Balancing the solution-quality/time trade-off and optimizing problems which feature offline and online phases can deliver significant improvements in efficiency and budget control. Offline/online integration yields benefits by achieving high quality solutions while reducing online computation time. This book considers multi-stage optimization problems under uncertainty and proposes various methods that have broad applicability. Due to the complexity of the task, the most popular approaches depend on the temporal granularity of the decisions to be made and are, in general, sampling-based methods and heuristics. Long-term strategic decisions that may have a major impact are typically solved using these more accurate, but expensive, sampling-based approaches. Short-term operational decisions often need to be made over multiple steps within a short time frame and are commonly addressed via polynomial-time heuristics, with the more advanced sampling-based methods only being applicable if their computational cost can be carefully managed. Despite being strongly interconnected, these 2 phases are typically solved in isolation. In the first part of the book, general methods based on a tighter integration between the two phases are proposed and their applicability explored, and these may lead to significant improvements. The second part of the book focuses on how to manage the cost/quality trade-off of online stochastic anticipatory algorithms, taking advantage of some offline information. All the methods proposed here provide multiple options to balance the quality/time trade-off in optimization problems that involve offline and online phases, and are suitable for a variety of practical application scenarios.

Artificial intelligence often has to deal with uncertain scenarios, such as a partially observed environment or noisy observations. Traditional probabilistic models, while being very principled approaches in these contexts, are incapable of dealing with both algebraic and logical constraints. Existing hybrid continuous/discrete models are typically limited in expressivity, or do not offer any guarantee on the approximation errors. This book, Learning and Reasoning in Hybrid Structured Spaces, discusses a recent and general formalism called Weighted Model Integration (WMI), which enables probabilistic modeling and inference in hybrid structured domains. WMI-based inference algorithms differ with respect to most alternatives in that probabilities are computed inside a structured support involving both logical and algebraic relationships between variables. While the research in this area is at an early stage, we are witnessing an increasing interest in the study and development of scalable inference procedures and effective learning algorithms in this setting. This book details some of the most impactful contributions in context of WMI-based inference in the last 5 years. Moreover, by providing a gentle introduction to the main concepts related to WMI, the book can be useful for both theoretical researchers and practitioners alike.