Gordon H. Bradley

Transformation of integer programs to knapsack problems

It is shown that any bounded integer linear programming problem can be trans- formed to an equivalent integer linear programming problem with a single constraint and the same number of variables. The original integer problem and the equivalent single constraint problem have identical feasible integer solutions, and thus the single constraint problem may be solved instead of the original problem. It is shown that any bounded integer programming problem is equivalent to a problem of optimizing a linear function subject only to bounds on the function and bounds on the variables. The results are extended to problems with nonlinear constraints.

Algorithms for Hermite and Smith normal matrices and linear Diophantine equations

New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to a system of linear diophantine equations is presented. A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables. Rough bounds for the number of operations are cubic polynomials involving the order of the matrix and the determinant of the matrix. The algorithms are valid if the elements of the matrix are in a principal ideal domain.

Coefficient reduction for inequalities in 0-1 variables

For a given inequality with 0–1 variables, there are many other “equivalent” inequalities with exactly the same 0–1 feasible solutions. The set of all equivalent inequalities is characterized, and methods to construct the equivalent inequality with smallest coefficients are described.

Algorithm and bound for the greatest common divisor of n integers

A new version of the Euclidean algorithm for finding the greatest common divisor of <italic>n</italic> integers <italic>a<subscrpt>i</subscrpt></italic> and multipliers <italic>x<subscrpt>i</subscrpt></italic> such that gcd = <italic>x</italic><subscrpt>1</subscrpt> <italic>a</italic><subscrpt>1</subscrpt> + ··· + <italic>x<subscrpt>n</subscrpt> a<subscrpt>n</subscrpt></italic> is presented. The number of arithmetic operations and the number of storage locations are linear in <italic>n</italic>. A theorem of Lamé that gives a bound for the number of iterations of the Euclidean algorithm for two integers is extended to the case of <italic>n</italic> integers. An algorithm to construct a minimal set of multipliers is presented. A Fortran program for the algorithm appears as <italic>Comm. ACM</italic> Algorithm 386.

Survey of Deterministic Networks

Abstract The state-of-the-art of deterministic networks is surveyed with a discussion of shortest path, transportation, assignment, transshipment, maximum flow, minimum spanning tree, Chinese postman, Euler path and multicommodity flow models. The computational complexity of network models is discussed. The survey concentrates on models and algorithms that can be used to solve large-scale problems.

Model integration with a typed executable modeling language

Contemporary executable modeling languages for mathematical programming are extended by defining a typing system for all the objects in a model and by specifying formal methods to manipulate the type information. The modelers intent to formulate consistent, meaningful constraints and functions can be automatically verified. It is shown how typing supports the development of integrated models from distinct model components. A library unit is proposed as an extension to modeling languages; it provides a mechanism for building integrated models from previously validated models.<<ETX>>

Structural Redundancy in Large-Scale Optimization Models

This paper discusses automatic detection and exploitation of structural redundancy in large-scale mathematical programming models. From our perspective, such redundancy represents embedded special structure which can give significant insight to the model proponent as well as greatly reduce solution effort. We report experiments with real-life linear programming (LP) and mixed-integer (MIP) models in which various methods are developed and tested as integral modules in an optimization system of advanced design. We seek to understand the modeling implications of these embedded redundancies as well as to exploit them during actual optimization. The latter goal places heavy emphasis on efficient, as well as effective, identification techniques for economic application to large models. Several (polynomially bounded) heuristic detection algorithms are presented from our work. In addition, bounds are reported for a maximum row dimension of the more complex structures. These bounds are useful for objectively estimating the quality of heuristically derived assessments of structural redundancy. Finally, some additional suggestions are made for analyzing nonlinear programming (NLP) models.

An Algorithm for Integer Linear Programming: A Combined Algebraic and Enumeration Approach

This paper develops an algorithm for pure integer programming problems. It first transforms the integer programming problem to an algebraically equivalent Hermite canonical problem, and then employs the Fourier-Motzkin elimination. These algebraic operations transform the problem into a form that leads to an efficient implicit enumeration scheme to calculate an optimal solution. The algorithm constructs, in a finite number of operations, an optimal solution to an integer program with n variables and n or n + 1 inequality constraints. If the original problem has more than n + 1 constraints, then the integer program with only the constraints that are binding at an optimal linear programming solution is solved in place of the original problem. Computational results are presented.

Network and Graph Markup Language (NaGML) - Data File Formats

The Network and Graph Markup Language (NaGML) is a family of Extensible Markup Language (xml) languages for network and graph data files. The topology, node properties, and arc properties are validated against the user’s specification for the data values. NaGML is part of a component architecture that reads, validates, processes, displays, and writes network and graph data. Because it implements a family rather than a single xml language, NaGML offers (1) flexibility in choosing property names, data types, and restrictions, (2) strong validation, and (3) a variety of data file formats. This paper demonstrates these points with a sampling of the possible data file formats.

Trading Rules for a Decentralized Exchange Economy

Constructive proofs of the existence of trading rules for a decentralized monetary economy and for a non-decentralized barter economy are developed. It is proven that there is no trading rule for a decentralized barter economy.