On the Approximation of Inconsistent Inequality Systems

In this paper is analyzed the minimal correction problem for an inconsistent linear inequality system. By the correction we mean avoiding its contradictory nature by means of relaxing the constraints. When the system of inequalities Ax ≤ b has no solutions, we are interested in a vector that satisfies the system in a Least Squares (LS) sense, i.e. a vector x ∈ R that minimizes the quantity ∥∥(Ax − b)+∥∥2 , where (Ax − b)+ is the vector whose i component is max { (Ax − b)i , 0 } . In fact, the right-hand side (RHS) vector is corrected. Often, in the real world it is more expedient to correct some submatrix of the augmented matrix (A, b), i.e. the RHS vector as well as some rows and some columns of the matrix A. 1. Correction of RHS vector. Least Squares problem for linear inequalities Consider the system of linear inequalities