Harlan P. Crowder

Validation of subgradient optimization

The “relaxation” procedure introduced by Held and Karp for approximately solving a large linear programming problem related to the traveling-salesman problem is refined and studied experimentally on several classes of specially structured large-scale linear programming problems, and results on the use of the procedure for obtaining exact solutions are given. It is concluded that the method shows promise for large-scale linear programming

Solving Large-Scale Zero-One Linear Programming Problems

In this paper we report on the solution to optimality of 10 large-scale zero-one linear programming problems. All problem data come from real-world industrial applications and are characterized by sparse constraint matrices with rational data. About half of the sample problems have no apparent special structure; the remainder show structural characteristics that our computational procedures do not exploit directly. By todays standards, our methodology produced impressive computational results, particularly on sparse problems having no apparent special structure. The computational results on problems with up to 2,750 variables strongly confirm our hypothesis that a combination of problem preprocessing, cutting planes, and clever branch-and-bound techniques permit the optimization of sparse large-scale zero-one linear programming problems, even those with no apparent special structure, in reasonable computation times. Our results indicate that cutting-planes related to the facets of the underlying polytope are an indispensable tool for the exact solution of this class of problem. To arrive at these conclusions, we designed an experimental computer system PIPX that uses the IBM linear programming system MPSX/370 and the IBM integer programming system MIP/370 as building blocks. The entire system is automatic and requires no manual intervention.

On Reporting Computational Experiments with Mathematical Software

Many papers appearing in journals reporting computational experiments use computer generated evidence to compare or rank competing mathematical software techmques. Unfortunately, to date there have been no standards or gmdehnes indicating how computer experiments should be conducted or how the results should be presented. An initial attempt is made to rectify this situation, and a summary of unportant points which should be considered when writing or evaluating a paper m which computational results are reported is provided.

Reporting computational experiments in mathematical programming

This paper presents a set of recommended standards for the presentation of computational experiments in mathematical programming.

Optimizing Customer Mail Streams at Fingerhut

Fingerhut mails up to 120 catalogs per year to each of its 7 million customers. With this dense mail plan and mailing decisions made independently for each catalog, many customers were receiving redundant and unproductive catalogs. To identify and eliminate this excessive operational expense, IBM and Fingerhut together developed an optimization system that selects the most profitable sequence of catalogs, called a mail stream, for each customer. With mail streams, Fingerhut makes better mailing decisions at the customer level, resulting in increased profits. Today, Fingerhut runs this application weekly to find the most profitable mail stream for each customer.

Use of Cyclic Group Methods in Branch and Bound

Publisher Summary The use of cyclic groups, cutting planes, and shortest paths to provide bounds is very much along the lines of Tomlins use of Gomorys mixed integer cut. That cut is what the cutting plane algorithm gives after one iteration. Initially, the linear program obtained by dropping all of the integer restrictions is solved as a linear program. The algorithm has two general steps: (1) choose some linear program and solve it as a linear program and (2) choose some variable that is required to be integer-valued but whose value is not integer. Then, create two new linear programs. There are two choices that must be made: the linear program to solve in step 1 and the variable to branch on in step 2. Once a variable in step 2 is chosen creating two new linear programs, one always chooses the next linear program in step 1 to be one of these two linear programs. Whenever one returns to step 1 from one of the three possible cases in step 1, one have not just created two new linear programs and the choice of linear program is completely open. Each such choice begins a new major cycle. The first major cycle consists of solving the linear program, branching on some variable, solving one of these two linear programs, and continuing until the linear program chosen is either infeasible or has an integer optimum solution.

APL2: getting started

APL is a concise and economical notation for expressing computational algorithms and procedures. This paper introduces the main ideas of APL2, an IBM implementation of APL, and illustrates the programming style with some graphical examples.

Mathematical Programming Algorithms in APL

The APL programming language offers a way of succinctly expressing data processing algorithms. We will introduce and demonstrate those APL concepts which we find useful for designing, implementing, and testing mathematical programming procedures.

Solving the installation scheduling problem using mixed integer linear programming

The installation scheduling problem involves finding a program for installing a large number of sizes and types of items (e.g., machines) over time so as to optimize some measure (e.g., initial capital investment), subject to various resource constraints. Examples of this problem are scheduling the installation of point-of-sale terminals in supermarket and retail chains, and teller terminals in banks. We have formulated the installation scheduling problem as a mixed integer linear program and developed a computer code for solving the model. By using techniques for exploiting the special structure of the model, our formulation allows rather quick solution times.