2017/07/13 by Patrick Mehlitz · 1 citation
Computer Science · Engineering · Mathematics · #Artificial intelligence #Bilevel optimization #Class (philosophy) #Computer science #Fixed Point Theorems Analysis #Mathematical economics #Mathematical optimization #Mathematics #Optimization and Mathematical Programming #Optimization and Variational Analysis #Optimization problem
paper · doi:10.1080/02331934.2017.1349123
crossref issued 2017/07/13 · crossref published 2017/07/13 · crossref published-online 2017/07/13 · openalex publication_date 2017/07/13 · crossref created 2017/07/13 · crossref deposited 2017/08/29 · crossref published-print 2017/10/03 · openalex created_date 2025/10/10 · crossref indexed 2026/08/03 · openalex updated_date 2026/08/04
We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.