Computable Error Bounds For Convex Inequality Systems In Reflexive Banach Spaces

Abstract

In 1952, A. J. Hoffman proved a fundamental result of an error bound on the distance from any point to the solution set of a linear system in $\hbox{{\bbb R}}^n$. In SIAM J. Control, 13 (1975), pp. 271--273, Robinson extended Hoffman's theorem to any system of convex inequalities in a normed linear space which satisfies the Slater constraint qualification and has a bounded solution set. This paper studies any system of convex inequalities in a reflexive Banach space which has an unbounded solution set. It is shown that Hoffman's error bound holds for such a system when a related convex system, which defines the recession cone of the solution set for the system, satisfies the Slater constraint qualification.