Quantified integer programming with polyhedral and decision-dependent uncertainty

The research field of optimization under uncertainty, even though already established in the middle of the last century, gained much attention in the last two decades. The necessity to deal with uncertain data remains a major challenge despite (or because of) the times of big data and cloud solutions. The most prominent modeling paradigms dealing with such optimization tasks are stochastic programming and robust optimization. Sometimes, in order to obtain an even more realistic description of the underlying problem, multistage models can be used. While there are some real multistage stochastic approaches, most multistage extensions to robust optimization hardly ever consider more than two stages. Even less attention is paid to the consideration of decision-dependent uncertainty. To overcome these shortcomings, we explored multistage optimization under decision-dependent uncertainty via quantified programming.
Quantified programs, which are linear programs with ordered variables that are either existentially or universally quantified, provide a convenient framework for multistage optimization under uncertainty. While considerable research has been conducted with regard to quantified linear programming (QLP) this thesis focused on quantified integer programming (QIP). Whereas most research in this area is concerned with complexity results regarding the QIP satisfiability problem, we concentrate on solution and modeling techniques for the QIP optimization problem, which we tested and implemented in our open-source solver.
One way to solve a QIP is to apply a game tree search, enhanced with non-chronological backjumping. We developed and theoretically substantiated further solution techniques for QIPs within a game tree search framework and established the strategic copy-pruning mechanism, which allows to implicitly deduce the existence of a strategy in linear time (by static examination of the QIP-matrix) without explicitly traversing the strategy itself. We also showed that the implementation of our findings can massively speed up the search process.
Furthermore, in order to enhance the expressive power of QIPs, we introduced QIPs with a polyhedral uncertainty set. We showed that by exploiting this extension, we were able to significantly speed up our solver on various multistage combinatorial optimization problems. Additionally, we established QIPs with interdependent domains and thereby provide a general framework for multistage optimization problems under decision-dependent uncertainty. We theoretically substantiated solution techniques, provided implementation details and derived polynomial-time reduction functions mapping both extensions to the basic QIP.