Constructing packings in Grassmannian manifolds via alternating projection

This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix. This approach is both powerful and versatile. In cases where experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini--Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal.Comment: 41 pages, 7 tables, 4 figure

Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

In this thesis, we provide (or review) new and effective algorithms based on Mixed-Integer Linear Programming (MILP) models and/or decomposition approaches to solve exactly various cutting and packing problems. The first three contributions deal with the classical bin packing and cutting stock problems. First, we propose a survey on the problems, in which we review more than 150 references, implement and computationally test the most common methods used to solve the problems (including branch-and-price, constraint programming (CP) and MILP), and we successfully propose new instances that are difficult to solve in practice. Then, we introduce the BPPLIB, a collection of codes, benchmarks, and links for the two problems. Finally, we study in details the main MILP formulations that have been proposed for the problems, we provide a clear picture of the dominance and equivalence relations that exist among them, and we introduce reflect, a new pseudo-polynomial formulation that achieves state of the art results for both problems and some variants. The following three contributions deal with two-dimensional packing problems. First, we propose a method using Logic based Benders’ decomposition for the orthogonal stock cutting problem and some extensions. We solve the master problem through an MILP model while CP is used to solve the slave problem. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. Then, we introduce TwoBinGame, a visual application we developed for students to interactively solve two-dimensional packing problems, and analyze the results obtained by 200 students. Finally, we study a complex optimization problem that originates from the packaging industry, which combines cutting and scheduling decisions. For its solution, we propose mathematical models and heuristic algorithms that involve a non-trivial decomposition method. In the last contribution, we study and strengthen various MILP and CP approaches for three project scheduling problems

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones