Stephan Dempe - Bilevel Optimization: Advances and Next Challenges

Aims and Scope

Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences.

The series The seriesSpringer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing.

Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS.

More information about this series at http://www.springer.com/series/7393

This Springer imprint is published by the registered company Springer Nature Switzerland AG.

The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Bilevel optimization refers to the area of optimization dealing with problems having a hierarchical structure, involving two decision-makers: a leader and a follower. This problem originated from the field of economic game theory and was introduced in the habilitation thesis of Heinrich Freiherr von Stackelberg (October 31, 1905, Moscow to October 12, 1946, Madrid) completed in 1934. This thesis, written in Cologne, on market structure and equilibrium (in German language: Marktform und Gleichgewicht) was published in the same year by Julius Springer, Berlin and Wien [8]. An English translation of the thesis was published in 2011 by Springer [9].

One of the central topics in von Stackelbergs habilitation thesis is a model of duopoly, now known as Stackelberg game. About 50 years later, mathematicians renamed the model into the bilevel optimization (or as synonym bilevel programming [1]) problem and its quick development within mathematical optimization started into different directions. One of the initial points of attention was the realization that the problem is not well-posed if the followers decision is not uniquely defined. Another issue resulted from different possibilities to transform the bilevel problem into single-level problems, which are not necessarily equivalent to the original one. It might be worth to note that later on, two-level (as a synonym for bilevel) optimization was one of the initial sparks of non-differentiable optimization.

Nowadays, bilevel optimization has further developed into a wide number of different directions (finite and infinite dimensional problems, instances with one or many objective functions in the lower- and/or upper-level problem, as well as problems with discrete or continuous variables in one or both levels). Although we can find deterministic algorithms as well as metaheuristics suggested to solve those problems, the bilevel optimization problem itself is NP-hard. The problem has a huge number of applications, and there is now a strong interaction between bilevel optimization theory and related applications.