2012/06/26 by Stephan Dempe, S. Dempe, Boris S. Mordukhovich +3 · 1 citation
Computer Science · Engineering · Mathematics · #Bellman equation #Bilevel optimization #Computer science #Convex optimization #Epistemology #Fixed Point Theorems Analysis #Law #Lipschitz continuity #Mathematical economics #Mathematical optimization #Mathematics #Operations research #Optimization and Mathematical Programming #Optimization and Variational Analysis #Optimization problem #Pessimism #Philosophy #Political science #Pure mathematics #Representation (politics) #Statistics #Subderivative #Value (mathematics)
paper · doi:10.1080/02331934.2012.696641
crossref issued 2012/06/26 · crossref published 2012/06/26 · crossref published-online 2012/06/26 · openalex publication_date 2012/06/26 · crossref created 2012/06/27 · crossref published-print 2014/04/01 · crossref deposited 2017/06/20 · openalex created_date 2025/10/10 · crossref indexed 2026/08/04 · openalex updated_date 2026/08/05
This article is devoted to the so-called pessimistic version of bilevel programming programs. Minimization problems of this type are challenging to handle partly because the corresponding value functions are often merely upper (while not lower) semicontinuous. Employing advanced tools of variational analysis and generalized differentiation, we provide rather general frameworks ensuring the Lipschitz continuity of the corresponding value functions. Several types of lower subdifferential necessary optimality conditions are then derived by using the lower-level value function approach and the Karush–Kuhn–Tucker representation of lower-level optimal solution maps. We also derive upper subdifferential necessary optimality conditions of a new type, which can be essentially stronger than the lower ones in some particular settings. Finally, certain links are established between the obtained necessary optimality conditions for the pessimistic and optimistic versions in bilevel programming.