Lipschitzian Properties

The notion of Lipschitz continuity is useful in many areas of analysis, but in variational analysis it takes on a fundamental role. To begin with, it singles out a class of functions which, although not necessarily differentiable, have a property akin to differentiability in furnishing estimates of the magnitudes, if not the directions, of change. For such functions, real-valued and vector-valued, subdifferentiation operates on an especially simple and powerful level. As a matter of fact, subdifferential theory even characterizes the presence of Lipschitz continuity and provides a calculus of the associated constants. It thereby supports a host of applications in which such constants serve to quantify the stability of a problem's solutions or the rate of convergence in a numerical method for determining a solution.

But the study of Lipschitzian properties doesn't stop there. It can be extended from single-valued mappings to general set-valued mappings as a means of obtaining quantitative results about continuity that go beyond the topological results obtained so far. In that context, Lipschitz continuity can be captured by coderivative conditions, which likewise pin down the associated constants. What's more, those conditions can be applied to basic objects of variational analysis such as profile mappings associated with functions, and this leads to important insights. For instance, the very concepts of normal vector and subgradient turn out to represent ‘manifestations of singularity’ in the Lipschitzian behavior of certain set-valued mappings.