The lecture notes provided below are preliminary and an ongoing work.
Course notes.
| LEC # |
TOPICS |
LECTURE NOTES |
| 1 |
Introduction
Review of Convexity and Linear Programming |
(PDF) |
| 2 |
PSD Matrices
Semidefinite Programming |
(PDF) |
| 3 |
Binary Optimization
Bounds: Goemans-Williamson and Nesterov
Linearly Constrained Problems |
(PDF) |
| 4 |
Review: Groups, Rings, Fields
Polynomials and Ideals |
(PDF) |
| 5 |
Univariate Polynomials
Root Bounds and Sturm Sequences
Counting Real Roots
Nonnegativity
Sum of Squares
Positive Semidefinite Matrices |
(PDF) |
| 6 |
Resultants
Discriminants
Applications
The set of Nonnegative Polynomials |
(PDF) |
| 7 |
Hyperbolic Polynomials
SDP Representability |
(PDF) |
| 8 |
SDP Representability
Convex Sets in R2
Hyperbolicity and the Lax Conjecture
Relating SDP-representable Sets and Hyperbolic Polynomials
Characterization |
(PDF) |
| 9 |
Binomial Equations
Newton Polytopes
The Bézout and BKK Bounds
Application: Nash Equilibria |
(PDF) |
| 10 |
Nonegativity and Sums of Squares
Sums of Squares and Semidefinite Programming
Applications and Extensions
Multivariate Polynomials
Duality and Density |
(PDF) |
| 11 |
SOS Applications
Moments
Bridging the Gap |
(PDF) |
| 12 |
Recovering a Measure from its Moments
A Probabilistic Interpretation
Duality and Complementary Slackness
Multivariate Case
Density Results |
(PDF) |
| 13 |
Polynomial Ideals
Algebraic Varieties
Quotient Rings
Monomial Orderings |
(PDF) |
| 14 |
Monomial Orderings
Gröbner Bases
Applications and Examples
Zero-dimensional Ideals |
(PDF) |
| 15 |
Zero-dimensional Ideals
Hilbert Series |
(PDF) |
| 16 |
Generalizing the Hermite Matrix
Parametric Versions
SOS on Quotients |
(PDF) |
| 17 |
Infeasibility of Real Polynomial Equations
Certificates
The Zero-dimensional Case
Optimization |
(PDF) |
| 18 |
Quantifier Elimination
Tarski-Seidenberg
Cylindrical Algebraic Decomposition (CAD) |
(PDF) |
| 19 |
Certificates
Psatz Revisited
Copositive Matrices and Pólya's Theorem
Positive Polynomials |
(PDF) |
| 20 |
Positive Polynomials
Schmüdgen's Theorem |
(PDF) |
| 21 |
Groups and their Representations
Algebra Decomposition |
(PDF) |
| 22 |
Sums of Squares Programs and Polynomial Inequalities |
(PDF) |