Experiments in mixed-integer linear programming

Bénichou, Michel; Gauthier, Jean-Michel; Girodet, Paul; Hentges, Gerard; Ribière, Gerard; Vincent, O.

doi:10.1007/bf01584074

Cited by 244 publications

(135 citation statements)

References 3 publications

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“…(where µ ∈ [0, 1] is a prescribed parameter usually bigger than [18] and strong-branching [67,7]. Next, we will present these two methods and how they can be combined.…”

Section: Branching Rulesmentioning

confidence: 99%

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Bonami

Kılınç

Linderoth

2011

The IMA Volumes in Mathematics and Its Applications

195

181

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Abstract. This paper provides a survey of recent progress and software for solving mixed integer nonlinear programs (MINLP) wherein the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in very years. By exploiting analogies to the case of well-known techniques for solving mixed integer linear programs and incorporating these techniques into the software, significant improvements have been made in our ability to solve the problems.Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.1. Introduction. Mixed-Integer Nonlinear Programs (MINLP) are optimization problems where some of the variables are constrained to take integer values and the objective function and feasible region of the problem are described by nonlinear functions. Such optimization problems arise in many real world applications. Integer variables are often required to model logical relationships, fixed charges, piecewise linear functions, disjunctive constraints and the non-divisibility of resources. Nonlinear functions are required to accurately reflect physical properties, covariance, and economies of scale.In all its generality, MINLP forms a particularly broad class of challenging optimization problems as it combines the difficulty of optimizing over integer variables with handling of nonlinear functions. Even if we restrict our model to contain only linear functions, MINLP reduces to a Mixed-Integer Linear Program (MILP), which is an NP-Hard problem [56]. On the other hand, if we restrict our model to have no integer variable but allow for general nonlinear functions in the objective or the constraints, then MINLP reduces to a Nonlinear Program (NLP) which is also known to be NP-Hard [91]. Combining both integrality and nonlinearity can lead to examples of MINLP that are undecidable [68].

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“…(where µ ∈ [0, 1] is a prescribed parameter usually bigger than [18] and strong-branching [67,7]. Next, we will present these two methods and how they can be combined.…”

Section: Branching Rulesmentioning

confidence: 99%

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Bonami

Kılınç

Linderoth

2011

The IMA Volumes in Mathematics and Its Applications

195

181

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“…When a node is branched, two nodes are generated by splitting a bound constraint. The selection of the variable whose bound will be splitted follows a rule based on pseudocosts [3]. Pseudocost is a measure associated with each variable of a problem.…”

Section: Solution Methodsmentioning

confidence: 99%

“…The rule for selecting a variable, whose explanation was delayed in the previous paragraph, is based on the global pseudocosts (see [3]) and merely says to select a variable x i * such that…”

Section: Solution Methodsmentioning

confidence: 99%

Orthogonal packing of identical rectangles within isotropic convex regions

Birgin

Lobato

2010

Computers & Industrial Engineering

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“…The concept of costs and estimations to generate the Branch-and-Bound tree has been introduced by Benichou et al (1971) and pseudo-costs based branching strategies are implemented in many integer programming solvers.…”

Section: Branching Strategiesmentioning

confidence: 99%

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

Fortz

Violin

2013

Electronic Notes in Discrete Mathematics

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This paper considers a pricing problem on a network with connected toll arcs and proposes a Dantzig-Wolfe reformulation for it. First, the relaxation of this formulation is theoretically shown to be at least as good as the reference proposed in the literature. Then, we detail the particularities of the implementation of a branch-and-price algorithm for solving it such as a primal heuristic, the branching rules, the stabilization process, and an algorithm for fixing variables using Lagrangian duality. Finally, numerical results on two types of instances, real and pseudo-randomly generated, are reported, confirming numerically the main theoretical results.

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Experiments in mixed-integer linear programming

Cited by 244 publications

References 3 publications

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Orthogonal packing of identical rectangles within isotropic convex regions

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

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