Equivalence of Robust Multiobjective Bilevel Decision Problem Under Multiple Perturbation Sets

The robust multiobjective bilevel decision problem (RMBDP) based on worst-case conditional value at risk (WCVaR) is a bilevel decision model composed of multiple loss objectives of upper- and lower-level decision-makers under multiple perturbation sets in a concave probability density distribution function cluster. First, a concave probability density distribution function cluster composed of differential multiple perturbation sets reflecting the objectives of upper- and lower-level decision-makers is defined, respectively. Given the corresponding multiple loss objective functions, a robust VaR vector and a robust WCVaR vector corresponding to the differential multiple perturbations convex sets are defined, respectively. It has been theoretically proven that the robust VaR and WCVaR vectors are the risk losses of the VaR and WCVaR vectors in the worst-case scenario corresponding to the differential multiple perturbation sets. Then, the RMBDP based on WCVaR under differential multiple perturbations sets in a concave probability density distribution function cluster is constructed. The main conclusion includes that under certain conditions, an efficient solution to WCVaR-based RMBDP is equivalent to an efficient solution to a semi-infinite multiobjective bilevel decision problem with multiple perturbation sets. Finally, for the case where both upper and lower levels are finite mixed distribution function clusters, it is proved that an efficient solution WCVaR-based RMBDP is equivalent to an efficient solution to a multiobjective bilevel decision problem with finite multiple constraints. In this case, when all objectives and constraint functions are linear, the equivalent bilevel problem can be transformed into solving an approximate bilevel linear programming.