Contributions to complementarity and bilevel programming in Banach spaces

• Main Author: Mehlitz, Patrick
• Other Authors: TU Bergakademie Freiberg, Mathematik und Informatik
• Format: Doctoral Thesis
• Language: English
• Published: Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola" 2017
• Subjects: Komplementaritätsoptimierung; Optimale Steuerung; Variationsanalysis; Zwei-Ebenen-Optimierung; Bilevel programming; Complementarity programming; Optimal control; Variational analysis; ddc:511; Banach-Raum; Komplementaritätsproblem; Variationsrechnung; Optimale Kontrolle
• Online Access: http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-227091; http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-227091; http://www.qucosa.de/fileadmin/data/qucosa/documents/22709/thesisMehlitz2017PDFA.pdf

In this thesis, we derive necessary optimality conditions for bilevel programming problems (BPPs for short) in Banach spaces. This rather abstract setting reflects our desire to characterize the local optimal solutions of hierarchical optimization problems in function spaces arising from several applications. Since our considerations are based on the tools of variational analysis introduced by Boris Mordukhovich, we study related properties of pointwise defined sets in function spaces. The presence of sequential normal compactness for such sets in Lebesgue and Sobolev spaces as well as the variational geometry of decomposable sets in Lebesgue spaces is discussed. Afterwards, we investigate mathematical problems with complementarity constraints (MPCCs for short) in Banach spaces which are closely related to BPPs. We introduce reasonable stationarity concepts and constraint qualifications which can be used to handle MPCCs. The relations between the mentioned stationarity notions are studied in the setting where the underlying complementarity cone is polyhedric. The results are applied to the situations where the complementarity cone equals the nonnegative cone in a Lebesgue space or is polyhedral. Next, we use the three main approaches of transforming a BPP into a single-level program (namely the presence of a unique lower level solution, the KKT approach, and the optimal value approach) to derive necessary optimality conditions for BPPs. Furthermore, we comment on the relation between the original BPP and the respective surrogate problem. We apply our findings to formulate necessary optimality conditions for three different classes of BPPs. First, we study a BPP with semidefinite lower level problem possessing a unique solution. Afterwards, we deal with bilevel optimal control problems with dynamical systems of ordinary differential equations at both decision levels. Finally, an optimal control problem of ordinary or partial differential equations with implicitly given pointwise state constraints is investigated.