MIT OpenCourseWare

Browse materialsMIT 6.972 · Spring 2006

Algebraic Techniques and Semidefinite Optimization

Subject: Algebra

Art: not stated, Fletcher's Combined Block Telegraph Apparatus and Bell, 1889 · Public domain

Instructor: Prof. Pablo Parrilo
Term: Spring 2006
Level: Graduate
What's inside: 1 lecture note · 1 reading · 25 other files · 3 web pages
Materials open: MIT OCW

Start here

Start course: Lecture 1: Introduction Review of Convexity and Linear…Find a topic

Progress is saved in this browser.

Export or import progress

About this course

Source: MIT OpenCourseWare 6.972 Algebraic Techniques and Semidefinite Optimization, Spring 2006, Prof. Pablo Parrilo — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

How this course was organized

What's here, and what isn't

A lecture sequence of 22 items is confirmed. It stays as organized materials rather than a full course order.

Not included here

16 items from the source could not be safely included:

MIT's page format for this item wasn't recognized clearly enough to place it safely. (16 items)

Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Lecture Notes

22 files

Assignments

MIT OpenCourseWare, Prof. Pablo Parrilo, 6.972 Algebraic Techniques and Semidefinite Optimization, Spring 2006.

License: CC BY-NC-SA 4.0 · Edition: Spring 2006 · Open the original course on OCW ↗

Organized by RedLectern, independent of MIT and not endorsed by MIT.

Per-resource rights are recorded on each resource; All-Rights-Reserved items are link-only, never mirrored.

Algebraic Techniques and Semidefinite Optimization

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Lecture 1: Introduction Review of Convexity and Linear Programming

Lecture 2: PSD Matrices Semidefinite Programming

Lecture 3: Binary Optimization Bounds: Goemans-Williamson and Nesterov Linearly Constrained Problems

Lecture 4: Review: Groups, Rings, Fields Polynomials and Ideals

Lecture 5: Univariate Polynomials Root Bounds and Sturm Sequences Counting Real Roots Nonnegativity Sum of Squares Positive Semidefinite Matrices

Lecture 6: Resultants Discriminants Applications The set of Nonnegative Polynomials

Lecture 7: Hyperbolic Polynomials SDP Representability

Lecture 8: SDP Representability Convex Sets in R² \ⁿ \ⁿHyperbolicity and the Lax Conjecture Relating SDP-representable Sets and Hyperbolic Polynomials Characterization

Lecture 9: Binomial Equations Newton Polytopes The Bézout and BKK Bounds Application: Nash Equilibria

Lecture 10: Nonegativity and Sums of Squares Sums of Squares and Semidefinite Programming Applications and Extensions Multivariate Polynomials Duality and Density

Lecture 11: SOS Applications Moments Bridging the Gap

Lecture 12: Recovering a Measure from its Moments A Probabilistic Interpretation Duality and Complementary Slackness Multivariate Case Density Results

Lecture 13: Polynomial Ideals Algebraic Varieties Quotient Rings Monomial Orderings

Lecture 14: Monomial Orderings Gröbner Bases Applications and Examples Zero-dimensional Ideals

Lecture 15: Zero-dimensional Ideals Hilbert Series

Lecture 16: Generalizing the Hermite Matrix Parametric Versions SOS on Quotients

Lecture 17: Infeasibility of Real Polynomial Equations Certificates The Zero-dimensional Case Optimization

Lecture 18: Quantifier Elimination Tarski-Seidenberg Cylindrical Algebraic Decomposition (CAD)

Lecture 19: Certificates Psatz Revisited Copositive Matrices and Pólya’s Theorem Positive Polynomials

Lecture 20: Positive Polynomials Schmüdgen’s Theorem

Lecture 21: Groups and their Representations Algebra Decomposition

Lecture 22: Sums of Squares Programs and Polynomial Inequalities