Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the \(\mathbf {x}\)-dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In this paper we assume that such a transformation exists globally. However, this approach may destroy convexity in the lower level, which is very important for numerical methods. We present in this paper a solution approach for GSIP problems, which cleverly combines the above mentioned two techniques. It is shown that the convergence results for discretization methods in the case of SIP problems can be transferred to our transformation-based discretization method under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on several examples, including a problem of volume-maximal inscription of multiple variable bodies into a larger fixed body, which has never before been considered as a GSIP test problem.

离散化方法通常用于解决标准的半无限优化（SIP）问题。由于无限索引集的\（\ mathbf {x} \）-依赖性，很难将这些方法转移到一般半无限优化（GSIP）问题的情况下。另一方面，在合适的条件下，GSIP问题可以转换为SIP问题。在本文中，我们假设这种转换在全球范围内存在。但是，此方法可能会破坏较低级别的凸度，这对于数值方法非常重要。我们在本文中提出了一种解决GSIP问题的方法，该方法巧妙地结合了上述两种技术。结果表明，在SIP问题的情况下，离散化方法的收敛结果可以传递给我们。基于变换的离散化方法在适当的假设下进行变换。最后，我们在几个示例中说明了我们的方法的操作及其性能，包括将多个变量的体积最大写入到较大的固定体内的问题，这以前从未被视为GSIP测试问题。