n = 40;
m = 15*n;
wa = 0.01*pi; wb = pi;
wl = logspace(log10(wa),log10(wb),m)';
D = 1./sqrt(wl);
Al = [ones(m,1) 2*cos(kron(wl,[1:n-1]))];
cvx_begin
variable r(n,1)
variable R(m,1)
minimize( max( max( [R./(D.^2) (D.^2).*inv_pos(R)]' ) ) )
subject to
R == Al*r;
cvx_end
disp(['Problem is ' cvx_status])
if ~strfind(cvx_status,'Solved')
return
end
h = spectral_fact(r);
figure(1)
H = exp(-j*kron(wl,[0:n-1]))*h;
loglog(wl,abs(H),wl,D,'r--')
set(gca,'XLim',[wa pi])
xlabel('freq w')
ylabel('mag H(w) and D(w)')
legend('optimized','desired')
Calling SDPT3: 4200 variables, 1841 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 1841
dim. of sdp var = 1200, num. of sdp blk = 600
dim. of linear var = 1800
dim. of free var = 600 *** convert ublk to lblk
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version predcorr gam expon scale_data
HKM 1 0.000 1 0
it pstep dstep pinfeas dinfeas gap prim-obj dual-obj cputime
-------------------------------------------------------------------
0|0.000|0.000|1.4e+04|1.9e+02|1.1e+07| 0.000000e+00 0.000000e+00| 0:0:00| spchol 1 1
1|0.908|0.977|1.3e+03|4.5e+00|3.5e+05|-6.779434e+02 -8.898649e+01| 0:0:00| spchol 1 1
2|0.933|0.963|8.7e+01|1.9e-01|2.1e+04|-3.493165e+01 -9.285894e+01| 0:0:00| spchol 1 1
3|0.986|0.997|1.2e+00|3.5e-03|3.8e+02|-1.025150e+00 -9.212412e+01| 0:0:00| spchol 2 2
4|0.966|0.471|4.1e-02|2.0e-03|6.4e+01|-3.813882e-01 -7.807639e+01| 0:0:01| spchol 2 2
5|0.490|0.167|2.1e-02|1.0e-02|6.4e+01|-3.414179e-01 -6.791926e+01| 0:0:01| spchol 2 2
6|0.657|0.889|7.4e-03|5.3e-03|1.2e+01|-5.162194e-01 -1.214283e+01| 0:0:01| spchol 2 2
7|1.000|0.914|9.2e-06|1.9e-03|1.5e+00|-8.557500e-01 -2.263165e+00| 0:0:01| spchol 2 2
8|1.000|0.714|5.3e-06|5.5e-04|5.6e-01|-9.800866e-01 -1.526531e+00| 0:0:01| spchol 2 2
9|1.000|0.172|3.1e-06|5.1e-04|4.7e-01|-1.022151e+00 -1.473349e+00| 0:0:01| spchol 2 2
10|1.000|0.481|1.1e-06|2.6e-04|2.9e-01|-1.063172e+00 -1.348411e+00| 0:0:01| spchol 2 2
11|1.000|0.327|2.7e-07|1.4e-04|1.9e-01|-1.111437e+00 -1.299973e+00| 0:0:01| spchol 2 2
12|1.000|0.900|2.5e-07|2.6e-05|6.4e-02|-1.140628e+00 -1.202063e+00| 0:0:02| spchol 2 2
13|0.768|0.913|8.3e-08|7.6e-06|3.4e-02|-1.160497e+00 -1.193737e+00| 0:0:02| spchol 2 2
14|1.000|0.945|8.7e-09|3.9e-06|1.1e-02|-1.178238e+00 -1.188675e+00| 0:0:02| spchol 2 2
15|0.951|0.840|5.5e-09|1.3e-06|2.1e-03|-1.185499e+00 -1.187516e+00| 0:0:02| spchol 2 2
16|0.822|0.854|2.2e-09|2.4e-07|5.2e-04|-1.186859e+00 -1.187373e+00| 0:0:02| spchol 2 2
17|0.929|0.894|5.5e-10|6.0e-08|5.9e-05|-1.187278e+00 -1.187336e+00| 0:0:02| spchol 2 2
18|0.979|0.969|4.5e-11|6.9e-09|2.8e-06|-1.187329e+00 -1.187331e+00| 0:0:02| spchol 3 3
19|0.993|0.988|5.2e-11|3.3e-10|4.9e-08|-1.187331e+00 -1.187331e+00| 0:0:02|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 19
primal objective value = -1.18733104e+00
dual objective value = -1.18733109e+00
gap := trace(XZ) = 4.87e-08
relative gap = 1.44e-08
actual relative gap = 1.43e-08
rel. primal infeas = 5.24e-11
rel. dual infeas = 3.31e-10
norm(X), norm(y), norm(Z) = 1.2e+01, 2.2e+02, 2.7e+02
norm(A), norm(b), norm(C) = 5.0e+02, 2.0e+00, 3.6e+01
Total CPU time (secs) = 2.45
CPU time per iteration = 0.13
termination code = 0
DIMACS: 5.2e-11 0.0e+00 5.9e-09 0.0e+00 1.4e-08 1.4e-08
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.18733
Problem is Solved