Quantified linear programming [book]

Jan Wolf
2022
In many real world problems, dealing with uncertainty is a significant challenge for mathematical programming and related areas, and typically, standard (mixed-integer) linear programming formulations are not well suited to model such problems. However, a powerful way to express uncertainty or adversarial situations is through the use of quantified variables. To pursue this approach, this thesis is concerned with quantified (integer) linear programming, a generalization of traditional (integer)
more » ... linear programming. Whereas in traditional linear programs (LPs) and integer programs (IPs) all variables are implicitly existentially quantified, they can be either existentially or universally quantified in quantified (continuous) linear programs (QLPs) and quantified integer programs (QIPs). Explicit quantification of variables yields a considerable increase in expressive power, suitable to compactly represent many problems from the field of optimization under uncertainty, a topic that has engaged much attention in the mathematical programming com- munity and beyond in the last decade. In the context of optimization under uncertainty robust optimization and stochastic programming are currently the most prominent modeling paradigms. However, the abilities of linear programming extensions by variable quantification are relatively unexplored. The present thesis is subdivided into two parts and its aim is to study quantified linear programming as an independent mathematical programming paradigm. However, we mainly focus on the continuous case. In the first part we give a detailed introduction into the concept of quantified (integer) linear programming and highlight the strong similarities to two-person zero-sum games and related work in the context of optimization under uncertainty. We furthermore present some illustrating examples to demonstrate the modeling power and broad applicability of this methodology. After a survey of the basic results of current research in this field both theoretical and applied, we also provide [...]
doi:10.26083/tuprints-00021382 fatcat:2twiritf7zajzjunxummpo2o2i