Facet of regular 0–1 polytopes

Hammer, Peter L.; Johnson, Ellis L.; Peled, Uri N.

doi:10.1007/bf01580442

Cited by 171 publications

(88 citation statements)

References 3 publications

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“…In particular lifting has been crucial in developing strong facet-defining inequalites for 0-1 knapsack sets Balas (1975); Hammer et al (1975); Wolsey (1975), and their mixed integer counterpart, called single node flow sets Gu et al (1999); Padberg et al (1985); Stallaert (1997); Van Roy and Wolsey (1986). What is more these inequalities have provided effective cuts for 0-1 programs Crowder et al (1963) and mixed 0-1 programs Van Roy and Wolsey (1987), and along with Gomory mixed integer cuts Gomory (1960) and mixed integer rounding inequalities Nemhauser and Wolsey (1990) form part of state-of-the-art commercial mixed integer programming systems such as Cplex and Xpress.…”

Section: Introductionmentioning

confidence: 99%

Lifting, superadditivity, mixed integer rounding and single node flow sets revisited

Louveaux

Wolsey

2007

Ann Oper Res

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In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs.The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0-1 single node flow set.

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Section: Introductionmentioning

confidence: 99%

Lifting, superadditivity, mixed integer rounding and single node flow sets revisited

Louveaux

Wolsey

2007

Ann Oper Res

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“…The set of valid inequality constraints are called Minimum Dependent Sets (MDS) according to Visoldilokpun [41] and was reported in Nemhauser and Wolsey [42]. According to Kellerer et al [47], the minimal dependent sets are also known as minimal cover inequalities, and defining facets for the knapsack polytope was originally studied back in 1975 by Balas [48], Hammer et al [49], and Wolsey [50]. Consider the following feasible set for the binary knapsack problem…”

Section: Maximin-whole Area-removing-total Threat Reductionmentioning

confidence: 99%

Unmanned aerial vehicle routing in the presence of threats

Alotaibi

Rosenberger

Mattingly

et al. 2018

Computers & Industrial Engineering

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“…This comes up to finding the minimum possible cardinality of a minimal cover set (Hammer et al, 1975) for the knapsack constraint (15). Suppose that we sort the probabilities of all the scenarios in S 1 in descending order and this ordered vector of probabilities are denoted byπ ′′ .…”

Section: Initializationmentioning

confidence: 99%

Mathematical programming approaches for generating p-efficient points

Lejeune¹,

Noyan²

2010

European Journal of Operational Research

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Probabilistically constrained problems, in which the random variables are finitely distributed, are nonconvex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.

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Facet of regular 0–1 polytopes

Cited by 171 publications

References 3 publications

Lifting, superadditivity, mixed integer rounding and single node flow sets revisited

Lifting, superadditivity, mixed integer rounding and single node flow sets revisited

Unmanned aerial vehicle routing in the presence of threats

Mathematical programming approaches for generating p-efficient points

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