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 sedumi: 70 variables, 66 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 66, order n = 26, dim = 116, blocks = 2
nnz(A) = 120 + 0, nnz(ADA) = 4356, nnz(L) = 2211
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.46E-01 0.000
1 : 1.74E-01 2.73E-01 0.000 0.3229 0.9000 0.9000 1.92 1 1 1.2E+00
2 : 7.18E-01 2.38E-02 0.000 0.0870 0.9900 0.9900 1.18 1 1 8.8E-02
3 : 7.50E-01 7.94E-05 0.000 0.0033 0.9990 0.9990 1.08 1 1 2.8E-04
4 : 7.50E-01 1.18E-11 0.000 0.0000 1.0000 1.0000 1.00 1 1 4.1E-11
iter seconds digits c*x b*y
4 0.0 10.6 7.5000000003e-01 7.5000000001e-01
|Ax-b| = 2.8e-11, [Ay-c]_+ = 5.8E-12, |x|= 2.8e+00, |y|= 5.2e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 2.000E-02 0.000E+00
Max-norms: ||b||=2.000000e-01, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.78423.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.75
------------------------------------------------------------------------
The transition probability matrix of the optimal Markov chain is:
0.0000 0.3750 0.0000 0.3750 0.2500
0.3750 0.0000 0.3750 0.0000 0.2500
0.0000 0.3750 0.0000 0.3750 0.2500
0.3750 0.0000 0.3750 0.0000 0.2500
0.2500 0.2500 0.2500 0.2500 0.0000
The optimal mixing rate is:
0.7500