Bilevel Optimization Advances and Next Challenges

2019 marked the 85th anniversary of Heinrich Freiherr von Stackelberg’s habilitation thesis “Marktform und Gleichgewicht,” which formed the roots of bilevel optimization. Research on the topic has grown tremendously since its introduction in the field of mathematical optimization. Besides the substantial advances that have been made from the perspective of game theory, many sub-fields of bilevel optimization have emerged concerning optimal control, multiobjective optimization, energy and electricity markets, management science, security and many more. Each chapter of this book covers a specific aspect of bilevel optimization that has grown significantly or holds great potential to grow, and was written by top experts in the corresponding area. In other words, unlike other works on the subject, this book consists of surveys of different topics on bilevel optimization. Hence, it can serve as a point of departure for students and researchers beginning their research journey or pursuing related projects. It also provides a unique opportunity for experienced researchers in the field to learn about the progress made so far and directions that warrant further investigation. All chapters have been peer-reviewed by experts on mathematical optimization.

Part I: Bilevel optimization, game theory, and applications.- Interactions between bilevel optimization and Nash games.- On Stackelberg-Nash equilibria in bilevel optimization games.- A short state of the art on Multi-Leader-Follower games.- Regularization and approximation methods in Stackelberg games and bilevel optimization.- Applications of bilevel optimization in energy and electricity markets.- Bilevel optimization of regularization hyperparameters in machine Learning.- Part II: Theory and methods for linear and nonlinear bilevel optimization.- Bilevel optimization and variational analysis.- Constraint qualifications and optimality conditions in bilevel optimization.- Algorithms for simple bilevel programming.- Algorithms for linear bilevel optimization.- Global search for bilevel optimization with quadratic data.- MPEC methods for bilevel optimization problems.- Approximate bilevel optimization with population-based evolutionary algorithms.- Methods for pessimistic bilevel optimization.- Part III: Extensions and uncertainty in bilevel optimization.- Methods for multiobjective bilevel optimization.- Bilevel optimal control: existence results and stationarity conditions.- Bilevel linear optimization under uncertainty.- A unified framework for multistage mixed integer linear optimization.- Part IV: Numerical and research tools for bilevel optimization.- BOLIB: Bilevel Optimization LIBrary of test problems.- Bilevel optimization: theory, algorithms and applications.- Appendix: Bilevel optimization: bibliography.

Stephan Dempe is a Professor of Mathematical Optimization at the Technical University Bergakademie Freiberg, Germany. He is one of the world’s leading experts on bilevel optimization and has published four books and more than 50 articles on the subject.