# 投影片 1 - ce.ncu.edu.tw

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_end cvx_begin variables 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, 2x x 8 , 2x x 8 , 1 1 x x 5 , and 5 x x 5 , please plot the maximum norm ball inside the pol 5

yhedron and show the center and the radius of it 1 1 2 1 2 1 2 2 10 10 -2x 1+x 2 8 8 -2x 1+x 2 8 8 2x 1+x 2 8 2x 1+x 2 8

6 -1/5x 1-x 2 5 6 -1/5x 1-x 2 5 4 1/5x 1+x 2 5 4 1/5x 1+x 2 5 2 2 0 0 -2 -2 -4

-4 -6 -6 -8 -8 x c =? r=? -10 -10 -5 0 5 10 -10 -10 -5 0

5 10 Q3:minimize the average sidelobe energy min w *Pw s.t. w*a d 1 where P a a* d , 2,l u , 2 . Q4: transmit beamforming (1/4) m in Q4: transmit beamforming (2/4) Total power minimization n s.t.

h w n s.t. i 1 w i 2 hTi w i T i min t 2 min i 1 w i 2 j 2 2 i j i

T 0 h w i 2 i Epigraph method 2 2 hTi w i 0 , i 1, , N h T i 2 t 2 w j 2

i 0 , i 1, , N j i I 0 2 hT0 wi I0 i Re hTi w i 0, Im hTi w i 0, i 1, , N Q4: transmit beamforming (3/4) min t n s.t. i 1 w i 2 2 t hTi

0 0 0 h T 0 2 w i I 0 norm , ' fro ' t 0 0 w1 hTi 0 w n 0

i 1 T hi w i , i 1, , N 0 2 norm I0 i Re hTi w i 0, Im hTi w i 0, i 1, , N {Ax b, y} I 0 In complex _ lorentz (m) Q4: transmit beamforming (4/4) Angle spectrum Q5: Power allocation (a) (1/3) Worst case design Gii pi max min 2 i=1n pi

G p ij j i j i s.t. 0 pi pmax , i 1 n 2 G p ij j i min max pi i=1K j i Gii pi s.t. 0 pi pmax , i 1 n Q5: Power allocation (a) (2/3) Epigraph form

min t pi ,t 2 G p ij j i s.t. j i Gii pi t , i 1 n 0 pi pmax , i 1 n Geometric Programming (GP) (refer to lecture 4, P4) min t pi ,t s.t. 1 1 1

2 1 1 1 G G p p t 1, i 1 n ij ii j i i Gii pi t j i 1 pmax pi 1, i 1 n Q5: Power allocation (a) (3/3) Variables of change assume e y1 p1 , e y2 p2 , e y3 p3 , e y4 t c1: e a11y b11 e a12 y b12 e a13y b13 1 ; c 4 : ec1y d1 1 c 2 : e a21y b11 e a22 y b22 e a23y b23 1 ; c5 : ec2 y d2 1 c3 : e a31y b31 e a32 y b32 e a33y b33 1 ; c6 : ec3y d3 1 Convex problem min log e[0,0,0,1]y y

n aTik y bik s.t. log e k 1 log e cTi y di 0, i 1 n 0, i 1 n Q5: Power allocation (b) Minimize the total power, subject to all the users SINRs are not less than 0 min pi s.t. n p i

i 1 Gii pi 0 , i 1 n 2 Gij p j i j i pi 0, i 1 n The problem can be represented as an Linear programming (LP) (refer to l ecture 3, P22) Q5: Power allocation (c) Take the worst users SINR in (a) in place of 0 in (b), please re-design the transmit power , a nd compare with (a)

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