Richard J. Forrester

Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs

We present a linearization strategy for mixed 0-1 quadratic programs that produces small formulations with tight relaxations. It combines constructs from a classical method of Glover and a more recent reformulation-linearization technique (RLT). By using binary identities to rewrite the objective, a variant of the first method results in a concise formulation with the level-1 RLT strength. This variant is achieved as a modified surrogate dual of a Lagrangian subproblem to the RLT. Special structures can be exploited to obtain reductions in problem size, without forfeiting strength. Preliminary computational experience demonstrates the potential of the new representations.

A simple recipe for concise mixed 0-1 linearizations

A new linearization method for mixed 0-1 polynomial programs is obtained by repeatedly applying a classical strategy introduced almost 30 years ago. Two important contributions are: the most concise known linear representations of cubic and higher-degree problems, and a simple argument for explaining and extending two alternate linearizations.

Linear forms of nonlinear expressions: New insights on old ideas

We show how recent linearization methods for mixed 0-1 polynomial programs can be improved and unified through an interpretation of classical techniques. We consider quadratic expressions involving the product of a linear function and a binary variable, and extensions having products of binary variables. Computational results are reported.

Improving the quality of the assignment of students to first-year seminars

Many post-secondary academic institutions in the United States have a First-Year Seminar Program. These seminars are designed to support the success of new incoming first-year students by combining writing, research and active discussion among small groups of students. At Dickinson College, students are required to select six seminars they find interesting from a list of approximately 42 seminars. The college then attempts to assign each student to a seminar on their list, while maintaining course capacities. Using standard commercial optimization software, we develop an approach that not only solves this basic assignment problem, but also seeks to balance both the gender and number of international students in the seminars. In addition, we utilize Monte Carlo simulation to study how the number of seminars each student is required to select affects the likelihood that a feasible assignment exists.

A Comparison of Algorithms for Finding an Efficient Theme Park Tour

The problem of efficiently touring a theme park so as to minimize the amount of time spent in queues is an instance of the Traveling Salesman Problem with Time-Dependent Service Times (TSP-TS). In this paper, we present a mixed-integer linear programming formulation of the TSP-TS and describe a branch-and-cut algorithm based on this model. In addition, we develop a lower bound for the TSP-TS and describe two metaheuristic approaches for obtaining good quality solutions: a genetic algorithm and a tabu search algorithm. Using test instances motivated by actual theme park data, we conduct a computational study to compare the effectiveness of our algorithms.

Tightening concise linear reformulations of 0-1 cubic programs

A common strategy for solving 0-1 cubic programs is to reformulate the non-linear problem into an equivalent linear representation, which can then be submitted directly to a standard mixed-integer programming solver. Both the size and the strength of the continuous relaxation of the reformulation determine the success of this method. One of the most compact linear representations of 0-1 cubic programs is based on a repeated application of the linearization technique for 0-1 quadratic programs introduced by Glover. In this paper, we develop a pre-processing step that serves to strengthen the linear programming bound provided by this concise linear form of a 0-1 cubic program. The proposed scheme involves using optimal dual multipliers of a partial level-2 RLT formulation to rewrite the objective function of the cubic program before applying the linearization. We perform extensive computational tests on the 0-1 cubic multidimensional knapsack problem to show the advantage of our approach.