David I. Steinberg

Occurrences of cycling and other phenomena arising in a class of linear programming models

An investigation into the average queue size for a certain class of queues has resulted in the formulation of linear programming problems which are ill-conditioned in some cases. In attempting to solve these linear programming models, using IBMs MPS package, instances of cycling were encountered. Small perturbations in the input data resulted in problems which did not cycle. This fact, plus several other observed phenomena suggest that the primary reason that cycling is not known to occur more frequently is that round-off errors in the computations perturb the problem sufficiently to prevent cycling (or at least to prevent indefinite cycling). In one case maximizing and minimizing an objective function subject to the same constraint set was attempted, but MPS solved only one of these while giving an indication of infeasibility for the other.

An implicit enumeration algorithm for the all integer programming problem

Abstract In this paper an implicit enumeration algorithm is presented for solving the general integer programming problem. The algorithm is a generalization of algorithms of Balas and Geoffrion for solving 0-1 problems. In order that each variable may be treated directly, a special data structure has been designed. Based on this data structure, a procedure is developed which efficiently keeps track of the status of the algorithm and of the potential actions to be taken at any stage in the procedure. The concept of a ‘best’ surrogate constraint is extended and incorporated into the algorithm. The algorithm can be modified for solving mixed integer programming problems.

Computational experience with an algorithm for 0–1 integer programming

Abstract In a previous paper in this journal, the authors described an implicit enumeration algorithm for the all integer programming problem. In this paper, a specialization of the aforementioned algorithm for 0–1 integer programming problems is developed. The computational efficiency of this specialization is investigated by solving a set of test problems, using a computer code of the algorithm written for this purpose.