The Moment-SOS Hierarchy

The Moment-SOS hierarchy is a powerful methodology that is used to solve the Generalized Moment Problem (GMP) where the list of applications in various areas of Science and Engineering is almost endless. Initially designed for solving polynomial optimization problems (the simplest example of the GMP), it applies to solving any instance of the GMP whose description only involves semi-algebraic functions and sets. It consists of solving a sequence (a hierarchy) of convex relaxations of the initial problem, and each convex relaxation is a semidefinite program whose size increases in the hierarchy.The goal of this book is to describe in a unified and detailed manner how this methodology applies to solving various problems in different areas ranging from Optimization, Probability, Statistics, Signal Processing, Computational Geometry, Control, Optimal Control and Analysis of a certain class of nonlinear PDEs. For each application, this unconventional methodology differs from traditional approaches and provides an unusual viewpoint. Each chapter is devoted to a particular application, where the methodology is thoroughly described and illustrated on some appropriate examples.The exposition is kept at an appropriate level of detail to aid the different levels of readers not necessarily familiar with these tools, to better know and understand this methodology.Contents: Notation, Definitions and PreliminariesPrinciple of the Moment-SOS HierarchyThe Moment-SOS Hierarchy for Applications in Probability and Statistics:Volume and Gaussian Measure of Semi-Algebraic SetsLebesgue Decomposition of a MeasureSuper Resolution on Semi-Algebraic DomainsSparse Polynomial InterpolationRepresentation of (Probabilistic) Chance-ConstraintsApproximate Optimal DesignThe Moment-SOS Hierarchy for Applications in Control, Optimal Control and Non-Linear Partial Differential Equations:Optimal ControlConvex Computation of Region of Attraction and Reachable SetNon-Linear Partial Differential EquationsMiscellaneous
Readership: Graduate students, academics and researchers interested in the methodology of the moment-SOS hierarchy and its applications in various fields of science and engineering.Moment-SOS Hierarchy;Generalized Moment Problem (GMP);Polynomial Optimization Problems;Convex Relaxation;Semidefinite Program;Optimal Control;Optimal Design; Lebesgue Decomposition;Nonlinear Partial Differential Equations0Key Features:It covers applications of a very different nature in a unified manner with an appropriate level of details for a general audienceFor all these applications (not covered in previous books of the third author), this methodology is new and unconventionalAll three authors are well-respected researchers in control and applied mathematics