Linear systems revisited

Abstract

An elegant and well-developed theory exists for solving systems of linear equations. It has been applied successfully for decades in industrial and scientific applications and it forms the basis for linear programming. However many systems of linear equations for industrial applications may require all of the variables to be nonnegative or in an even more restricted area, and/or to be whole numbers. These conditions can create difficulties for the standard approach. So presented here is a new absolute value transformation and a multi-stage Monte Carlo simulation solu tion technique for dealing with the nonnegativity solution re quirement and other conditions. The new approach has the ad ditional advantage of working on nonlinear problems. The system of equations is transformed into a statistic (where the minimum of the statistic is the solution of the system). Then solving the system of equations becomes a problem of finding a way across the sampling distribution to the minimum, which will solve the system of equations. Regular Monte Carlo simulation to solve a system or optimiza tion problem suffers from only being able to roughly approx imate the answer. Multi-stage Monte Carlo simulation overcomes this obstacle by using a more sophisticated sampling scheme and additionally exploiting the well-behaved nature of the sampling distributions of transformed systems of equations.