On the stability of regularization methods in linear programming. (Q5942473)

A standard problem of linear programming \[ f(x) = \langle c, x \rangle \to \inf, x \in X = \{x\in E^n:x\geq0, \quad Ax \leq b\}, \tag{1} \] where \(E^n\) is the \(n\)-dimensional Euclidean vector space with a scalar product \(\langle x,y\rangle = \sum_{i=1}^n x^i y^i\) for arbitrary \(x = (x^1, \dots, x^n)^T, y = (y^1,\dots, y^n)\), \(A= \{a_{ij}\}\) is an \(m \times n\) matrix, \(c \in E^n, b \in E^n\), is considered. The stability of the principal methods of regularization (stabilization method, difference method, method of quasi-solutions, etc.) is investigated. The basic idea of extension of sets is used to prove stability of the methods discussed.