On reduction of exhausters in nonsmooth analysis

In nonsmooth analysis and optimization, some generalizations of the concept of derivative such as directional derivatives (such as Dini, Hadamard directional derivatives), Clarke subdifferential, Michel-Penot subdifferential and quasidifferential are employed, in addition to the concept of gradient used in the analysis of differentiable functions and the concept of subdifferential, which is used instead of gradient in convex analysis. The concept of exhauster is one of the effective tools used in the analysis and optimization of nonsmooth functions. In this thesis, firstly Dini, Hadamard directional derivatives and the optimality conditions given by them are examined. Then, the concept of the subdifferential in convex analysis and optimality conditions obtained by subdifferential are reminded. Then, upper and lower exhausters are introduced and these dual concepts are examined on various examples. The optimality conditions given by upper and lower exhausters are examined and their connection with directional derivatives is explained. After these studies, the "reduction techniques" of exhausters, which have an important role in this theory, were investigated in detail and compiled into two separate groups according to their reduction approaches. First, some techniques for making exhausters smaller by inclusion or minimal by inclusion are examined and these techniques are explained with various examples. Then, the conditions and techniques available in the literature for making exhausters smaller by shape or minimal by shape are examined, and analyzed on illustrative examples.