First-Order and Second-Order Conditions for Error Bounds
Zili Wu, Jane J. Ye · SIAM Journal on Optimization · 2003
For a lower semicontinuous function f on a Banach space X, we study the existence of a positive scalar $\mu$ such that the distance function d S associated with the solution set S of $f(x)\leq 0$ satisfies \[ d_S(x)\leq \mu \max\{ f(x),0\} \] for each point x in a neighborhood of some point x 0 in X with $f(x)<\epsilon$ for some $0<\epsilon \leq +\infty.$ We give several sufficient conditions for this in terms of an abstract subdifferential and the Dini derivatives of f. In a Hilbert space we further present some second-order conditions. We also establish the corresponding results for a system of inequalities, equalities, and an abstract constraint set.