Column generation for extended formulations

Sadykov, Ruslan; Vanderbeck, François

doi:10.1007/s13675-013-0009-9

Cited by 29 publications

(24 citation statements)

References 23 publications

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“…Our objective (19) is to find the combination of patterns of highest cost. Constraint (20) ensures that only one pattern is selected and constraint set (21) requires that the selected pattern does not imply item overproduction. Note that our definition of patterns allows for an overproduction.…”

Section: Using Column Generationmentioning

confidence: 99%

Combining dynamic programming with filtering to solve a four-stage two-dimensional guillotine-cut bounded knapsack problem

Clautiaux

Sadykov

Vanderbeck

et al. 2018

Discrete Optimization

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The two-dimensional knapsack problem consists in packing a set of small rectangular items into a given large rectangle while maximizing the total reward associated with selected items. We restrict our attention to packings that emanate from a k-stage guillotine-cut process. We introduce a generic model where a knapsack solution is represented by a flow in a directed acyclic hypergraph. This hypergraph model derives from a forward labeling dynamic programming recursion that enumerates all non-dominated feasible cutting patterns. To reduce the hypergraph size, we make use of further dominance rules and a filtering procedure based on Lagrangian reduced costs fixing of hyperarcs. Our hypergraph model is (incrementally) extended to account for explicit bounds on the number of copies of each item. Our exact forward labeling algorithm is numerically compared to solving the max-cost flow model in the base hyper-graph with side constraints to model production bounds. Benchmarks are reported on instances from the literature and on datasets derived from a real-world application.

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Section: Using Column Generationmentioning

confidence: 99%

Combining dynamic programming with filtering to solve a four-stage two-dimensional guillotine-cut bounded knapsack problem

Clautiaux

Sadykov

Vanderbeck

et al. 2018

Discrete Optimization

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“…Similarly, when only the number of variables is large, the formulations can be used in column generation or branch-and-price procedures [15]. These procedures can be very effective and so can extensions such as branchand-cut-and-price [48] and branch-and-price for extended formulations [142]. For this reason, when both large and small (usually extended) formulations are available it is not always clear which is more convenient.…”

Section: Large Formulationsmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

259

143

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1. Introduction. Throughout more than 50 years of existence, mixed integer linear programming (MIP) theory and practice have been significantly developed, and it is now an indispensable tool in business and engineering [68,94,104]. Two reasons for the success of MIP are linear programming (LP) based solvers and the modeling flexibility of MIP. We now have several extremely effective state-of-the-art solvers [82,69, 52,171] that incorporate many advanced techniques [1,2,25,23,92,112,24] and, since its early stages, MIP has been used to model a wide range of applications [44,45].While in many cases constructing valid MIP formulations is relatively straightforward, some care should be taken in this construction as certain formulation attributes can significantly reduce the effectiveness of LP-based solvers. Fortunately, constructing formulations that behave well with state-of-the-art solvers can usually be achieved by following simple guidelines described in standard textbooks. However, more advanced techniques can often perform significantly better than textbook formulations and are sometimes a necessity. The main objective of this survey is to summarize the state of the art of such formulation techniques for a wide range of problems. To keep the length of this survey under control, we concentrate on formulations for sets of a mixed integer nature that require both integer constrained and continuous variables. We hence purposefully place less emphasis on some related areas such as combinatorial optimization, quadratic and polynomial 0/1 optimization, and polyhedral approximations of convex sets. These topics are certainly areas of important and active research, so we cover them succinctly in section 12.

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“…This process allows the implicit combination of arcs into paths without having to generate these. Sadykov and Vanderbeck (2013) describe this in generality.…”

Section: Dynamic Dantzig-wolfe Decompositionmentioning

confidence: 99%