Error bounds for an inequality system

Abstract

For an inequality system, an error bound is an estimation for the distance from any point to the solution set of the inequality. The Ekeland variational principle (EVP) is an important tool in the study of error bounds. We prove that EVP is equivalent to an error bound result and present several sufficient conditions for an inequality system to have error bounds. In a metric space, a condition is similar to that of Takahashi. In a Banach space we express conditions in terms of an abstract subdifferential and the lower Dini derivative. We then discuss error bounds with exponents by a relation between the lower Dini derivatives of a function and its power function. For an l.s.c. convex function on a reflexive Banach space these conditions turn out to be equivalent. Furthermore a global error bound closely relates to the metric regularity.