John J. H. Forrest

Updated triangular factors of the basis to maintain sparsity in the product form simplex method

In recent years triangular factorization of the basis has greatly enhanced the efficiency of linear programming inversion routines, leading to greater speed, accuracy and a sparser representation. This paper describes a new product form method for updating the triangular factors at each iteration of the simplex method which has proved extremely effective in reducing the rate of growth of the transformation (eta) files, thus reducing the amount of work per iteration and the frequency of re-inversion. We indicate some of the programming measures required to implement the method and give computational experience on real problems of up to 3500 rows.

Global optimization using special ordered sets

The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick [4] points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland [2] or Soland [6], or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin [1], have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin [3] show how they have benefitted from the vast improvements in branch and bound integer programming capabilities over the last few years, as a result of being incorporated in a general mathematical programming system.Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis.This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two by-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval.The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.

Steepest-edge simplex algorithms for linear programming

We present several new steepest-edge simplex algorithms for solving linear programming problems, including variants of both the primal and the dual simplex method. These algorithms differ depending upon the space in which the problem is viewed as residing, and include variants in which this space varies dynamically. We present computational results comparing steepest-edge simplex algorithms and approximate versions of them against simplex algorithms that use standard pivoting rules on truly large-scale realworld linear programs with as many as tens of thousands of rows and columns. These results demonstrate unambiguously the superiority of steepest-edge pivot selection criteria to other pivot selection criteria in the simplex method.

IBM Solves a Mixed-Integer Program to Optimize Its Semiconductor Supply Chain

IBM Systems and Technology Group uses operations research models and methods extensively for solving large-scale supply chain optimization (SCO) problems for planning its extended enterprise semiconductor supply chain. The large-scale nature of these problems necessitates the use of computationally efficient solution methods. However, the complexity of the models makes developing robust solution methods a challenge. We developed a mixed-integer programming (MIP) model and supporting heuristics for optimizing IBMs semiconductor supply chain. We designed three heuristics, driven by practical applications, for capturing the discrete aspects of the MIP. We leverage the model structure to overcome computational hurdles resulting from the large-scale problem. IBM uses the model and method daily for operational and strategic planning decisions and has saved substantial costs.

Vector processing in simplex and interior methods for linear programming

We discuss the application of vector processing to various phases of simplex and interior point methods for linear programming. Preliminary computational results of experiments on the IBM 3090 vector facility will be presented.

Implementing interior point linear programming methods in the Optimization Subroutine Library

This paper discusses the implementation of interior point (barrier) methods for linear programming within the framework of the IBM Optimization Subroutine Library. This class of methods uses quite different computational kernels than the traditional simplex method. In particular, the matrices we must deal with are symmetric and, although still sparse, are considerably denser than those assumed in simplex implementations. Severe rank deficiency must also be accommodated, making it difficult to use off-the-shelf library routines. These features have particular implications for the exploitation of the newer IBM machine architectural features. In particular, interior methods can benefit greatly from use of vector architectures on the IBM 3090™ series computers and “super-scalar“ processing on the RISC System/6000™ series.

A Column-Generation Approach to the Multiple Knapsack Problem with Color Constraints

In this paper, we study a new problem that we refer to as the multiple knapsack with color constraints (MKCP). Motivated by a real application from the steel industry, the MKCP can be formulated by generalizing the multiple knapsack problem. A real-life instance (called mkc) of this problem class is available through MIPLIB (Bixby 2004) and a larger instance (mkc7) is downloadable from the COIN site (IBM 2004). The focus of this paper is to present improved computational results for the two mentioned instances of this problem using a column-generation approach. We solve mkc to optimality and use Dantzig-Wolfe decomposition for upper bounding the other instance. Solving mkc to optimality took less time than it takes to solve the LP relaxation of the original formulation. The larger instance is solved to near optimality (within 0.5 of optimality) in a fraction of the time required to solve the original relaxed LP.