THE A POSTERIORI APPROACH FOR DISCRETE OPTIMIZATION

KOLINJUWADI SANKARAN NAGARAJ, Purdue University

Abstract

This dissertation analyzes and computationally tests a novel approach to improving the performance of Branch-and-Bound Algorithms, namely the a posteriori approach. We consider the class of branch-and-bound algorithms for Integer Programming that use the Bounding test to fathom subproblems. The Bounding test for a maximization is, "If u(P) (LESSTHEQ) l(P), then fathom P", where P is the Integer Program to be solved, P is the subproblem being tested, u is the upper bound to the optimal value of P and l is a lower bound to the optimal value of P. Typically, a relaxation mechanism is used to determine the upper bound and the lower bound is provided by a heuristic. Many such algorithms perform poorly because the Bounding test is the weakest where it should be the strongest, namely, at the top of the enumeration tree where each unfathomed subproblem leads to numerous descendents. This is precisely the issue addressed in this dissertation. Specifically, we propose strengthening the Bounding test by raising the valid right-hand-side of l(P) to a bigger non-a priori valid value. Thus potentially more nodes can be fathomed and the algorithm becomes more efficient. However, in the process, we have made the algorithm a priori invalid and the final incumbent is no longer guaranteed to be optimal. Instead, if we now are able to establish validity by termination using information gained during the course of execution, then not only do we still have a valid algorithm but also have improved its efficiency. This is precisely the aim of the a posteriori approach. In this study, we first study both analytically and experimentally the a posteriori approach of modifying the Bounding test. We then propose a general a posteriori approach for any algorithm for discrete optimization.

Degree

Ph.D.

Subject Area

Industrial engineering

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