Pivoting and extensions: In honor of A.W. Tucker (Mathematical Programming Study 1): M.L. Balinski

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This volume initiates the series MATHEMATICAL PROGRAMMING STUDIES. Conceived as a companion to the journal MATHEMATICAL PROGRAMMING, each STUDY is to be centered about a unifying subject matter and to consist of either a collection of papers, a single monograph, the proceedings of a specialized symposium, a guide to computational practice, or any clearly focused and useful piece of work.

The conjunction of the launching of the STUDY series and the retirement from Princeton University of A.W. Tucker proved to be too neighborly to withstand the logic, sentiment and pleasure of combining the events to honor him. Accordingly, his friends, colleagues, students and sons were invited to add their contributions to a volume devoted to an area pioneered and vigorously promoted by A. W. Tucker throughout the last decade: PIVOTING AND EXTENSIONS. His pervasive influence on the field of mathematical programming, and on the people who work in it, is clearly displayed by the contents of this volume. The affection and respect for him is explicit in this volume, but was also expressed in the many letters of regret received from invited contributors who were precluded by the subject matter from submitting their own papers. There is still another fitting coincidence: this STUDY series carries further an idea initiated by Professor Tucker which led to the establishment of the Annals of Mathematics Studies.

Pivoting is an essential computational and -theoretical tool in mathematical programming. Although based on the Gauss-Jordan complete elimination or replacement idea of linear algebra, its computational significance was only fully realized with the development of the simplex method for linear programming in 1947. Since then, the use of the pivoting idea in theory and computation has proliferated and has led to developments in linear, nonlinear and integer programming, the structure of convex polytopes, matrix and bi-matrix games, the computation of fixed points and economic equilibria, the complementarity problems, and still other domains. The papers of this STUDY, many of which were presented at the 8 a International Symposium on Mathematical Programming held at Stanford University in August ,1973, bear witness to this statement.

The first four papers deal with the structure of convex polytopes. Paper 1 establishes improved lower bounds for the maximum diameter of polytopes. Papers 2 and 3 introduce and study "abstract polytopes", a combinatorial construct which generalizes the (pivotal) structure of extreme points and their adjacencies but is a particular type of pseudomanifold. In Paper 4 the Hirsch conjecture is established constructively for two classes of transportation polytopes.

Several papers are related to the complementary pivoting ideas first introduced by Lemke and since extended in many directions. Paper 5 characterizes the existence of a solution ray to a linear complementarity problem when the "structured" matrix is copositive plus. Paper 7 describes an algorithm for solving piecewise linear convex equations. Paper 10 provides an algorithm which proves the fundamental theorem of algebra by building a labeling procedure, pivotal in character, upon a combinatorial lemma concerning labeled triangulations of the complex plane. Paper 12 gives an expository account of the original Lemke-Howson idea in terms of a geometric labeling system for bi-matrix games and presents an orientation theory for the equilibrium points and the complementary pivot paths connecting them for such games. Paper 6 modifies Fourier's original method of elimination of variables for solving systems of linear inequalities thus obtaining a practical computational algorithm for a class of parametric linear programs. Paper 8 shows that optimal integer solutions exist to certain types of linear programs given a "balanced" structural matrix and nonnegative integer right-hand sides, thus generalizing previous results of Berge, by using the theory of blocking and anti-blocking pairs of matrices. Paper 9, an application to coding theory, uses integer programming cuts and pivoting to establish a lower bound on the uniform length of word required for a binary linear error-correcting code satisfying a minimum distinguishability criterion.

Paper 11 uses Tucker's pivotal algebra to introduce determinants in a new, simpler and computationally efficient way. Finally, Paper 12 presents a complementary pivot algorithm for the problem of finding the point of a polytope in Euclidean space having smallest Euclidean norm, a special form of quadratic program which admits a special algorithm having a transparent geometry.

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