Global Weak Sharp Minima on Banach Spaces

Abstract

We consider a proper lower semicontinuous function $f$ on a Banach space X with $\lambda=\inf\{f(x):\;x\in X\}>-\infty$. Let $\alpha\geq\lambda$ and $S_\alpha=\{x\in X:\; f(x)\leq\alpha\}$. We define the lower derivative of f at the set $S_\alpha$ by $$\underline{D}(f,S_\alpha)=\liminf_{x\rightarrow S_\alpha}\frac{f(x)-\alpha}{dist(x,S_\alpha)},$$ where $x\rightarrow S_\alpha$ can be interpreted in various ways. We show that, when f is convex and $\alpha = \lambda$, it is equal to the largest weak sharp minima constant. In terms of these derivatives and subdifferentials, we present several characterizations for convex f to have global weak sharp minima. Some of these results are also shown to be valid for nonconvex f. As applications, we give error bound results for abstract linear inequality systems.