Acceleration of cutting‐plane and column generation algorithms: Applications to network design

Ben‐Ameur, Walid; Neto, José

doi:10.1002/net.20137

Cited by 66 publications

(64 citation statements)

References 61 publications

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“…The substitution of x * withx in the separation problem suggests some similarities with stabilization techniques used in column generation where dual multipliers are appropriately modified in the pricing subproblem, see, e.g., [DL05], and with the in-out search strategy for cutting plane methods [BAN07,FS10].…”

Section: Revised Coordinated Cut Separation Problemmentioning

confidence: 99%

Coordinated cutting plane generation via multi-objective separation

2012

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In cutting plane methods, the question of how to generate the "best possible" set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0-1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating Stable Set and Cut Set inequalities for, respectively, the Max Clique and Min Steiner Tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for Max Clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.

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Section: Revised Coordinated Cut Separation Problemmentioning

confidence: 99%

Coordinated cutting plane generation via multi-objective separation

2012

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“…Referring for example to [6] for a wide overview, we restrict here our attention to the following stabilization scheme. Instead of θ k , some other function is maximized, the dual restricted master (1.14) being replaced by…”

Section: Stabilized Column Generationmentioning

confidence: 99%

Comparison of bundle and classical column generation

Briant¹,

Lemaréchal

Meurdesoif

et al. 2006

Math. Program.

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Abstract. When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley's method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

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“…Computational results are presented in Sect. 4. Section 5 concludes the work with a short summary and a discussion of future work.…”

Section: Figure 1 Depicts An Instance Of the Stprbh And Its Optimal Smentioning

confidence: 99%

“…This may be explained by the fact that only the y-variables appear in the objective function, and the continuous addition of violated inequalities mainly influences the values of the x and y h variables (without significantly changing the values of y-variables). A more sophisticated cut-loop scheme, like, e.g., in-out separation considered in [4,12], could theoretically further improve the performance, however, we did not investigate this further, since the current tailing-off control already …”

Section: Studying the Influence Of The Valid Inequalitiesmentioning

confidence: 99%

A node-based layered graph approach for the Steiner tree problem with revenues, budget and hop-constraints

Sinnl

Ljubić

2016

Math. Prog. Comp.

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The Steiner tree problem with revenues, budget and hop-constraints (STPRBH) is a variant of the classical Steiner tree problem. The goal is to find a tree maximizing the collected revenue, which is associated with nodes, subject to a given budget for the edge cost of the tree and a hop-limit for the distance between the given root node and any other node in that tree. In this work, we introduce a novel generic way to model hop-constrained tree problems as integer linear programs and apply it to the STPRBH. Our approach is based on the concept of layered graphs that gained widespread attention in the recent years, due to their computational advantage when compared to previous formulations for modeling hop-constraints. Contrary to previous MIP formulations based on layered graphs (that are arc-based models), our model is node-based. Thus it contains much less variables and allows to tackle largescale instances and/or instances with large hop-limits, for which the size of arc-based layered graph models may become prohibitive. The aim of our model is to provide a good compromise between quality of root relaxation bounds and the size of the underlying MIP formulation. We implemented a branch-and-cut algorithm for the STPRBH based on our new model. Most of the instances available for the DIMACS challenge, including 78 (out of 86) previously unsolved ones, can be solved to proven optimality within a time limit of 1000 s, most of them being solved within a few seconds only. These instances contain up to 500 nodes and 12,500 edges, with hop-limit up to 25.

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Acceleration of cutting‐plane and column generation algorithms: Applications to network design

Cited by 66 publications

References 61 publications

Coordinated cutting plane generation via multi-objective separation

Coordinated cutting plane generation via multi-objective separation

Comparison of bundle and classical column generation

A node-based layered graph approach for the Steiner tree problem with revenues, budget and hop-constraints

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