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 $x$-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 \emph{transformation-based discretization method} under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on three 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.
Citation
Mathematical Methods of Operations Research (2020), 1-32; DOI 10.1007/s00186-020-00724-8 (http://link.springer.com/article/10.1007/s00186-020-00724-8)