Well-Behavior, Well-Posedness and Nonsmooth Analysis

AMS subject classification: 90C30, 90C33.

We survey the relationships between well-posedness and well-behavior. The latter notion means that any critical sequence (xn) of a lower semicontinuous function f on a Banach space is minimizing. Here “critical” means that the remoteness of the subdifferential ∂f(xn) of f at xn (i.e. the distance of 0 to ∂f(xn)) converges to 0. The objective function f is not supposed to be convex or smooth and the subdifferential ∂ is not necessarily the usual Fenchel subdifferential. We are thus led to deal with conditions ensuring that a growth property of the subdifferential (or the derivative) of a function implies a growth property of the function itself. Both qualitative questions and quantitative results are considered.