The Smallest Correction of an Inconsistent System of Linear Inequalities

Abstract

The problem of calculating a point x that satisfies a given system of linear inequalities, A x #x2265; b, arises in many applications. Yet often the system to be solved turns out to be inconsistent due to measurement errors in the data vector, b. In such a case it is useful to find the smallest perturbation of b that recovers feasibility. This motivates our interest in the least correction problem and its properties.

Let A x #x2265; b be an inconsistent system of linear inequalities. Then it is always possible to find a correction vector y such that the modified system A x #x2265; b #x2212; y is solvable. The smallest correction vector of this type is obtained by solving the least correction problem $\begin{gathered} {\text{minimize}}\quad P\left( {{\text{x,y}}} \right) = \frac{1}{2}\left\| {\text{y}} \right\|_2^2 \hfill \\ {\text{subject}}{\text{ to}}\quad {\text{Ax}}{\text{ + }}{\text{ y}} \geqslant {\text{b}}{\text{.}} \hfill \\ \end{gathered}$Let $\mathbb{U}$ denote the convex cone which consists of all the points $\mathbb{R}^m$ for which the system A x #x2265; u is solvable. Let $\mathbb{Y}$ denote the polar cone of $\mathbb{U}$. It is shown that the least correction problem has a simple geometric interpretation which is based on the polar decomposition of $\mathbb{R}^m$ into $\mathbb{U}$ and $\mathbb{Y}$. A further insight into the least correction concept is gained by exploring the duality relations that characterize such problems.