Automatic Identification of Generalized Upper Bounds in Large-Scale Optimization Models

Abstract

To solve contemporary large-scale linear, integer and mixed integer programming problems, it is often necessary to exploit intrinsic special structure in the model at hand. One commonly used technique is to identify and then to exploit in a basis factorization algorithm a generalized upper bound (GUB) structure. This report compares several existing methods for identifying GUB structure. Computer programs have been written to permit comparison of computational efficiency. The GUB programs have been incorporated in an existing optimization system of advanced design and have been tested on a variety of large-scale real-life optimization problems. The identification of GUB sets of maximum size is shown to be among the class of NP-complete problems; these problems are widely conjectured to be intractable in that no polynomial-time algorithm has been demonstrated for solving them. All the methods discussed in this report are polynomial-time heuristic algorithms that attempt to find, but do not guarantee, GUB sets of maximum size. Bounds for the maximum size of GUB sets are developed in order to evaluate the effectiveness of the heuristic algorithms.