Nonlinear error bounds for lower semicontinuous functions on metric spaces

Error Bounds: Necessary and Sufficient Conditions

“…A different subdifferential criterion was obtained in [18,19] as a by-product in the investigation of calmness of a standard constraint system in the sense of [48]. Another criterion was worked out in [20] on the basis of a primal-type estimate; further important results of this nature can be found in [13,42,43,45]. This paper goes in the same direction by employing several groups of derivativelike objects both from the primal as well as from the dual space.…”

Ekeland's Variational Principle for Set-Valued Functions

“…Since the pioneering work of Hoffman [19], this notion (and its local version) have played an important role in many areas in mathematical programming and variational analysis (see the excellent surveys [23,31] for results before 1997, and for more recent results see [1,2,7,10,12,27,28,29,30,36] and references therein). The earlier results all are either under certain convexity assumption or restricted to the special case when n = 1.…”

A Note on Well-posed Problems

“…The main assumption of Proposition 2.1 was already established in [CM,Proposition 4.1] (see also [AC2,Proposition 5.1]): it provides a socalled error bound estimate for f in a neighborhood of a "critical set" C, which was required to be nonempty for the purposes of [CM]. Allowing C = ∅ does not affect the proof (that we give for completeness), and fits our present purposes.…”