Moment-SOS Hierarchy, The: Lectures in Probability, Statistics, Computational Geometry, Control and Nonlinear Pdes (Optimization and Its Applications)

Contents
Preface
Bibliography
Acknowledgements
1. Notation, Definitions and Preliminaries
	1.1 Notation and definitions
		1.1.1 Semidefinite programming
		1.1.2 Measures and functions
		1.1.3 Polynomials
	1.2 Some background results
		1.2.1 Conditions for existence of a representing measure
		1.2.2 Putinar’s Certificate of Positivity
		1.2.3 Stokes’ Theorem
		1.2.4 The Christoffel function
	1.3 Notes and sources
Bibliography
2. Principle of the Moment-SOS Hierarchy
	2.1 The Moment-SOS hierarchy
	2.2 Illustration in polynomial optimization
	2.3 Notes and sources
Bibliography
Part I. The Moment-SOS Hierarchy for Applications in Probability and Statistics
	3. Volume and Gaussian Measure of Semi-Algebraic Sets
		3.1 Introduction
		3.2 An infinite-dimensional LP
		3.3 The Moment-SOS hierarchy
			3.3.1 Upper bounds on μ(Ω)
			3.3.2 Lower bounds on μ(Ω)
		3.4 Improving convergence by using Stokes’ Theorem
			3.4.1 Additional information from Stokes’ Theorem
			3.4.2 Tighter semidefinite relaxations
		3.5 Some numerical experiments
			3.5.1 On compact sets Ω ⊂ R2
			3.5.2 On non-compact sets Ω ⊂ R2
		3.6 Volume of sublevel sets of polynomials
		3.7 Concluding remarks
		3.8 Notes and sources
	Bibliography
	4. Lebesgue Decomposition of a Measure
		4.1 Introduction
		4.2 Lebesgue decomposition and convex optimization
			4.2.1 Lebesgue decomposition as a convex optimization problem
		4.3 A numerical approximation scheme
			4.3.1 The Moment-SOS hierarchy
			4.3.2 Recovering an (atomic) singular part
			4.3.3 Some numerical experiments
		4.4 Concluding remarks
		4.5 Notes and sources
	Bibliography
	5. Super Resolution on Semi-Algebraic Domains
		5.1 Introduction
		5.2 Super resolution as an instance of GMP
			5.2.1 The Moment-SOS hierarchy
			5.2.2 The Moment-SOS hierarchy in action
		5.3 Examples
		5.4 Concluding remarks
		5.5 Notes and sources
	Bibliography
	6. Sparse Polynomial Interpolation
		6.1 Introduction
		6.2 Sparse interpolation as a super-resolution problem
			6.2.1 A uniqueness result
		6.3 The Moment-SOS hierarchy
			6.3.1 A rigorous compressed sensing LP approach
		6.4 Numerical experiments
		6.5 Notes and sources
	Bibliography
	7. Representation of (Probabilistic) Chance-Constraints
		7.1 Introduction
		7.2 An infinite-dimensional LP
		7.3 The Moment-SOS hierarchy
			7.3.1 Outer approximations
			7.3.2 Inner approximations
			7.3.3 Accelerating convergence
			7.3.4 Numerical examples
		7.4 Concluding remarks
		7.5 Notes and sources
	Bibliography
	8. Approximate Optimal Design
		8.1 Introduction
		8.2 The ideal approximate optimal design problem
		8.3 The Moment-SOS hierarchy for optimal design
			8.3.1 The first step
			8.3.2 The second step
		8.4 Numerical experiments
		8.5 Concluding remarks
		8.6 Notes and sources
	Bibliography
Part II. The Moment-SOS Hierarchy for Applications in Control, Optimal Control and Non-Linear Partial Differential Equations
	9. Optimal Control
		9.1 Introduction
		9.2 Polynomial optimal control
		9.3 LP formulation
			9.3.1 Relaxed controls
			9.3.2 Occupation measure
			9.3.3 Primal LP on measures
			9.3.4 Dual LP on functions
		9.4 Approximation results
			9.4.1 Lower bound on value function
			9.4.2 Uniform approximation of the value function along trajectories
		9.5 Optimal control over a set of initial conditions
		9.6 A numerical scheme via the Moment-SOS hierarchy
		9.7 Numerical illustrations
			9.7.1 Uniform approximation along an optimal trajectory
			9.7.2 Uniform approximation over a set
		9.8 Notes and sources
	Bibliography
	10. Convex Computation of Region of Attraction and Reachable Set
		10.1 Introduction
		10.2 Occupation measures and Liouville equation
		10.3 Infinite dimensional LP characterization of ROA
			10.3.1 Dual LP
		10.4 Finite-dimensional SDP approximations
			10.4.1 Outer approximation
		10.5 Numerical examples
			10.5.1 Van der Pol oscillator
			10.5.2 Double integrator
			10.5.3 Acrobot
		10.6 Notes and sources
	Bibliography
	11. Non-Linear Partial Differential Equations
		11.1 Introduction
		11.2 Notions of solutions
			11.2.1 Weak and entropy solutions
			11.2.2 Measure-valued solutions
			11.2.3 An emphasis on compact sets
		11.3 A convex optimization approach for mv solutions on compact sets
			11.3.1 Moment constraints for the entropy mv solution
				11.3.1.2 Second step: enforcing (11.16) by moment constraints
			11.3.2 The Generalized Moment Problem
			11.3.3 Interpretation of the moment solutions
		11.4 The Riemann problem for the Burgers equation
			11.4.1 Shock waves
			11.4.2 Rarefaction waves
		11.5 Notes and sources
	Bibliography
	12. Miscellaneous
		12.1 Scalability and sparsity
			12.1.1 Structured sparsity pattern
			12.1.2 A sparse Moment-SOS hierarchy for optimization
			12.1.3 An alternative sparse Positivstellensatz
		12.2 A distinguished SOS: The Christoffel polynomial
		12.3 Notes and sources
	Bibliography
Glossary
Index