Karla L. Hoffman

LP-Based Combinatorial Problem Solving

A tutorial outline of the polyhedral theory that underlies linear programming (LP)-based combinatorial problem solving is given. Design aspects of a combinatorial problem solver are discussed in general terms. Three computational studies in combinatorial problem solving using the polyhedral theory developed in the past fifteen years are surveyed: one addresses the symmetric traveling salesman problem, another the optimal triangulation of input/output matrices, and the third the optimization of large-scale zero-one linear programming problems.

Improving LP-Representations of Zero-One Linear Programs for Branch-and-Cut

We present various techniques for automatically improving the LP-representation of general zero-one linear programming problems. These include detection of redundant rows and blatant infeasibilities, coefficient reduction using the Euclidean algorithm, optimality fixing and variable elimination. Extensions to the case where special-ordered-set constraints are present are discussed as well. A summary of the branch-and-cut approach to general zero-one problems (including flowcharts) is given. We report numerical experiments to test the effect of such preprocessing within a branch-and-cut algorithm for eleven large-scale real-world zero-one linear-programming problems. An illustrative example is included in the Appendix. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A method for globally minimizing concave functions over convex sets

A method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear. The algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region. The algorithm does not involve cuts of the feasible region, requires only simplex pivot operations and univariate search computations to be performed, allows the objective function to be lower semicontinuous and nonseparable, and is guaranteed to converge to the global solution. Computational aspects of the algorithm are discussed.

Concave minimization via collapsing polytopes

We present a procedure for globally minimizing a concave function over a bounded polytope by successively minimizing the function over polytopes containing the feasible region, and collapsing to the feasible region. The initial containing polytope is a simplex, and, at the kth iteration, the procedure chooses the most promising vertex of the current containing polytope to refine the approximation. The method generates a tree whose ultimate terminal nodes coincide with the vertices of the feasible region, and accounts for the vertices of the containing polytopes.

Methodology and analysis for comparing discrete linear l1 approximation codes

This is the first of a projected series of papers dealing with computational experimentation in mathematical programming. This paper provides early results of a test case using four discrete linear L1 approximation codes. Variables influencing code behavior are identified and measures of performance are specified. More importantly, an experimental design is developed for assessing code performance and is illustrated using the variable “problem size”.

Documentation for a model: a hierarchical approach

A set of documents and their organization according to functional requirements in order to produce information that will facilitate the use of models are described. The authors discuss the role of models in the policy process and of documentation in the assessment of such models.

A Test Problem Generator for Discrete Linear L 1 Approximation Problems

Described here are the theoretmal development and computer implementation of a procedure that generates test problems for L~ esth-nation of the linear model y= Xfl + u. The generation procedure allows the user flembdlty m specifying the problem dimensions, the LI solution vector fl*, the distribution of the observed residuals ~, as well as the column rank, row repetitions, and degree of degeneracy of the matrix X The user can also specify the distributional form, mean, and variance for each independent variable An lraportant feature of the generator is that any problem it creates is guaranteed to have a umque solutmn ~ whenever X has full rank

Comparison of mathematical programming software: a case study using discrete L 1

Abstract This paper presents the methodology and results of a computational experiment which compares the performance of four computer codes which determine the best discrete L1 approximation to a continuous nonlinear function. The experiment utilizes 320 test problems created by a test problem generator. Several performance measures describe solution quality as well as computational effort.

The internal revenue service post-of-duty location modeling system - final report

This report documents a project undertaken by the National Bureau of Standards to develop a mathematical model which identifies optimal locations of Internal Revenue Service Posts-of-Duty . The mathematical model used for this problem is the uncapacitated, fixed charge, location-allocation model which minimizes travel and facility costs, given a specified level of activity. The report includes a discussion of the location problem and the mathematical model developed. Data sources identified and used are also described. Brief descriptions of the mathematical techniques used and the interactive, user-friendly computer system built to solve the problem are also provided. The system is microcomputer -based and uses menus and graphically displayed maps of tax districts for interactive inputs and solution outputs .