Linear Machine Solution to Point Location Problem

Linear machine has been recently proposed as an elegant solution for solving the point location problem arising in multi-parametric programming (mp-P) based online optimization. Linearmachine associates a linear discriminant function with each polytopic region in the parametric space. The solution to the point location problem is then obtained by simply evaluating these discriminant functions and finding their maximum value. In this technical note, we rigorously establish the correctness of the linear machine generation procedure and identify a necessary condition for existence of linear machine. A modified procedure, involving systematic subdivision of the parametric space, is proposed when this condition is not satisfied. Analysis of complexity and storage requirements, along with computational experiments on a large sized example, indicate that linear machine can be an efficient tool for solving the point location problem.