Netzer T. Geometry of Linear Matrix Inequalities. A Course...Optimization 2023

Textbook in PDF format

This textbook provides a thorough introduction to spectrahedra, which are the solution sets to linear matrix inequalities, emerging in convex and polynomial optimization, analysis, combinatorics, and algebraic geometry. Including a wealth of examples and exercises, this textbook guides the reader in helping to determine the convex sets that can be represented and approximated as spectrahedra and their shadows (projections). Several general resuts obtained in the last 15 years by a variety of different methods are presented in the book, along with the necessary background from algebra and geometry.
Introduction and Preliminaries
Introduction
Preliminaries
Linear Matrix Inequalities and Spectrahedra
Spectrahedra
First Properties of Spectrahedra
Hyperbolic Polynomials
Definite Determinantal Representations and Interlacing
Hyperbolic Curves and the Helton-Vinnikov Theorem
Hyperbolic Polynomials from Graphs
Derivative Cones
Free Spectrahedra
Spectrahedral Shadows
Spectrahedral Shadows
Operations on Spectrahedral Shadows
Positive Polynomials and the Lasserre-Parrilo Relaxation
Convex Hulls of Curves
General Exactness Results: The Helton-Nie Theorems
Hyperbolicity Cones as Spectrahedral Shadows
Necessary Conditions for Exactness
Generalized Relaxations and Scheiderer's Counterexamples
Real Algebraic Geometry
Semialgebraic Sets, Semialgebraic Mappings and Dimension
Positive Polynomials and Quadratic Modules
Positive Matrix Polynomials
Model-Theoretic Characterization of Stability
Sums of Squares on Compact Curves and Base Extension
Convexity
Convex Cones and Duality
Faces and Dimension
Semidefinite Programming
Lagrange Multipliers and Convex Optimization
References