Section 4.6.3: Find the fastest mixing Markov chain on a graph
n = 5;
E = [0 1 0 1 1; ...
1 0 1 0 1; ...
0 1 0 1 1; ...
1 0 1 0 1; ...
1 1 1 1 0];
cvx_begin
variable P(n,n) symmetric
minimize(norm(P - (1/n)*ones(n)))
P*ones(n,1) == ones(n,1);
P >= 0;
P(E==0) == 0;
cvx_end
e = flipud(eig(P));
r = max(e(2), -e(n));
disp('------------------------------------------------------------------------');
disp('The transition probability matrix of the optimal Markov chain is: ');
disp(P);
disp('The optimal mixing rate is: ');
disp(r);
Calling SDPT3: 68 variables, 9 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 9
dim. of sdp var = 10, num. of sdp blk = 1
dim. of linear var = 8
dim. of free var = 5 *** 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 mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|2.5e+01|1.0e+01|3.0e+03|-6.622169e-10| 0:0:00| chol 1 1
1|0.921|0.963|2.0e+00|4.5e-01|6.4e+01| 2.781077e+00| 0:0:00| chol 1 1
2|1.000|0.957|7.0e-07|2.8e-02|5.2e+00|-3.520294e-01| 0:0:00| chol 1 1
3|1.000|0.717|7.3e-07|8.7e-03|1.1e+00|-5.103355e-01| 0:0:00| chol 1 1
4|0.965|0.369|1.0e-07|5.5e-03|4.6e-01|-7.432146e-01| 0:0:00| chol 1 1
5|0.990|0.944|3.9e-08|3.2e-04|1.9e-02|-7.491908e-01| 0:0:00| chol 1 1
6|0.988|0.988|1.7e-09|4.9e-06|2.3e-04|-7.499892e-01| 0:0:00| chol 1 1
7|0.989|0.989|2.8e-10|3.0e-06|7.8e-06|-7.499999e-01| 0:0:00| chol 1 1
8|1.000|0.989|2.2e-12|9.9e-08|2.4e-07|-7.500000e-01| 0:0:00| chol 1 2
9|1.000|0.989|7.7e-13|3.1e-09|7.4e-09|-7.500000e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 9
primal objective value = -7.49999998e-01
dual objective value = -7.50000000e-01
gap := trace(XZ) = 7.36e-09
relative gap = 2.95e-09
actual relative gap = 7.57e-10
rel. primal infeas = 7.73e-13
rel. dual infeas = 3.09e-09
norm(X), norm(y), norm(Z) = 1.1e+00, 1.2e+00, 2.8e+00
norm(A), norm(b), norm(C) = 1.0e+01, 2.0e+00, 4.5e+00
Total CPU time (secs) = 0.3
CPU time per iteration = 0.0
termination code = 0
DIMACS: 7.7e-13 0.0e+00 6.9e-09 0.0e+00 7.6e-10 2.9e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.75
------------------------------------------------------------------------
The transition probability matrix of the optimal Markov chain is:
0 0.3750 0 0.3750 0.2500
0.3750 0 0.3750 0 0.2500
0 0.3750 0 0.3750 0.2500
0.3750 0 0.3750 0 0.2500
0.2500 0.2500 0.2500 0.2500 0
The optimal mixing rate is:
0.7500