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 results 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.
Author(s): Tim Netzer, Daniel Plaumann
Series: Compact Textbooks in Mathematics
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 166
City: Basel
Contents
1 Introduction and Preliminaries
1.1 Introduction
1.2 Preliminaries
2 Linear Matrix Inequalities and Spectrahedra
2.1 Spectrahedra
2.2 First Properties of Spectrahedra
2.3 Hyperbolic Polynomials
2.4 Definite Determinantal Representations and Interlacing
2.5 Hyperbolic Curves and the Helton-Vinnikov Theorem
2.6 Hyperbolic Polynomials from Graphs
2.7 Derivative Cones
2.8 Free Spectrahedra
3 Spectrahedral Shadows
3.1 Spectrahedral Shadows
3.2 Operations on Spectrahedral Shadows
3.3 Positive Polynomials and the Lasserre-Parrilo Relaxation
3.4 Convex Hulls of Curves
3.5 General Exactness Results: The Helton-Nie Theorems
3.6 Hyperbolicity Cones as Spectrahedral Shadows
3.7 Necessary Conditions for Exactness
3.8 Generalized Relaxations and Scheiderer's Counterexamples
A Real Algebraic Geometry
A.1 Semialgebraic Sets, Semialgebraic Mappings and Dimension
A.2 Positive Polynomials and Quadratic Modules
A.3 Positive Matrix Polynomials
A.4 Model-Theoretic Characterization of Stability
A.5 Sums of Squares on Compact Curves and Base Extension
B Convexity
B.1 Convex Cones and Duality
B.2 Faces and Dimension
B.3 Semidefinite Programming
B.4 Lagrange Multipliers and Convex Optimization
References
