An Algorithm for the Solution of Linear Inequalities

The problem of solving a system of linear inequalities is central to pattern classification where a solution to the system, consistent or not, is required. In this paper, an algorithm is developed using the method of conjugate gradients for function minimization. Specifically, it is shown that the algorithm converges to a solution in both the consistent and inconsistent cases in a finite number of steps: this is the main result. A related criterion function which has significance in pattern classification problems is derived and a variant of the algorithm to minimize the same is given along with computationally convenient modifications. A linear minimization algorithm which makes complete use of the problem structure is given: this is a part of the main algorithm. Computer simulation results for switching problems are presented and the algorithm is compared with Ho–Kashyap and accelerated relaxation algorithms; the results show that the proposed algorithm is faster than the latter algorithms with respect to both the number of iterations and time for convergence.