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 in control theory
Example: dierential inclusion (x(t) i
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
Examples
Nonsmooth optimization
minimize
subject to
f cvx, dom
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 minimization
minimize
subject to
f (x)
xC
Gradient update
x
Agenda
Interior Point Methods
1
Barrier functions
2
Analytic center
3
Central path
4
Barrier method
5
Primal-dual path following algorithms
6
NesterovTodd scaling
7
Complexity analysis
Interior point methods
Primal (P)
minimize
subject to
cT x
Gx + s = h
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 is symmetric and B is
antisymmetric. Then
(
)
A B
P 0
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 conic form:
T
0
1
min
x,t
x
t
x
ai
+
t
0
s.t.
0, i
(b) Th
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 the Lagrangian
L=t
N
(yi ) (x ai )
,
ti
t
i=1
=t
N
(yi
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
subject to
f0 (x)
fi (x) 0 i = 1, . . . , m
hi (x) = 0 i = 1,
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 ), does there exist x Rn
such that f (x) < 0?
If not, f (