Connected level sets, minimizing sets, and uniqueness in optimization
Connected level sets, minimizing sets, and uniqueness in optimization
复制标题
连接水平集、最小化集和优化的唯一性
DOI：
10.1007/bf00934339
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发表时间：
1982
影响因子：
1.9
通讯作者：
D. H. Martin
中科院分区：
文献类型：
--
作者：
D. H. Martin
研究了拓扑空间X上真实的函数f的低水平集的连通性与f的适当定义的极小集的唯一性之间的密切关系。提出了两种不同的理论，简单的一个是关于LE-水平集
$$LE_\alpha（f）= \{ x \in X| f（x）{\displaystyle f（x）}
另一个是LT级集合
$$LT_\alpha（f）= \{ x \in X| f（x）\leqslane\alpha \} .$$
在每一个理论中，最小化集的一个特定概念被定义为使得具有连通水平集的函数f最多可以有一个最小化集。匡威结果表明，如果X是Hausdorff的，并且集合LEα（f）都是紧的，那么在每个理论中，f只有当它有连通的水平集时才有唯一的极小化集。本文的结论表明，连接LT-水平集的功能自然出现在参数线性规划。
AbstractIntimate relationships are investigated between connectedness properties of the lower level sets of a real functionf on a topological spaceX and the uniqueness of suitably defined minimizing sets forf. Two distinct theories are presented, the simpler one pertaining to the LE-level sets
$$LE_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} $$
and the other to the LT-level sets
$$LT_\alpha (f) = \{ x \in X|f(x) \leqslant \alpha \} .$$
In each theory, a specific notion of minimizing set is defined in such a way that a functionf having connected level sets can have at most one minimizing set. That this uniqueness is not trivial, however, is shown by the converse result that, ifX is Hausdorff and the sets LEα(f) are all compact, then, in each theory,f has a unique minimizing set only if it has connected level sets. The paper concludes by showing that functions with connected LT-level sets arise naturally in parametric linear programming.