Error Bounds for Abstract Linear Inequality Systems

• Authors: Kung Fu Ng;Wei Hong Yang
• Venue: SIAM Journal on Optimization
• Year: 2002

In this paper we study error bounds of the abstract linear inequality system (A,C,b): $Ax \leqslant b$, where A is a bounded linear operator from a Banach space X to a Banach space Y partially ordered by a closed convex cone C. We also give some general results on the existence of error bounds for a convex function F; we show in particular that F has an error bound if and only if the directional derivative of the distance function (to the solution set S) at each boundary point of S along any nontangential direction is bounded by the derivative of F. As an application, we prove that if C is a polyhedral cone, then the system (A,C,b) has an error bound. When Y is a Hilbert space, our results can be expressed in terms of the angles between Ax-b-P-C(Ax-b) and Im(A), or in terms of the angles between Im(A) and the nonvertex supporting hyperplanes of C. In the case in which $X=\mathbb{R}^{n}$ and C is an "ice-cream" cone, we identify exactly when (A,C,b) has an error bound.