5 research outputs found

    Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method

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    This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of non-negative real numbers satisfying certain summation conditions. Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure. First, we give new results on the diameters of certain classes of transportation polytopes and their relation to the Hirsch Conjecture, which asserts that the diameter of every dd-dimensional convex polytope with nn facets is bounded above by n−dn-d. In particular, we prove a new quadratic upper bound on the diameter of 33-way axial transportation polytopes defined by 11-marginals. We also show that the Hirsch Conjecture holds for p×2p \times 2 classical transportation polytopes, but that there are infinitely-many Hirsch-sharp classical transportation polytopes. Second, we present new results on subpolytopes of transportation polytopes. We investigate, for example, a non-regular triangulation of a subpolytope of the fourth Birkhoff polytope B4B_4. This implies the existence of non-regular triangulations of all Birkhoff polytopes BnB_n for n≥4n \geq 4. We also study certain classes of network flow polytopes and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California, Davis. 183 pages, 49 figure

    Subject Index Volumes 1–200

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    Optimal Partitioning and Coordination Decisions in Decomposition-based Design Optimization.

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    Successful design of complex modern products is a grand challenge for design organizations. The task is becoming increasingly important due to economic competition and concern over safety, reliability, and energy efficiency. Automotive and aerospace products, for example, are composed of numerous interdependent subsystems with a level of complexity that surpasses the capability of a single design group. A common approach is to partition complex design problems into smaller, more manageable design tasks that can be solved by individual design groups. Effective management of interdependency between these subproblems is critical, and a successful design process ultimately must meet the needs of the overall system. Decomposition-based design optimization techniques provide a mathematical foundation and computational tools for developing such design processes. Two tasks must be performed so that decomposition-based design optimization can be used to solve a system design problem: partitioning the system into subproblems, and determining a coordination method for guiding subproblem solutions toward the optimal system design. System partition and coordination strategy have a profound impact on the design process. The effect of partitioning and coordination decisions have been studied independently, while interaction between these decisions has been largely ignored. It is shown here that these two sets of decisions do interact: how a system is partitioned influences appropriate coordination decisions, and vice versa. Consequently, addressing partitioning and coordination decisions simultaneously leads to improved system design processes. The combined partitioning and coordination decision problem is a difficult combinatorial problem. An evolutionary algorithm that solves this decision problem effectively is presented. The set of all partitioning and coordination options for a specific formulation framework, augmented Lagrangian coordination (ALC), is derived, and a method for choosing Pareto-optimal solutions from amongst these options is described. Concepts and techniques are demonstrated using several engineering example problems. A detailed model for an electric vehicle design problem is presented that considers three vehicle systems: powertrain, chassis, and structure, and partitioning and coordination decisions for this problem are analyzed.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58449/1/jtalliso_1.pd
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