In the setting of normed spaces ordered by a convex cone not necessarily solid, we use six set scalarization functions, which are extensions of the oriented distance of Hiriart-Urruty, and we discuss convexity and continuity properties of their composition with two set-valued maps. Furthermore, as an application, we derive a multiplier rule for weak minimal solutions of a convex set optimization problem, with respect to the lower set less preorder of Kuroiwa. Some illustrative examples are also given.

中文翻译：

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基于定向距离的六组标量：连续性，凸性及其在凸集优化中的应用

在由凸锥（不一定是实心）排序的赋范空间的设置中，我们使用六个集合标量函数，它们是Hiriart-Urruty的定向距离的扩展，并且我们使用两个集合值映射图讨论了其组成的凸性和连续性。此外，作为一种应用，相对于Kuroiwa的较低集较少的序数，我们针对凸集优化问题的弱最小解导出了一个乘数规则。还给出了一些说明性的例子。