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 sedumi: 68 variables, 9 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
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 = 9, order n = 21, dim = 115, blocks = 3
nnz(A) = 50 + 0, nnz(ADA) = 81, nnz(L) = 45
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.32E+01 0.000
1 : -4.78E-01 4.45E+00 0.000 0.3379 0.9000 0.9000 2.21 1 1 5.0E+00
2 : -6.85E-01 1.30E+00 0.000 0.2921 0.9000 0.9000 1.34 1 1 1.3E+00
3 : -7.51E-01 3.93E-02 0.000 0.0302 0.9900 0.9900 1.13 1 1 3.5E-02
4 : -7.50E-01 1.28E-06 0.000 0.0000 1.0000 1.0000 1.01 1 1 1.1E-06
5 : -7.50E-01 1.31E-13 0.000 0.0000 1.0000 1.0000 1.00 1 1 1.2E-13
iter seconds digits c*x b*y
5 0.0 13.5 -7.5000000000e-01 -7.5000000000e-01
|Ax-b| = 9.0e-14, [Ay-c]_+ = 2.8E-15, |x|= 1.9e+00, |y|= 8.3e-01
Detailed timing (sec)
Pre IPM Post
1.000E-02 2.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 4.000000e-01,
Cholesky |add|=0, |skip| = 1, ||L.L|| = 1.
------------------------------------------------------------
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