Error bounds for inconsistent linear inequalities and programs

For any system of linear inequalities, consistent or not, the norm of the violations of the inequalities by a given point, multiplied by a condition constant that is independent of the point, bounds the distance between the point and the nonempty set of points that minimize these violations. Similarly, for a dual pair of possibly infeasible linear programs, the norm of violations of primal-dual feasibility and primal-dual objective equality, when multiplied by a condition constant, bounds the distance between a given point and the nonempty set of minimizers of these violations. These results extend error bounds for consistent linear inequalities and linear programs to inconsistent systems. Error bounds are playing an increasingly important role in mathematical programming. Beginning with Hooman's classical error bound for linear inequalities 3], many papers have examined error bounds for linear and convex inequalities, linear and nonlinear programs, as well as linear and nonlinear complementarity problems and aane These error bounds can be traced to the following classical elementary error bound for an arbitrary point in the n-dimensional real space relative to a nonsingular system of equations Bx = c with solution x = B ?1 c: kx ? xk = kx ? B ?1 ck = kB ?1 (Bx ? c)k 5 kB ?1 k kBx ? ck Here kk denotes some norm, and the residual kBx?ck, when multiplied by the \condition constant" kB ?1 k, bounds the distance between x and the solution x. In a similar, but considerably more general manner, mathematical programming research has concentrated on establishing bounds for computable residuals for various problems. Such residuals evaluated at an arbitrary point, bound a \distance" to the solution set from the given point. Essentially all such research has assumed that the underlying mathematical programming problem, just like Bx = c with nonsingular B, is solvable. In this work we shall not make this rather restrictive assumption, because we want to handle more general situations that correspond to a singular or even rectangular matrix B, for which no exact solution may exist. In line with this objective, we will begin by showing, for the possibly inconsistent system of linear inequalities Ax 5 b;