In this paper, we consider a class of semi-infinite programming problems with a parameter. As the parameter increases, we prove that the optimal values decrease monotonically. Moreover, the limit of the sequence of optimal values exists as the parameter tends to infinity. In finding the limit, we decompose the original optimization problem into a series of subproblems. By calculating the maximum optimal values to the subproblems and applying a fixed-point theorem, we prove that the obtained maximum value is exactly the limit of the sequence of optimal values under certain conditions. As a result, the limit can be obtained efficiently by solving a series of simplified subproblems. Numerical examples are provided to verify the limit obtained by the proposed method.

中文翻译：

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一类半无限规划问题的最优性分析

在本文中，我们考虑一类带有参数的半无限规划问题。随着参数的增加，我们证明了最优值单调减少。此外，由于参数趋于无穷大，因此存在最优值序列的限制。在寻找极限时，我们将原始优化问题分解为一系列子问题。通过计算子问题的最大最优值并应用不动点定理，我们证明了得到的最大值正是特定条件下最优值序列的极限。因此，通过求解一系列简化的子问题，可以有效地获得极限。提供了数值例子来验证所提出的方法获得的极限。