Correcting an Inconsistent System of Linear Inequalities by Nonlinear Programming

We consider the problem of correcting an inconsistent system of linear inequalities, Ax ≤ b, subject to nonnegativity constraints, x ≥ 0. We formulate this problem as a nonlinear program and derive the corresponding Karush-Kuhn-Tucker conditions. These conditions reveal several interesting properties that solutions must satisfy and allow us to derive several equivalent problems that involve fewer decision variables and are more amenable to solution. We propose using a gradient projection method to minimize an objective function φ(x), subject only to x ≥ 0. We also propose a hybrid approach that exploits an interesting relation between the correction problem and the method of total least squares.