Agenda
Applications of semidenite programming
1
Control and system theory
2
Combinatorial and nonconvex optimization
Spectral estimation & super-resolution
3
Control and system theory
SDP in wide use
Agenda
Nesterovs method for the minimization of nonsmooth functions
1
Smoothing
2
Conjugate functions
3
Properties of conjugate functions
Smoothing by conjugation
4
5
Nesterov 2005 algorithm
6
Example
Agenda
An introduction to dual methods: a personal view
1
Status
2
Templates for convex cone problems
3
Proximal dual methods
4
Smoothing and augmented Lagrangian methods
Projections?
Constrained mini
This solution has
been provided by
Weijie Su
Homework II, Math 301
Weijie Su
March 11, 2013
1.
(a) We rst prove a lemma.
Lemma 0.1. Write a Hermitian matrix P = A + iB , where both A, B are real and A
This solution has
been provided by
Will Fithian
Math 301, HW # 1
Will Fithian
March 4, 2013
I discussed some problems with Josh Loftus.
1. (a) As SOCP:
min t
x,t
s.t. x ai
2
t, i
In more explicit con
1
This solution has
been provided by
Weijie Su
(a) We can pose this problem as
minimize
subject to
t
|x ai |2 t, i = 1, . . . , N,
which is an SOCP.
(b) Note that
(
)
x ai
|x ai |2 t
K 0.
t
Consider
Agenda
1
Duality
2
Dual cones
3
Conic duality
4
Examples
5
Geometric view of cone programs
6
Conic duality theorem
7
Examples
Duality
Problem in standard form (do not assume convexity)
minimize
subjec
Agenda
1
2
Polynomial nonnegativity
Sum of squares decomposition and SDP
3
Examples
4
Global optimization
5
Application
Polynomial nonnegativity
Given a polynomial in n variables f (x1 , . . . , xn ),