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 following statements as true of false. A proof or
counterexam
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 real roots of a univariate
polynomial. Today we will expa
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 hyperbolic. These
polynomials were originally studied
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 nonnegative polynomials, we encountered convex sets in
R2 that la
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, namely binomial
equations. As we will see, in this case
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 questions regarding the existence of
roots and nonnegativity
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 abstract algebra. We also introduce and begin
studying the mai
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:
(a) Show that the discriminant of a polynomial p(x) sa
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 for linearly and quadratically constrained
quadratic pr
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 aim to understand the properties, both mathematical and
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 applications of semidenite programming,
namely its use in
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 n . A matrix A S n is called positive
semidenite if xT
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, namely the relationships
between polynomials that are sum
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 couple of applications,
we focus here on the dual side,
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 then recall the
Positivstellensatz and its relations wit
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 equivalently, on
u
the representation of positive polynomi
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 in semidenite
programming problems, particularly those a
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
parrilo@mit.edu
Submitted to SIAG/OPT News and Views
1
Introduction
Sum
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 rstorder formulas
over real elds. The cylindrical algeb
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 the techniques available
for polynomial systems over the
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 producing a univariate atomic measure with a given set
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 and varieties over the
complex numbers. This dual view
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 Groebner bases, followed
by examples and applications. For b
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 matrix Hq (p) in the univariate case, whose signature
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 we will develop the
zerodimensional case.
1
Zerodimensio