论文标题
在优化理论中的隐式变量上
On implicit variables in optimization theory
论文作者
论文摘要
数学程序的隐式变量是不需要优化但用于建模可行性条件的变量。它们经常出现在几个不同的优化理论类别中，其中包括双重编程，评估的多目标优化或带有松弛变量的非线性优化问题。为了处理隐式变量，它们通常被解释为明确的变量。在这里，我们首先指出，这是一种头脑头的方法，可引起人造本地最佳解决方案。之后，我们得出了各种Mordukhovich-Stationarity-type必要的最佳条件，这些条件与将隐式变量视为显式变量，或者一方面仅将它们隐式使用它们对另一方面的约束进行建模。将提供对获得的平稳性条件以及相关的基础约束资格的详细比较。总体而言，我们在相当普遍的环境中依靠现代分析工具。最后，我们将发现应用于数学优化的不同知名问题类别，以可视化所获得的理论。
Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory.
