Semidenite Programming
OR 6327 Spring 2012
Scribe: Mike Todd
Lecture 1
January 24, 2012
This course will be concerned with semidenite programming, the study of optimization
problems that include constraints that certain matrices be positive semidenite. We

OR 6327: Semidenite Programming. Spring 2012.
Homework Set 2. Due: Thursday March 8.
1. Consider the dynamical system x = Ax, where x
A=
4
0
6 1
2
and
.
a) Show that xT x may not be decreasing, by choosing a suitable x(0).
b) Give a certicate that x conv

OR 6327: Semidenite Programming. Spring 2012.
Takehome Final Exam. Pickup Tuesday, May 15th, 9 am (also available electronically); due
12 m (noon) Wednesday.
This is to be your work solely. You can refer to the class notes and to any results from
homework

Semidenite Programming
OR 6327 Spring 2012
Scribe: Joyjit Roy
Lecture 11
February 28, 2012
Polynomial Optimization
Today, we will be looking at the problem of globally optimizing polynomials, either with no
constraints or with polynomial constraints. Lets

Semidenite Programming
OR 6327 Spring 2012
Scribe: Alex Fix
1
Lecture 10
February 23, 2012
Concluding discussion of the Lovsz -Function
a
We will use the insights of the previous lectures to end up with a general way to use SDP for
combinatorial optimizat

Semidenite Programming
OR 6327 Spring 2012
Scribe: Dmitriy Drusvyatskiy
Lecture 12
March 1, 2012
Consider the primal-dual pair of semidenite programs,
minX
(P )
C X
AX = b,
X
0,
maxy,S
(D)
bT y
A y + S = C,
S
0.
We have seen that weak duality holds, meani

Semidenite Programming
OR 6327 Spring 2012
Scribe: Daniel Fleischman
1
Lecture 8
February 16, 2012
Shannon Capacity - Motivation
Suppose you have a channel where you can send one symbol per time interval, from a nite
set of n symbols. Upon proper encoding

Semidenite Programming
OR 6327 Spring 2012
Scribe: SinShuen Cheung
Lecture 9
Febrary 21, 2012
Today we talk about SDP formulations of the Lovsz theta function.
a
1
SDP formulations of the Lovsz theta function
a
Recall that in the last lecture, we introduc

Semidenite Programming
OR 6327 Spring 2012
Scribe: Patrick Steele
1
Lecture 7
February 14, 2012
Applications of semidenite programming to combinatorial optimization
We consider the Max Cut problem and the Lovaszs theta function (discussed in the next lect

Semidenite Programming
OR 6327 Spring 2012
Scribe: Chaoxu Tong
Lecture 5
February 7, 2012
Convex Quadratically-Constrained Quadratic Programming .
Consider the problem
min 0 ()
() 0,
= 1, . . . , ,
where each is a convex quadratic function of .
We can a

Semidenite Programming
OR 6327 Spring 2012
Scribe: Shanshan Zhang
Lecture 6
February 9, 2012
Today, we will talk about control theory. For more information, please see Carsten Scherers
notes (see the link on the home page). Consider the linear system
x =

Semidenite Programming
OR 6327 Spring 2012
Scribe: Ilker Birbil
Lecture 4
February 2, 2012
Consider the matrix R Rmn from the last lecture and its singular value decomposition
given by R = P QT , where P Rmm and Q Rnn are orthogonal matrices, and =
Diag (

Semidenite Programming
OR 6327 Spring 2012
Scribe: Xiaoting Zhao
Lecture 3
January 31, 2012
Fact 10: U is psd (respectively, pd) if and only if all its principal minors (2n of them) are
nonnegative (positive); U is pd if and only if all its leading princi

Semidenite Programming
OR 6327 Spring 2012
Scribe: Tia Sondjaja
Lecture 2
January 26, 2012
Today, we will cover a few important facts about symmetric matrices and look at the problem
of minimizing the maximum eigenvalue of a matrix as an SDP problem in du

OR 6327: Semidenite Programming. Spring 2012.
Homework Set 4. Due: Thursday April 19.
1. Assume X
0, S 0, and P is invertible. Suppose that the scaled matrices
T
T
1
P XP and P SP commute. Show that M XS, where M = P T P , is symmetric
and positive denite