Preface

While the scope of a book is to present the problem discussed as esoterically as possible we also have to acknowledge its roots in “real life.” This is the case here with “control problems with contradictory conditions” which are readily found in everyday life. They may not be stated so, but each time we balance between mutually exclusive goals or have to choose a direction in the detriment of another we actually solve (optimally or not) such a control problem. An “either/or” decision is always difficult for human beings, the same goes for optimization algorithms with discrete variables. We have thus strong reasons to pursue solutions for this class of problems. As is usually the case, the solution is better or faster if we understand the problem’s underlying structure. Therefore, the scope of this book is to provide efficient constructions which are subsequently put under a mixed-integer form which can be solved by a computer in a “reasonable” time. This book represents the culmination of over 5 years of collaboration work of the authors. The main contributions are the result of work started during the Ph.D. theses of the first two authors and by subsequent advancements in the following years. This book was inspired by our desire to bring to light the importance of the analysis and control of dynamical systems with conflicting objectives and the effective usage of the associated mixed-integer formulations. It is worth mentioning that the topic is not new and monographs exists covering mixed-integer optimization. However, most of them assume a specific background in mathematics and optimization. The present book is dedicated to a generic class of constraints and their use in optimization problems and, in this respect, goes deeper into the details of their construction, representation and computational complexity. It is important to mention that the present manuscript is mainly dedicated to the problem description and not to the numerical optimization routines. It focuses on the mixed-integer aspects of the constraints formulation and their relationship with the optimization-based control design. To our knowledge, another textbook is not