Paul D. Domich

Hermite normal form computation using modulo determinant arithmetic

This paper describes a new class of Hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation. This class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string. Computational results are also given.

An interior point method for general large-scale quadratic programming problems

In this paper, we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems efficiently solves a wide class of largescale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a three-dimensional cost improvement subproblem, which is solved at every interation. We have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the sucess of the algorithm. We describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented.

Algorithmic Enhancements to the Method of Centers for Linear Programming Problems

Interior point algorithms for solving linear programming problems are considered. The techniques are derived from a continuous version of Huards method of centers that yields a family of trajectories in the feasible region that all converge to an optimal solution. The tangential direction of these trajectories is the dual affine direction. Deficiencies in some of these trajectories are discussed, and the need to recenter is argued. Several new algorithms that use the dual affine direction and a recentering direction in a multidirection approach are then derived. The most promising of these algorithms is based on minimizing the cost function on a sequence of two-dimensional cross sections of the feasible region. Numerical results are presented. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Residual hermite normal form computations

This paper extends the class of Hermite normal form solution procedures that use modulo determinant arithmetic. Given any relatively prime factorization of the determinant value, integral congruence relations are used to compute the Hermite normal form. A polynomial-time complexity bound that is a function of the length of the input string exists for this class of procedures. Computational results for this new approach are given.

Optimizing Over Three-Dimensional Subspaces in an Interior-Point Method for Linear Programming*

Abstract Interior-point algorithms for solving linear programming problems are considered. A three-dimensional method is developed that, at each iteration, solves a subproblem based on minimizing the cost function on low-dimensional cross sections of the feasible region. The generators for the three-dimensional subproblem include the dual affine search direction and two higher-order search directions. One of the higher-order directions is a third-order correction to the Newton recentering direction, and the other is a correction to the dual affine direction that is motivated by the use of rank-one updates of the second-derivative information. Numerical results are presented for this method that indicate a nearly 20% reduction in CPU time compared to our best dual affine implementation.

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 .