Approximate Subdifferentials and Applications . I : the Finite Dimensional Theory

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 derívate 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. Introduction. The concept of a subdifferential is usually associated with convex functions for which, to a large extent, subdifferentials proved to be one of the most useful and porweful instruments responsible for the success of convex analysis in the 1960's [17]. Many of the good properties of convex subdifferentials were inherited by generalized gradients introduced by Clarke [3] for lower semicontinuous functions on Banach spaces. Rockafellar gave an alternative definition which applies to arbitrary functions on arbitrary locally convex spaces [19]. Here we study another class of objects called approximate subdifferentials denoted by 3a/(x). They appeared for the first time in a finite dimensional situation as by-products of certain approximative optimization techniques developed by Mordukhovich [14]. Some subsequent attempts to extend the definition to a more general situation were successful only for Banach spaces with an equivalent Gâteaux [7] or Fréchet [12] differentiable norm. A new definition that applies in arbitrary locally convex spaces was offered by Ioffe in [8] where many nice analytic properties of approximate subdifferentials were first announced. An infinite dimensional theory of approximate subdifferentials will be considered in the second ("General theory") and the third ("Banach theory", which is especially rich) parts of the paper approximately corresponding respectively to the first announcement [8] and the first version of the paper mimeographically distributed as [10]. Here we present the first part where all spaces are assumed finite dimensional. There have been two reasons to consider the finite dimensional case separately. On the one hand, definitions and proofs are much simpler in this case and many Received by the editors November 20, 1981 and, in revised form, March 8, 1983. 1980 Mathematics Subject Classification. Primary 49B27. ©1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page 389 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use