Reduction of Exhausters by Set Order Relations and Cones

The notions of upper and lower exhausters are effective tools for the study of non smooth functions. There are many studies presenting optimality conditions for unconstrained and constrained cases. One can observe that optimality conditions in terms of both proper and adjoint exhausters are related to all elements of the exhausters. Moreover, in the constrained case the conditions that must be provided for a particular cone determined by constraint set and the point (to be checked whether it is optimal) are rather challenging to check. Thus it is advantageous to reduce the number of sets in the exhauster for constrained case. In this work, we first consider constrained optimization problems and deal with the problem of reducing generalized exhausters of the directional derivative of the objective function. We present some results to reduce generalized lower (upper) exhausters by using set order relations $\preceq^{m_1}$ and $\preceq^{m_2}$, respectively. Furthermore, we show that a generalized exhauster $E$ can be reduced to the set of minimal elements of $E$ with respect to $\preceq^{m_1}$ or $\preceq^{m_2}$. Then considering unconstrained optimization problems, lower exhausters are reduced by using cones.