Canonical Cuts on the Unit Hypercube

Balas, Egon; Jeroslow, Robert G.

doi:10.1137/0123007

Cited by 217 publications

(118 citation statements)

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“…Let g be locally Lipschitz at every point in an open set containing X, let D be a finite box such that ∂g(x) ⊆ D for all x ∈ X, and let g(x) = g(x) + min d∈D d(x −x) as in (4). Then the inequality g(x) ≤ 0 is valid for P .…”

Section: Interval-gradient and Interval-subgradient Cutsmentioning

confidence: 99%

“…Moreover, we consider the extension to MINLP of a classical family of MILP cuts mostly known as No-good cuts (or Farkas cuts) and originally introduced, to the best of our knowledge, in [4] with the name of canonical cuts. These cutting planes are generated with respect to a specific solutionx by imposing a positive distance betweenx and any new solution 1 .…”

Section: Introductionmentioning

confidence: 99%

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On interval-subgradient and no-good cuts

d'Ambrosio

Frangioni

Liberti

et al. 2010

Operations Research Letters

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Interval-gradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x,x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that no-good cuts (which have the form x−x ≥ ε for some norm and positive constant ε) are a special case of interval-subgradient cuts whenever the 1-norm is used. We then briefly discuss what happens if other norms are used.

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Section: Interval-gradient and Interval-subgradient Cutsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On interval-subgradient and no-good cuts

d'Ambrosio

Frangioni

Liberti

et al. 2010

Operations Research Letters

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“…first mentioned by Balas and Jeroslow [3]. Thus, constraints (11) for x binary are equivalent to the following set of canonical cuts i∈x * (1)…”

Section: Implicit Reformulationmentioning

confidence: 99%

The robust vehicle routing problem with time windows

Agra

Christiansen

Figueiredo

et al. 2013

Computers & Operations Research

165

107

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This paper addresses the robust vehicle routing problem with time windows. We are motivated by a problem that arises in maritime transportation where delays are frequent and should be taken into account. Our model only allows routes that are feasible for all values of the travel times in a predetermined uncertainty polytope, which yields a robust optimization problem. We propose two new formulations for the robust problem, each based on a different robust approach. The first formulation extends the well-known resource inequalities formulation by employing adjustable robust optimization. We propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. The second formulation generalizes a path inequalities formulation to the uncertain context. The uncertainty appears implicitly in this formulation, so that we develop a new cutting plane technique for robust combinatorial optimization problems with complicated constraints. In particular, efficient separation procedures are discussed. We compare the two formulations on a test bed composed of maritime transportation instances. These results show that the solution times are similar for both formulations while being significantly faster than the solutions times of a layered formulation recently proposed for the problem.

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“…Balas and Jeroslow (1972) propose "no-good" cuts for the case of binary master variables (Eq. 23) X i:…”

Section: £mentioning

confidence: 99%

Perspectives in multilevel decision-making in the process industry

Brunaud¹,

Grossmann²

2017

Front. Eng

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Decisions in supply chains are hierarchically organized. Strategic decisions involve the long-term planning of the structure of the supply chain network. Tactical decisions are mid-term plans to allocate the production and distribution of materials, while operational decisions are related to the daily planning of the execution of manufacturing operations. These planning processes are conducted independently with minimal exchange of information between them. Achieving a better coordination between these processes allows companies to capture benefits that are currently out of their reach and improve the communication among their functional areas. We propose a network representation for the multilevel decision structure and analyze the components that are involved in finding integrated solutions that maximize the sum of the benefits of all nodes of the decision network. Although such task is very challenging, significant research progress has been made in each component of this structure. An overview of strategic models, mid-term planning models, and scheduling models is presented to address the solution of each node in the decision network. Coordination mechanisms for converging the integrated solutions are also analyzed, including solving large-scale models, multiobjective optimization, bi-level programming, and decomposition. We conclude by summarizing the challenges that hinder the full integration of multilevel decision making in supply chain management.

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Canonical Cuts on the Unit Hypercube

Cited by 217 publications

References 0 publications

On interval-subgradient and no-good cuts

On interval-subgradient and no-good cuts

The robust vehicle routing problem with time windows

Perspectives in multilevel decision-making in the process industry

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