K. Aardal and S. van Hoesel. Polyhedral techniques in combinatorial optimization ii: Computations. Statistica Neerlandica, 50:3, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....3.1. Branch and bound. One way to obtain a lower bound is to take an integer linear programming formulation of the problem and to drop the integrality requirements on the variables. The bound provided by this linear relaxation of the problem is usually too weak. Polyhedral lower bounds [1] [2] [23] are obtained by identifying additional linear inequalities that strengthen the linear relaxation, and in the best case are necessary in the linear description of the convex hull of feasible solutions. If we would know the linear description completely, then we could solve the problem as a ....

K. I. Aardal, C. P. M. van Hoesel (1999). Polyhedral techniques in combinatorial optimization II: computations and applications. Statistica Neerlandica 53, 129-178.

....3.1. Branch and bound. One way to obtain a lower bound is to take an integer linear programming formulation of the problem and to drop the integrality requirements on the variables. The bound provided by this linear relaxation of the problem is usually too weak. Polyhedral lower bounds [1] [2] 23] are obtained by identifying additional linear inequalities that strengthen the linear relaxation, and in the best case are necessary in the linear description of the convex hull of feasible solutions. If we would know the linear description completely, then we could solve the problem as ....

K. I. Aardal, C. P. M. van Hoesel (1996). Polyhedral techniques in combinatorial optimization I: theory. Statistica Neerlandica 50, 3-26.

....set containing all points in X . The advantage with this approach is that if the convex hull of X is known, we can solve minfcx : x 2 conv(X)g as a linear programming problem, which is computationally easy, but gives the same solution as optimizing over X . As discussed in Part I of this article (Aardal and Van Hoesel (1996)) it is hard in general to describe the convex hull of X by concise families of inequalities even if we allow for classes containing exponentially many inequalities. In a practical setting, however, the complete description of the convex hull of X is not needed. What is important is that we have a ....

K. Aardal and C.P.M. van Hoesel (1996) "Polyhedral techniques in combinatorial optimization I: theory", Statistica Neerlandica 50 3--26.

....decade. During the past ten years, however, an enormous amount of more problem specific results have been obtained. Moreover, surprisingly large instances have been solved using a mixture of cutting plane algorithms and branch and bound. For recent surveys we refer to Aardal and Van Hoesel [1] [2], and to Chapter 3 of [22] Similar developments have been attained for column generation methods, which can be viewed as dual to cutting plane techniques. For a survey we refer to Barnhart et al. 11] A new development of the last decade is the theoretical quality analysis of cutting planes. ....

K. Aardal, S. van Hoesel (1995). Polyhedral Techniques in Combinatorial Optimization II: Computations, Report UU-CS-1995-42, Utrecht University.

....past decade. During the past ten years, however, an enormous amount of more problem specific results have been obtained. Moreover, surprisingly large instances have been solved using a mixture of cutting plane algorithms and branch and bound. For recent surveys we refer to Aardal and Van Hoesel [1], 2] and to Chapter 3 of [22] Similar developments have been attained for column generation methods, which can be viewed as dual to cutting plane techniques. For a survey we refer to Barnhart et al. 11] A new development of the last decade is the theoretical quality analysis of cutting ....

K. Aardal, C.P.M. van Hoesel (1996). Polyhedral techniques in combinatorial optimization I: Theory. Statist. Neerlandica 50, 3--26.

....faces of the convex hull of feasible solutions to the initial formulation strengthens the lower bound obtained from the linear relaxation. This so called cutting plane technique has made it possible to solve large scale instances of several hard combinatorial optimization problems, see Aardal and Van Hoesel (1995,1996). When developing valid inequalities for a certain problem type one typically starts by investigating the facial structure of various relaxations of the problem. The advantage is that the inequalities that are valid for a relaxation RP of a polyhedron P is also valid for P , and in several cases ....

K. Aardal and C.P.M. Van Hoesel (1996). Polyhedral techniques in combinatorial optimization I: Theory.

....faces of the convex hull of feasible solutions to the initial formulation strengthens the lower bound obtained from the linear relaxation. This so called cutting plane technique has made it possible to solve large scale instances of several hard combinatorial optimization problems, see Aardal and Van Hoesel (1995,1996) When developing valid inequalities for a certain problem type one typically starts by investigating the facial structure of various relaxations of the problem. The advantage is that the inequalities that are valid for a relaxation RP of a polyhedron P is also valid for P , and in several ....

....4 Computational Experience Here we consider the computational effect of adding knapsack cover inequalities to the linear relaxation of the single level formulation, and knapsack cover, flow cover and fixed charge path inequalities to the two level formulation. The interested reader is referred to Aardal (1995) for more computational results regarding the single level problem, and to Aardal (1992) for results regarding the two level problem. We have considered single level instances of 5 different sizes a Theta b, where a denotes the number of clients and b the number of facilities. The problem ....

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K. Aardal and C.P.M. Van Hoesel (1995). Polyhedral techniques in combinatorial optimization II: Computations.

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K. Aardal and C.P.M. van Hoesel (1995b) "Polyhedral techniques in combinatorial optimization I: theory" (to appear in The Golden Jubilee Issue of Statistica Neerlandica.

.... problems such as the traveling salesman problem (TSP) the linear ordering problem and the node packing problem (which is very close to the frequency assignment problem) A recent survey of the approach and a list of successful practical applications using it can be found in Aardal and van Hoesel [1]. An overview of implementational issues of branch and cut can be found in Junger et al. 4] A generic branch and cut code (MINTO) for integer linear programming problems has been developed by Nemhauser et al. 9] One of the main features of this package is that it provides the possibility of ....

K. Aardal and S. van Hoesel, "Polyhedral Techniques in Combinatorial Optimization," to appear in Statistica Neerlandica, 1995.

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K. Aardal and S. van Hoesel. Polyhedral techniques in combinatorial optimization ii: Computations. Statistica Neerlandica, 50:3, 1996.

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