=5]-=-, is restricted to the presolve phase of solving a linear program (LP); cf. also Brearly et al. [1]. It contains the observation that rounding errors in presolve may change the status of an LP from fe... |

13 | A simple derivation of the Hansen-Bliek-Rohn-NingKearfott enclosure for linear interval equations
- Neumaier
- 1999
(Show Context)
Citation Context ...ore expensive to achieve, since one needs tosnd an interval enclosure for the solution of a linear system for the active part (and then check the validity of the nonactive inequalities). See Neumaier =-=[15, 17]-=- for appropriate techniques in the dense case, and Rump [19] for the sparse case. Very recent work by Jansson [9] improves on this in certain cases. Fortunately, rigorous upper bounds are not required... |

11 |
M.: Combining and strengthening Gomory cuts
- Ceria, Cornuéjols, et al.
- 1995
(Show Context)
Citation Context ...h zeros in all positions except for a single component a k = 1 for some k with nonintegral 15 z k , we get the traditional Gomory cuts; other choices yield generalized Gomory cuts; cf. Ceria et al. [=-=2]-=-. In the presence of rounding errors, Gomory cuts and generalized Gomory cuts are no longer valid, since the arguments in the derivation are based on exact arithmetic. However, by a small amendment of... |

10 |
Continuous constraints – updating the technology, WWW-site, http://www.mat.univie.ac.at/∼neum/glopt/coconut.html [6
- COCONUT
(Show Context)
Citation Context ...K denotes the subvector of x indexed by the indices from K. Boxes (interval vectors) are written in bold face. Acknowledgment We acknowledge support by the European Community research project COCONUT =-=[4]-=-, project reference IST-2000-26063. We also want to thank ILOG for providing us with a CPLEX license while working on this project. 2 Failure of MIP solvers The commercial MIP solver CPLEX [7] is a pr... |

9 | Optimization Environments and the NEOS Server
- Gropp, Moré
- 1997
(Show Context)
Citation Context ...the rounding errors introduced through this ill-conditioning. We also ran the original problem on all MIP solvers (and the only MINLP solver with AMPL input) available (on June 18, 2002) through NEOS =-=[3, 6]-=-, namely BONSAIG, FortMP, GLPK, XPRESS, XPRESS-MP/INTEGER, and MINLP. Only FortMP solved the original problem correctly. The othersve solvers reported 'no solution found' (BONSAIG), 'global search com... |

4 |
Fehlerabschätzung bei linearer Optimierung
- Krawczyk
- 1975
(Show Context)
Citation Context ...ing routine is the result of an approximate calculation and hence is itself approximate. Obtaining rigorous error bounds for the solution of linear programming problems is a dicult task (cf. Krawczyk =-=[11-=-], Jansson [8], Jansson & Rump [10]), especially in the ill-conditioned case, where the active set might not be identied cor5 rectly. Fortunately, it is possible to postprocess the approximate result ... |

3 |
Rigorous solution of linear programming problems with uncertain data
- Jansson, Rump
- 1991
(Show Context)
Citation Context ...proximate calculation and hence is itself approximate. Obtaining rigorous error bounds for the solution of linear programming problems is a dicult task (cf. Krawczyk [11], Jansson [8], Jansson & Rump =-=[10-=-]), especially in the ill-conditioned case, where the active set might not be identied cor5 rectly. Fortunately, it is possible to postprocess the approximate result to obtain rigorous bounds for the ... |

2 |
Zur linearen Optimierung mit unscharfen Daten
- Jansson
- 1985
(Show Context)
Citation Context ... the result of an approximate calculation and hence is itself approximate. Obtaining rigorous error bounds for the solution of linear programming problems is a dicult task (cf. Krawczyk [11], Jansson =-=[8-=-], Jansson & Rump [10]), especially in the ill-conditioned case, where the active set might not be identied cor5 rectly. Fortunately, it is possible to postprocess the approximate result to obtain rig... |

2 |
Interval Arithmetic Solves Nonlinear Problems While Providing Guaranteed Results
- Walster
- 2001
(Show Context)
Citation Context ...nly one of the corresponding bounds can be active, hence one has y i = 0 or z i = 0.) 7 On computers with fast outward rounding interval arithmetic (such as the SUN FORTE Fortran 95 and C++ compilers =-=[20]-=-), (8) and (10) can be used directly. On other machines, it is preferable to rewrite the expressions so that simple directed rounding applies. Since r vanishes in exact arithmetic, it will be tiny eve... |

1 |
Aggregation and mixed integer rounding to solve MIPs, CORE Discussion paper 9839. http://www.core.ucl.ac.be/wolsey/mir.ps
- Marchand, Wolsey
(Show Context)
Citation Context ...ual lower and upper bound, as soon as a rounding error is introduced.) 6 Mixed integer rounding Both MIR cuts and generalized Gomory cuts may be generated by mixed integer rounding (Marchand & Wolsey =-=[12]-=-). Mixed integer rounding (MIR) is a technique for constructing cuts based on the following (or an equivalent) rounding lemma. It exploits the knowledge that some or all the variables involved in an i... |