Extension of Hoffman’s Error Bound to Polynomial Systems

Given any linear system (defined by linear inequalities/equalities) in ⁿ, Hoffman’s error bound says that the distance from any point x ∈ ⁿ to the solution set of the linear system is bounded by a constant (independent of x) times a certain residual function evaluated at x. This paper considers arbitrary (possibly nonconvex) polynomial systems (defined by polynomial inequalities/equalities). It is shown that the solution sets of such systems are, in general, Hölder continuous as the right-hand side changes, in contrast to the Lipschitzian continuity for the linear case. Also, for any convex quadratic inequality system possessing an interior solution, the authors show that its solution set behaves in a Lipschitzian manner as the right-hand side varies. This result sharpens the earlier error bound results by Mangasarian and Robinson for general convex differentiable inequality systems, which, in addition to the interiority assumption, require either an asymptotic constraint qualification condition or that the solution set be bounded.