Set Convergence

The precise meaning of such basic concepts in analysis as differentiation, integration and approximation is dictated by the choice of a notion of limit for sequences of functions. In the past, pointwise limits have received most of the attention. Whether ‘uniform’ or invoked in an ‘almost everywhere’ sense, they underlie the standard definitions of derivatives and integrals as well as the very meaning of a series expansion. In variational analysis, however, pointwise limits are inadequate for such mathematical purposes. A different approach to convergence is required in which, on the geometric level, limits of sequences of sets have the leading role.

Motivation for the development of this geometric approach has come from optimization, stochastic processes, control systems and many other subjects. When a problem of optimization is approximated by a simpler problem, or a sequence of such problems, for instance, it's of practical interest to know what might be expected of the behavior of the associated sets of feasible or optimal solutions. How close will they be to those for the given problem? Related challenges arise in approximating functions that may be extended-real-valued and mappings that may be set-valued. The limiting behavior of a sequence of such functions and mappings, possibly discontinuous and not having the same effective domains, can't be well understood in a framework of pointwise convergence. And this fundamentally affects the question of how ‘differentiation’ might be extended to meet the demands of variational analysis, since that's inevitably tied to ideas of local approximation.