Safe bounds in linear and mixed-integer linear programming

Neumaier, Arnold; Shcherbina, Oleg

doi:10.1007/s10107-003-0433-3

Cited by 137 publications

(120 citation statements)

References 16 publications

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“…The generation of numerically safe GMI cuts has been investigated in [24] when all the variables are bounded, and in [12] in general. These safe cuts are generated in floating-point arithmetic using a clever rounding scheme and they are guaranteed to be satisfied by all feasible solutions of the MILP.…”

Section: Related Workmentioning

confidence: 99%

On the safety of Gomory cut generators

2013

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Gomory mixed-integer cuts are one of the key components in Branch-and-Cut solvers for mixed-integer linear programs. The textbook formula for generating these cuts is not used directly in open-source and commercial software that work in finite precision: Additional steps are performed to avoid the generation of invalid cuts due to the limited numerical precision of the computations. This paper studies the impact of some of these steps on the safety of Gomory mixed-integer cut generators. As the generation of invalid cuts is a relatively rare event, the experimental design for this study is particularly important. We propose an experimental setup that allows statistically significant comparisons of generators. We also propose a parameter optimization algorithm and use it to find a Gomory mixedinteger cut generator that is as safe as a benchmark cut generator from a commercial solver even though it generates many more cuts.

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Section: Related Workmentioning

confidence: 99%

On the safety of Gomory cut generators

2013

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“…Thus, the computed minimizer might be wrong. The unsafe MILP solver is made safe thanks to the correction procedure introduced in [23]. It consists in computing a safe lower bound of the global minimizer.…”

Section: Getting a Safe Minimizermentioning

confidence: 99%

“…E cient MILP solvers rely on FP computations and thus, might miss some solutions. In order to ensure a safe behavior of our algorithm, correct rounding directions are applied to the relaxation coe cients [19,4] and a procedure [23] to compute a safe minimizer from the unsafe result of the MILP solver is also applied. Preliminary experiments are promising and this new ltering technique should really help to scale up all veri cations tools that uses a FP solver.…”

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confidence: 99%

Boosting Local Consistency Algorithms over Floating-Point Numbers

Belaid¹,

Michel²,

Rueher³

2012

Lecture Notes in Computer Science

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Abstract. Solving constraints over oating-point numbers is a critical issue in numerous applications notably in program veri cation. Capabilities of ltering algorithms over the oating-point numbers (F) have been so far limited to 2b-consistency and its derivatives. Though safe, such ltering techniques su er from the well known pathological problems of local consistencies, e.g., inability to e ciently handle multiple occurrences of the variables. These limitations also have their origins in the strongly restricted oating-point arithmetic. To circumvent the poor properties of oating-point arithmetic, we propose in this paper a new ltering algorithm, called FPLP, which relies on various relaxations over the real numbers of the problem over F. Safe bounds of the domains are computed with a mixed integer linear programming solver (MILP) on safe linearizations of these relaxations. Preliminary experiments on a relevant set of benchmarks are promising and show that this approach can be e ective for boosting local consistency algorithms over F. IntroductionCritical systems are more and more relying on oating-point (FP) computations. For instance, embedded systems are typically controlled by software that store measurements and environment data as oating-point number (F). The initial values and the results of all operations must therefore be rounded to some nearby oat. This rounding process can lead to signi cant changes, and, for example, can modify the control ow of the program. Thus, the veri cation of programs performing FP computations is a key issue in the development of critical systems.Methods for verifying programs performing FP computations are mainly derived from standard program veri cation methods. Bounded model checking (BMC) techniques have been widely used for nding bugs in hardware design [3] and software [11]. SMT solvers are now used in most of the state-of-the-art BMC tools to directly work on high level formula (see [2,9,11]). The bounded model checker CBMC encodes each FP operation of the program with a set of logic This work was partially supported by ANR VACSIM (ANR-11-INSE-0004), ANR AEOLUS (ANR-10-SEGI-0013) and OSEO ISI PAJERO projects.

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“…Recently, Neumaier and Shcherbina [19] have investigated rigorous error bounds for mixed-integer linear programming problems. In their paper, in addition to rigorous cuts and a certificate of infeasibility, a rigorous lower bound for linear programming problems with exact input data and finite simple bounds is presented.…”

Section: Rigorous Error Boundsmentioning

confidence: 99%

Computational Experience with Rigorous Error Bounds for the Netlib Linear Programming Library

Keil

Jansson

2006

Reliable Comput

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Abstract. The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordóñez and Freund that 71% of these problems are ill-conditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. In addition to the original input data of these problems we also consider interval input data. The computed rigorous bounds and the performance of the algorithms are related to the distance to the next ill-posed linear programming problem.

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Safe bounds in linear and mixed-integer linear programming

Cited by 137 publications

References 16 publications

On the safety of Gomory cut generators

On the safety of Gomory cut generators

Boosting Local Consistency Algorithms over Floating-Point Numbers

Computational Experience with Rigorous Error Bounds for the Netlib Linear Programming Library

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