Approximate subdifferentials and applications. I. The finite-dimensional theory
Александр Давидович Иоффе · Transactions of the American Mathematical Society · 1984
We introduce and study a new class of subdifferentials associated with arbitrary functions. Among the questions considered are: connection with other derivative-like objects (e.g. derivatives, convex subdifferentials, generalized gradients of Clarke and derivate containers of Warga), calculus of approximate subdifferentials and applications to analysis of set-valued maps and to optimization. It turns out that approximate subdifferentials are minimal (as sets) among other conceivable subdifferentials satisfying some natural requirements. This shows that certain results involving approximate subdifferentials are the best possible and, at the same time, marks certain limitations of nonsmooth analysis. Another important property of approximate subdifferentials is that, being essentially nonconvex, they admit a rich calculus that covers the calculus of convex subdifferentials and leads to more precise and sometimes new results for generalized gradients of Clarke.