Abstract

We establish conditions for stability, metric regularity, and a pseudo-Lipschitz property of the solution maps of parametric inequality systems involving nonsmooth (not necessarily locally Lipschitz) continuous functions and closed convex sets. We also derive open mapping and inverse mapping theorems for nonsmooth continuous functions, Lagrange multiplier rules for nonsmooth cone-constrained optimization problems, and conditions for the continuity of the optimal value functions of optimization problems. The main tool used is a generalized Jacobian, called approximate Jacobian. It provides a flexible nonsmooth local analysis of continuous functions and often gives sharp calculus rules for locally Lipschitz functions. The regularity condition, which plays a key role in the local analysis, is a new extension of the Robinson regularity condition for continuous functions.