Lecture 1 Notes: Convexity
2
Topics
To understand better what is going on, we will embark in a journey to learn a wide variety of methods
used to approach these problems. Some of our stops along the way will include:
Linear optimization, second order con
Lecture 7 Notes: Hyperbolic Polynomials
1
Hyperbolic polynomials
Consider a homogeneous multivariate polynomial p R[x1 , . . . , xn ] of degree d. Here homogeneous of
degree d means that the sum of degrees of each monomial is constant and equal to d, i.e.
Lecture 4 Notes: Polynomials
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Review: groups, rings, elds
Denition 1 A group consists of a set G and a binary operation dened on G, for which the
following conditions are satised:
1. Associative: (a b) c = a (b c), for all a, b, c G.
2. Identity: There e
Lecture 2 Notes: PSD Matrices
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PSD matrices
There are several equivalent conditions for a matrix to be positive (semi)denite. We present below
some of the most useful ones:
Proposition 1 The following statements are equivalent:
The matrix A S n is posit
Lecture 8 Notes: Lax Conjecture
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Convex sets in R2
In this lecture we will study conditions that a set S R2 must satisfy for it to be semidenite representable, i.e., to admit a characterization of the type
cfw_(x, y) R2 | I + xB + yC 0,
(1)
where B, C S
Lecture 6 Notes: Resultants
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Resultants
Consider two polynomials p(x) and q(x), of degree n, m, respectively. We want to obtain an easily
checkable criterion to determine whether they have a common root, that is, there exists an x0 C for
which p(x0 ) = q
Lecture 10 Notes: Semidefinite Programming
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Nonegativity and sums of squares
Recall from a previous lecture the denition of a polynomial being a sum of squares.
Denition 1. A univariate polynomial p(x) is a sum of squares (SOS) if there exist q1 , . . .
Lecture 9 Notes: Binomial Equations
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Binomial equations
We introduce in this section a particular kind of polynomial equations, that have nice computational
properties. A binomial system of polynomial equations is one where each equation has only two ter
Lecture 5 Notes: Real Roots
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Univariate polynomials
A univariate polynomial p(x) R[x] of degree n has the form:
p(x) = pn xn + pn1 xn1 + + p1 x + p0 ,
(1)
where the coecients pk are real. We normally assume pn = 0, and occasionally we will normalize it t
Lecture 3 Notes: Binary Optimization
In this lecture, we will discuss one of the most important applications of semidenite programming,
namely its use in the formulation of convex relaxations of nonconvex optimization problems. We will
present the results