Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

An analog of Karmarkar's algorithm for inequality constrained linear programs, with a "new" class of projective transformations for centering a polytope

Bibliography: p. 12.Research supported in part by ONR contract N00014-87-K-0212.by Robert M. Freund

An analog of Karmarkar's algorithm for inequality constrained linear programs, with a "new" class of projective transformations for centering a polytope

Also issued as: Working Paper (Sloan School of Management) ; WP 1921-87.Includes bibliographical references (leaf 12).Supported in part by ONR N00014-87-K-0212by Robert M. Freund

A dual version of Tardos's algorithm for linear programming

Bibliography: p. 11.by James B. Orlin

The Dikin-Karmarkar Principle for Steepest Descent

This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/16565Steepest feasible descent methods for inequality constrained optimization problems have commonly been plagued by short steps. The consequence of taking short steps is slow convergence to non-stationary points (zigzagging). In linear programming, both the projective algorithm of Karmarkar (1984) and its affined-variant, originally proposed by Dikin (1967), can be viewed as steepest feasible descent methods. However, both of these algorithms have been demonstrated to be effective and seem to have overcome the problem of short steps. These algorithms share a common norm. It is this choice of norm, in the context of steepest feasible descent, that we refer to as the Dikin-Karmarkar Principle. This research develops mathematical theory to quantify the short step behavior of Euclidean norm steepest feasible descent methods and the avoidance of short steps for steepest feasible descent with respect to the Dikin-Karmarkar norm. While the theory is developed for linear programming problems with only nonnegativity constraints on the variables. Our numerical experimentation demonstrates that this behavior occurs for the more general linear program with equality constraints added. Our numerical results also suggest that taking longer steps is not sufficient to ensure the efficiency of a steepest feasible descent algorithm. The uniform way in which the Dikin-Karmarkar norm treats every boundary is important in obtaining a satisfactory convergence

On the convergence of the affine-scaling algorithm

Cover title.Includes bibliographical references (p. 20-22).Research partially supported by the National Science Foundation. NSF-ECS-8519058 Research partially supported by the U.S. Army Research Office. DAAL03-86-K-0171 Research partially supported by the Science and Engineering Research Board of McMaster University.by Paul Tseng and Zhi-Quan Luo

Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem

Includes bibliographical references.Robert M. Freund

Some Modifications and Extensions of Karmarkar's Main Algorithm with Computational Experiences

The Karmarkar algorithm and its modifications are studied in this thesis. A modified line search algorithm with extended searching bound to the facet of the simplex is developed and implemented. Using this modification, a modified row partition method is tested. Both algorithms are coded in Fortran 77 and compared their performances with the original Karmarkar algorithm. The modifications are promising and other extensions are encouraged.Computing and Information Scienc