cvx_class 2014. cvx tool setup search for cvx tool ( ) dezip to your assigned directory key...
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CVX_class 2014
Cvx tool setup• Search for CVX tool ( http://cvxr.com/cvx/ )• Dezip to your assigned directory• Key cvx_setup in the matlab command window
No errors! cvx has been successfully installed.
Cvx programming• Between cvx_begin & cvx_endcvx_beginvariables w(x,y) (complex, symmetric,…..)(refer 3.2)
minimize (convex function) or Maximize (concave function) (refer 3.3)
subject to …… constraints(refer to 3.4)cvx_end• Some special variables
– Cvx_optval– Cvx_status– Cvx_slvtol– Cvx_slvitr
Some cvx functions
• Quadprog• Linprog• Norm
– Norm(*,Inf)– Norm(*,1)
• Refer to 3.5 and appendix B
Others• Set (refer to 3.6 and appendix B.3)
• Dual variables (refer to 3.7)• Expression holders (refer to 3.8)• DCP ruleset (refer to 4)• Semidefinite programming using cvx (refer to 6)• Geometric programming using cvx(refer to 7)
Q2: Chebyshev Center• Consider a polyhedron composed of the halfspaces, , , , and , please plot the maximum norm ball inside the
polyhedron and show the center and the radius of it
1 22 8x x 1 22 8x x
1 2
15
5x x 1 2
15
5x x
-10 -5 0 5 10-10
-8
-6
-4
-2
0
2
4
6
8
10
-2x1+x
2 8
2x1+x
2 8
-1/5x1-x
2 5
1/5x1+x
2 5
-10 -5 0 5 10-10
-8
-6
-4
-2
0
2
4
6
8
10
xc=?
r=?
-2x1+x
2 8
2x1+x
2 8
-1/5x1-x
2 5
1/5x1+x
2 5
Q3:minimize the average sidelobe energy
*w Pw
* 1d w a
* d
P a a
2, , 2l u
min
s.t.
where ,
.
Q4: transmit beamforming (1/4)
Q4: transmit beamforming (2/4)
• Total power minimization • Epigraph method
2
21
2
02 2
2
0
min
. . , 1, ,
n
ii
Ti i
Ti j i
j i
Ti 0
i
s t i N
I
w
h w
h w
h w
min
2
21
2
02 2
2
0
min
. .
, 1, ,
Re 0, Im 0, 1, , T Ti i
n
ii
Ti i
Ti j i
T
i
i
j
i
i
i
0
t
s t t
i N
I
i N
w
h w
h w
h w
h w h w
Q4: transmit beamforming (3/4)
2
21
1
0
2
2
0
min
. .
00
0 1 , 1, ,
00
0 0
Re 0, Im 0, 1, ,
n
ii
Ti
Ti i
Ti
ni
Ti 0
i
T Ti i i i
t
s t t
i N
I
i N
w
hw
h w
hw
h w
h w h w
0norm I
, ' 'norm fro t
{ , } _ ( )0y I In complex lorentz m Ax b
Q4: transmit beamforming (4/4)
• Angle spectrum
Q5: Power allocation (a) (1/3)• Worst case design
2i=1
max
max min
. . 0 , 1
i
ii i
npij j i
j i
i
G p
G p
s t p p i n
2
i=1
max
min max
. . 0 , 1
i
ij j ij i
p Kii i
i
G p
G p
s t p p i n
Q5: Power allocation (a) (2/3)• Epigraph form
• Geometric Programming (GP) (refer to lecture 4, P4)
,
2
max
min
. . , 1
0 , 1
ip t
ij j ij i
ii i
i
t
G p
s t t i nG p
p p i n
,
1 1 1 2 1 1 1
1max
min
. . 1, 1
1, 1
ip t
ij ii j i i ii ij i
i
t
s t G G p p t G p t i n
p p i n
Q5: Power allocation (a) (3/3)• Variables of change
• Convex problem
31 2 4
13 1311 11 12 12 1 1
23 2321 11 22 22 2 2
31 31 32 32 33 33 3 3
1 2 3 , , ,
1: 1 ; 4 : 1
2 : 1 ; 5 : 1
3 : 1 ; 6 : 1
yy y y
a ba b a b c d
a ba b a b c d
a b a b a b c d
assume e p e p e p e t
c e e e c e
c e e e c e
c e e e c e
yy y y
yy y y
y y y y
[0,0,0,1]
1
min log
. . log 0, 1
log 0, 1
Tikik
Tii
na b
k
c d
e
s t e i n
e i n
y
y
y
y
Q5: Power allocation (b)• Minimize the total power, subject to all the users’ SINRs are not less than
• The problem can be represented as an Linear programming (LP) (refer to lecture 3, P22)
0
1
02
min
. . , 1
0, 1
i
n
ipi
ii i
ij j ij i
i
p
G ps t i n
G p
p i n
Q5: Power allocation (c)
• Take the worst user’s SINR in (a) in place of in (b), please re-design the transmit power ,
and compare with (a)
0