Partially smooth variational principles and applications

摘要

Smooth variational analysis has been highly successful in providing tools for the study of non-smooth analysis and optimization problems, especially when married to viscosity concepts. Outside of smoothable Banach spaces (thus, notably in L1 spaces) general constructions such as those of Ioffe require a largely nonconstructive intersection over smooth or finite-dimensional subspaces. Equally, outside of Asplund or Frechet spaces the most puissant results fail. Nonetheless, many problems inevitably lie in large (nonsmooth or non-Frechet) spaces, X . In such settings the `target' set may be significantly smaller and so lie in a much more richly renormable space, Y . For example, in most contexts existence results in control will require some measure of weak compactness of an associated lower level set, S. This set perforce lies in a weakly compactly generated and so smoothable subspace Y , and it is often the case that only variations in that subspace need be examined. Our intention in this paper is to provide variational tools in such settings and to present some of the possible applications. The organization of the paper is as follows. In the next section we define the underlying concept of partial viscosity subdifferentials that allows us to derive the variational results alluded to above. Then in Section 3 we study sufficient conditions ensuring the existence of the appropriately "nice" renorms of the underlying space. In Section 4 we give a version of the Borwein-Preiss smooth variational principle in a Banach spaces with a partially smooth equivalent norm: this is the basis for our variational arguments. Section 5 is devoted to a non-local fuzzy sum rule extending the result in for smooth Banach spaces. This non-local fuzzy sum rule marries the smooth variational principle with a decoupling technique of Crandall and Lions. We then use this rule to deduce and extend several other important results in nonsmooth analysis: (i) the local fuzzy sum rule in Section 6, (ii) Zagrodny's approximate mean value theorem in Section 7, (iii) the Clarke-Ledyaev mean value inequality in Section 8 and the Kruger-Mordukhovich extremal principle in Section 9. In Section 10 we discuss the relationship between the partial viscosity subdifferential and other frequently used subdifferential concepts. A brief application to distance functions and best approximations in Section 11 concludes the paper.