The structure and solution of various quadratic pseudo-Boolean optimization problems
This thesis concerns the structure and solution of certain quadratic pseudo-Boolean optimization problems. In Chapters 3, 4, 5, and 6 we investigate the structure and solution of the quadratic 0-1 knapsack problem, i.e. the optimization of a quadratic pseudo-Boolean function subject to a linear ineq...
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| Format: | Dissertation |
| Sprache: | Englisch |
| Veröffentlicht: |
ProQuest Dissertations & Theses
01.01.1997
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| ISBN: | 0591496577, 9780591496574 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | This thesis concerns the structure and solution of certain quadratic pseudo-Boolean optimization problems. In Chapters 3, 4, 5, and 6 we investigate the structure and solution of the quadratic 0-1 knapsack problem, i.e. the optimization of a quadratic pseudo-Boolean function subject to a linear inequality. First, we consider the structure of the polytope that arises from a linearization of the quadratic 0-1 knapsack problem. In Chapter 3, we describe some valid inequalities and characterize conditions under which they are facets of certain subpolytopes. In Chapter 4, we consider the lifting of the inequalities given in Chapter 3 and give bounds on the optimal increases in the values of the coefficients for special classes of these inequalities. In Chapter 5, we investigate certain complexity questions relating to the lifting of the inequalities given in Chapter 3. Finally, in Chapter 6, we are interested in the solution of quadratic 0-1 knapsack problems where the objective function is supermodular. In Chapter 7, we examine the solutions of unconstrained quadratic pseudo-Boolean problems where the quadratic function can be written as the product of two linear functions. We show that a continuous relaxation can be solved quickly, and we develop methods for fixing variables at their optimal integer values. Finally, extensive computational results are given. In Chapter 8, we are interested in finding two solutions to a family of set covering constraints that are "as distinct as possible." For these problems, we give a constrained quadratic pseudo-Boolean formulation and describe three heuristics for obtaining good approximations to the optimal solutions. Computational results are given to demonstrate the effectiveness of the methods. In Chapter 9, we define a partial order between variables of a pseudo-Boolean function and examine the class of those pseudo-Boolean functions for whom this ordering is complete. We investigate certain properties of these functions, and show that many constrained and unconstrained problems involving such functions can be easily solved. |
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| Bibliographie: | SourceType-Dissertations & Theses-1 ObjectType-Dissertation/Thesis-1 content type line 12 |
| ISBN: | 0591496577 9780591496574 |