  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.
