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: 70 variables (0 free), 66 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
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) = 4114, nnz(L) = 2090
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.25E+000 0.000
1 : 6.09E-002 8.00E-001 0.000 0.3558 0.9000 0.9000 2.48 1 1 2.0E+000
2 : 6.58E-001 1.83E-001 0.000 0.2286 0.9000 0.9000 1.55 1 1 3.7E-001
3 : 7.50E-001 1.49E-003 0.000 0.0081 0.9990 0.9990 1.19 1 1 2.7E-003
4 : 7.50E-001 1.22E-009 0.000 0.0000 1.0000 1.0000 1.00 1 1 1.9E-009
iter seconds digits c*x b*y
4 0.0 9.5 7.4999999999e-001 7.4999999978e-001
|Ax-b| = 3.3e-009, [Ay-c]_+ = 3.3E-010, |x|= 2.8e+000, |y|= 4.1e+000
Detailed timing (sec)
Pre IPM Post
1.001E-002 4.006E-002 0.000E+000
Max-norms: ||b||=8.000000e-001, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.50211.
------------------------------------------------------------------------
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