Fall 2010. Lecture . 7. N. Harvey. TexPoint. fonts used in EMF. . Read the . TexPoint. manual before you delete this box. .: . A. A. A. A. A. A. A. A. A. A. Covering Hemispheres by Ellipsoids. Let . ID: 644733
DownloadNote - The PPT/PDF document "C&O 355 Mathematical Programming" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
C&O 355Mathematical ProgrammingFall 2010Lecture 7
N. Harvey
TexPoint
fonts used in EMF.
Read the
TexPoint
manual before you delete this box
.:
A
A
A
A
A
A
A
A
A
A
Slide2Covering Hemispheres by EllipsoidsLet B
= { unit ball }.Let Hu = { x : x
T
u
¸0 }, where kuk=1.Find a small ellipsoid B’ that covers BÅH.
B
B’
u
H
u
Slide3Rank-1 UpdatesDef: Let z be a column vector and
® a scalar.A matrix of the form is called a rank-1 update matrix.
Claim 1:
Suppose
® -1/zTz. Then where ¯ = -®/(1+®z
Tz).Claim 2: If ®¸
-1/zT z then is PSD.
If ®>-1/zT
z then is PD.Claim 3:
Slide4Main Theorem:Let B = { x : k
xk·1 } and Hu = { x : x
T
u
¸0 }, where kuk=1.Let and .Let B’ = E( M, b ). Then: 1) B
Å Hu µ B’. 2)
Remark:
This notation only makes sense if M is positive definite. Claim 2 on rank-1 updates shows that it is, assuming n
¸
2.
Slide5Covering Half-ellipsoids by Ellipsoids
Let
E
be an ellipsoid centered at
z
Let
H
a
= { x :
a
T
x
¸
a
T
z
}
Find a small ellipsoid
E’
that covers
E
Å
H
a
E
E’
z
H
a
Slide6Use our solution for hemispheres!
Goal
Find an affine map
f
and choose u such that: f(B) = E and
f(Hu) =
HaDefine
E’ = f(B’).
Claim: E’ is an ellipsoid.Claim: E
Å Ha µ E’.
E
E’
z
B
B’
H
u
H
a
Slide7Choosing u
E
E’
z
B
B’
H
u
H
a
Assume E=E(
N,z
) and consider the map
f
(x) = N
1/2
x
+
z.
In Lecture 6 we showed that
E
=
f
(
B
).
Now choose
u
such that
f
(
H
u
) =
H
a
.
H
a
= { x :
a
T
(x-z)
¸
0 }
H
u
= { x :
u
T
x
¸
0 }
)
take u = N
1/2
a
Slide8Slide9Today's Top Docs
Related Slides