A Generalized Sylvester Identity and Fraction-free Random Gaussian Elimination

AbstractSylvester’s identity is a well-known identity that can be used to prove that certain Gaussian elimination algorithms are fraction free. In this paper we will generalize Sylvester’s identity and use it to prove that certain random Gaussian elimination algorithms are fraction free. This can be used to yield fraction free algorithms for solving Ax=b(x≥ 0) and for the simplex method in linear programming

Exact Arithmetic at Low Cost - a Case Study in Linear Programming

We describe a new exact-arithmetic approach to linear programming when the number of variables n is much larger than the number of constraints m (or vice versa). The algorithm is an implementation of the simplex method which combines exact (multiple precision) arithmetic with inexact (floating point) arithmetic, where the number of exact arithmetic operations is small and usually bounded by a function of min(n; m). Combining this with a "partial pricing" scheme (based on a result by Clarkson [8]) which is particularly tuned for the problems under consideration, we obtain a correct and practically efficient algorithm that even competes with the inexact state-of-the-art solver CPLEX 1 for small values of min(n; m) and and is far superior to methods that use exact arithmetic in any operation. 1 Introduction Linear Programming (LP) -- the problem of maximizing a linear objective function in n variables subject to m linear (in)equality constraints -- is the most prominent optimization ..

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201