Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
Prof. P.A. Parrilo
Spring 2006
MIT 6.972
Algebraic methods and semidenite programming
Homework assigment # 1
Date Given:
Date Due:
February 20th, 2006
March 2nd, 4PM
P1. [20 pts ] Classify the follow
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
February 28, 2006
Lecture 6
Lecturer: Pablo A. Parrilo
Scribe: ?
Last week we learned about explicit conditions to determine the number of re
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 2, 2006
Lecture 7
Lecturer: Pablo A. Parrilo
Scribe: ?
In this lecture we introduce a special class of multivariate polynomials, called
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 7, 2006
Lecture 8
Lecturer: Pablo A. Parrilo
1
Scribe: ?
SDP representability
A few lectures ago, when discussing the set of nonnegativ
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 9, 2006
Lecture 9
Lecturer: Pablo A. Parrilo
Scribe: ?
In this lecture, we study rst a relatively simple type of polynomial equations,
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
February 23, 2006
Lecture 5
Lecturer: Pablo A. Parrilo
Scribe: Noah Stein
In this lecture we study univariate polynomials, particularly quest
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
February 16, 2006
Lecture 4
Lecturer: Pablo A. Parrilo
Scribe: Pablo A. Parrilo
In this lecture we will review some basic elements of abstrac
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
Prof. P.A. Parrilo
Spring 2006
MIT 6.972
Algebraic methods and semidenite programming
Homework assigment # 2
Date Given:
Date Due:
March 23rd, 2006
April 4th, 4PM
P1. [20 pts ] Prove the following:
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
Prof. P.A. Parrilo
Spring 2006
MIT 6.972
Algebraic methods and semidenite programming
Homework assigment # 3
Date Given:
Date Due:
May 12th, 2006
May 19th, 12PM
P1. [30 pts ] Recall the relaxations
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
February 7, 2006
Lecture 1
Lecturer: Pablo A. Parrilo
1
Scribe: Pablo A. Parrilo
Introduction: what is this course about?
In this course we a
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
February 14, 2006
Lecture 3
Lecturer: Pablo A. Parrilo
Scribe: Pablo A. Parrilo
In this lecture, we will discuss one of the most important ap
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
February 9, 2006
Lecture 2
Lecturer: Pablo A. Parrilo
Scribe: Pablo A. Parrilo
Notation: The set of real symmetric n n matrices is denoted S
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 14, 2006
Lecture 10
Lecturer: Pablo A. Parrilo
Scribe: ?
In this lecture we begin our study of one of the main themes of the course, na
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 16, 2006
Lecture 11
Lecturer: Pablo A. Parrilo
Scribe: ?
In this lecture we continue our study of SOS polynomials. After presenting a c
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
May 2, 2006
Lecture 19
Lecturer: Pablo A. Parrilo
Scribe: ?
Today we continue with some additional aspects of quantier elimination. We will t
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
May 4, 2006
Lecture 20
Lecturer: Pablo A. Parrilo
Scribe: ?
In this lecture we introduce Schmdgens theorem about the K moment problem (or equ
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
May 9, 2006
Lecture 21
Lecturer: Pablo A. Parrilo
Scribe: ?
In this lecture we study techniques to exploit the symmetry that can be present i
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
Sum of Squares Programs and Polynomial Inequalities
Pablo A. Parrilo
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, MA 021394307, USA
[email protected]
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
April 27, 2006
Lecture 18
Lecturer: Pablo A. Parrilo
Scribe: ?
Quantier elimination (QE) is a very powerful procedure for problems involving
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
April 13, 2006
Lecture 17
Lecturer: Pablo A. Parrilo
Scribe: ?
One of our main goals in this course is to achieve a better understanding of t
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 21, 2006
Lecture 12
Lecturer: Pablo A. Parrilo
1
Scribe: ?
Recovering a measure from its moments
We review next a classical method for
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
March 23, 2006
Lecture 13
Lecturer: Pablo A. Parrilo
Scribe: ?
Today we introduce the rst basic elements of algebraic geometry, namely ideals
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
April 4, 2006
Lecture 14
Lecturer: Pablo A. Parrilo
Scribe: ?
After a brief review of monomial orderings, we develop the basic ideas of Groeb
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
April 11, 2006
Lecture 16
Lecturer: Pablo A. Parrilo
1
Scribe: ?
Generalizing the Hermite matrix
Recall the basic construction of the Hermite
Algebraic Techniques and Semidefinite Optimization
EECS 6.972

Spring 2006
MIT 6.972 Algebraic techniques and semidenite optimization
April 6, 2006
Lecture 15
Lecturer: Pablo A. Parrilo
Scribe: ?
Today we will see a few more examples and applications of Groebner bases, and w