Open Access

Jonas Schwinn

• In the context of this dissertation we consider two mathematical optimization problems. The first constitutes a classical problem of linear optimization, called the transportation problem; a detailed introduction is given in the first chapter. This problem motivated significant contributions in research on linear optimization and network flow problems and plays an important role in praxis to this day. Besides the logistic applications that motivated the work of Tolstoi, Kantorovich, Hitchcock, Koopmans, Dantzig, Ford and Fulkerson in the first half of the 20th century, the problem is now also attracting renewed interest in areas such as manufacturing industry, computer science, image processing, data analysis and machine learning. The main goal of this thesis is to apply the concept of Column Generation to a popular solution method originally devised by Dantzig (1951) called the Transportation Simplex. To the best of our knowledge, this approach has not yet been taken into accountIn the context of this dissertation we consider two mathematical optimization problems. The first constitutes a classical problem of linear optimization, called the transportation problem; a detailed introduction is given in the first chapter. This problem motivated significant contributions in research on linear optimization and network flow problems and plays an important role in praxis to this day. Besides the logistic applications that motivated the work of Tolstoi, Kantorovich, Hitchcock, Koopmans, Dantzig, Ford and Fulkerson in the first half of the 20th century, the problem is now also attracting renewed interest in areas such as manufacturing industry, computer science, image processing, data analysis and machine learning. The main goal of this thesis is to apply the concept of Column Generation to a popular solution method originally devised by Dantzig (1951) called the Transportation Simplex. To the best of our knowledge, this approach has not yet been taken into account in the literature. Our method is based on a recent publication by Schuhmacher and Gottschlich (2014) and introduces low-level changes in the implementation of the Transportation Simplex. In order to develop a numerically competitive version of the Column Generation approach, we need to build on an efficient implementation of this algorithm, which is described in Chapter 1. In this course, we further consider heuristic approaches to the transportation problem in Chapter 3. These are used to determine initial solutions for the Transportation Simplex and have significant impact on overall performance. As a by-product of our analysis, we compare the Transportation Simplex with the more general Network Simplex. While we consider symmetric as well as asymmetric transportation problems in the theoretical part of this work, the numerical analysis is focused on the symmetric case. Second, we develop another specifically tailored Column Generation method for a long-standing problem in the field of multivariate statistics in Chapter 6. Testing a given matrix for membership in the family of Bernoulli matrices is of high practical relevance due to the manifold applications of Bernoulli vectors in areas such as information technology, finance, medicine and operations research. The problem is known to be NP-complete, due to its relation to membership testing on the well-known correlation polytope, cf. Pitowsky (1991). For this purpose, we propose to solve a linear formulation of the problem by Column Generation and deal with the issue of exponentially many primal variables by applying a straightforward, yet novel, solution of the arising subproblems.…