Browsing by Author "Boyd, E. Andrew"

Now showing 1 - 11 of 11
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    A Combinatorial Abstraction of One Shortest Path Problem and Its Relationship to Greedoids
    (1988-05) Boyd, E. Andrew
    A natural generalization of the shortest path problem to arbitrary set systems is presented that captures a number of interesting problems, including the usual graph-theoretic shortest path problem and the problem of finding a minimum weight set on a matroid. Necessary and sufficient conditions for the solution of this problem by the greedy algorithm are then investigated. In particular, it is noted that it is necessary but not sufficient for the underlying combinatorial structure to be a greedoid, and the three extremely diverse collections of sufficient conditions taken from the greedoid literature are presented.
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    A Pseudopolynomial Network Flow Formulation for Exact Knapsack Separation
    (1991-03) Boyd, E. Andrew
    The NP-complete separation problem for the knapsack polyhedron P is formulated as a side-constrained network flow problem with a pseudopolynomial number of vertices and edges. It is demonstrated that the primal polyhedron associated with this formulation can be projected onto an appropriate subspace to yield P and that the dual polyhedron can be projected onto an appropriate subspace to yield the polar of P. Practical consequences of the formulation are discussed.
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    A Test Set of Real-World Mixed Integer Programming Problems
    (1991-11) Bixby, Robert E.; Boyd, E. Andrew; Indovina, Ronni R.
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    An Algorithmic Characterization of Antimatroids
    (1987-12) Boyd, E. Andrew
    In an article entitled "Optimal sequencing of a single machine subject to precedence constraints," E.L. Lawler presented a now classical minmax result for job scheduling. In essence, Lawler's proof demonstrated that the properties of partially ordered sets were sufficient to solve the posed scheduling problem. These properties are, in fact, common to a more general class of combinatorial structures known as antimatroids, which have recently received considerable attention in the literature. It is demonstrated that the properties of antimatroids are not only sufficient but necessary to solve the scheduling problem posed by Lawler, thus yielding an algorithmic characterization of antimatroids. Examples of problems solvable by the general result are provided.
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    Fenchel Cutting Planes for Linear Integer Programming Problems
    (1990-10) Boyd, E. Andrew
    The author recently introduced a new class of cutting planes for integer programs called Fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than through the use of explicit knowledge of the polyhedral structure of the integer program. The theory of Fenchel cuts is outlined and their polyhedral characteristics are briefly discussed. A general algorithm for generating Fenchel cuts is described and an instantiation of this algorithm is then presented for the separation problem associated with knapsack polyhedra. The paper concludes by provably optimizing a linear function over the intersection of the knapsack polyhedra defined by individual constraints in a collection of integer programs first introduced by Crowder, Johnson, and Padberg.
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    Resolving Degeneracy in Linear Programs: Steepest Edge, Steepest Ascent, and Closest Ascent
    (1991-07) Boyd, E. Andrew
    While variants of the steepest edge pivoting rule are commonly used in linear programming codes they are not known to have the theoretically attractive property of avoiding an infinite sequence of pivots at points of degeneracy. A natural extension of the steepest edge pivoting rule based on steepest ascent is developed and shown to be provably finite. An alternative finite pivoting procedure that is computationally more attractive than steepest ascent is then introduced and it is argued that with probability 1 the procedure has the same computational requirements as steepest edge independentof the linear program being solved. Both procedures have the unique advantage that they choose the pivot element without explicit knowledge of the set of all active constraints at a point of degeneracy, thus making them attractive in combinatorial settings where the linear program is never explicitly written out.
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    Solving Integer Programs with Enumeration Cuts
    (1992-03) Boyd, E. Andrew
    A cutting plane technique with applicability to the solution of general integer programs is presented and the computational value of this technique is demonstrated by applying it to a collection of seven difficult integer programs arising from real-world applications. Four of the seven problems are solved to optimality without the aid of branch and bound, and six of the seven problems have the gap between the value of the integer program and its linear programming relaxation closed by over 98%.
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    Solving very large scale school/student assignment problems
    (1994) Elizondo, Rodolfo; Boyd, E. Andrew; Tapia, Richard A.
    Currently, the Houston Independent School District has approximately 175 elementary schools providing education for more than 110,000 students. A question of major logistical impact is how to assign students to schools in an optimal fashion. Many conventional methods exist to deal with such problems, yet the sheer magnitude of the HISD student assignment problem presents new computational challenges which must be dealt with effectively if the problem is to be solved. This monograph examines issues related to finding the solution of school/student assignment problems on a workstation taken from real problem data giving rise to problems with over 20 million variables and 110,000 constraints.
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    The Lagrangian as a Primal Cutting Plane Method for Linear Integer Programming Problems
    (1988-12) Boyd, E. Andrew
    Lagrangian relaxation and more recently cutting plane techniques have both proven to be powerful methods in the solution of integer problems. This paper explores the relationship between these techniques by interpreting Lagrangian relaxation as a primal cutting plane method. Properties of the cuts generated by the Lagrangian are discussed and practical ramifications of the interpretation are emphasized. Computational results are presented.
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    The MIPLIB Mixed Integer Programming Library
    (1992-11) Bixby, R.E.; Boyd, E. Andrew; Dadmehr, Shireen S.; Indovina, Ronni R.