用集序关系和锥约简排废器
Reduction of Exhausters by Set Order Relations and Cones
摘要：
上、下穷举子的概念是研究非光滑函数的有效工具。有许多研究提出了无约束和约束情况下的最优性条件。我们可以观察到，关于真穷举和伴随穷举的最优性条件都与穷举的所有元素有关。此外，在受约束的情况下，必须为由约束集和点（要检查它是否是最优的）确定的特定锥提供的条件是相当具有挑战性的检查。因此，对于受约束的情况，减少聚类器中的集合的数量是有利的。在这项工作中，我们首先考虑约束最优化问题和处理的问题，减少广义穷举的方向导数的目标函数。本文分别给出了利用集序关系$\preceq^{m_1}$和$\preceq^{m_2}$约简广义下（上）穷举集的结果.进一步证明了广义离散元$E$可约化为$E$关于$\preceq^{m_1}$或$\preceq^{m_2}$的极小元集.然后考虑无约束最优化问题，利用锥来减少下穷举器。
英文摘要：
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.
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