Section 8.2.2: Separating polyhedra in 2D
randn('seed',0);
n = 2;
m = 2*n;
A1 = [1 1; 1 -1; -1 1; -1 -1];
A2 = [1 0; -1 0; 0 1; 0 -1];
b1 = 2*ones(m,1);
b2 = [5; -3; 4; -2];
fprintf(1,'Finding a separating hyperplane between the 2 polyhedra...');
cvx_begin
variables lam(m) muu(m) z(n)
maximize ( -b1'*lam - b2'*muu)
A1'*lam + z == 0;
A2'*muu - z == 0;
norm(z) <= 1;
-lam <=0;
-muu <=0;
cvx_end
fprintf(1,'Done! \n');
disp('------------------------------------------------------------------');
disp('The distance between the 2 polyhedra C and D is: ' );
disp(['dist(C,D) = ' num2str(cvx_optval)]);
t = linspace(-3,6,100);
p = -z(1)*t/z(2) + (muu'*b2 - lam'*b1)/(2*z(2));
figure;
fill([-2; 0; 2; 0],[0;2;0;-2],'b', [3;5;5;3],[2;2;4;4],'r')
axis([-3 6 -3 6])
axis square
hold on;
plot(t,p)
title('Separating 2 polyhedra by a hyperplane');
Finding a separating hyperplane between the 2 polyhedra...
Calling SDPT3: 11 variables, 5 equality constraints
------------------------------------------------------------
num. of constraints = 5
dim. of socp var = 3, num. of socp blk = 1
dim. of linear var = 8
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version predcorr gam expon scale_data
NT 1 0.000 1 0
it pstep dstep pinfeas dinfeas gap mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|3.7e-01|2.0e+00|2.7e+02| 3.394113e+01| 0:0:00| chol 1 1
1|0.660|1.000|1.2e-01|4.2e-02|6.0e+01| 2.006953e+01| 0:0:00| chol 1 1
2|0.936|1.000|8.0e-03|4.2e-03|3.8e+00|-2.107672e+00| 0:0:00| chol 1 1
3|0.917|0.954|6.6e-04|2.2e-03|4.7e-01|-2.010964e+00| 0:0:00| chol 1 1
4|1.000|1.000|6.4e-08|1.7e-04|6.7e-02|-2.111658e+00| 0:0:00| chol 1 1
5|0.982|0.982|2.1e-09|7.2e-06|1.2e-03|-2.121127e+00| 0:0:00| chol 1 1
6|0.985|0.988|8.5e-10|5.0e-07|1.7e-05|-2.121315e+00| 0:0:00| chol 1 1
7|0.954|0.984|3.4e-10|8.3e-09|6.1e-07|-2.121320e+00| 0:0:00| chol 2 2
8|1.000|1.000|5.4e-11|6.7e-11|6.6e-08|-2.121320e+00| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 8
primal objective value = -2.12132030e+00
dual objective value = -2.12132037e+00
gap := trace(XZ) = 6.56e-08
relative gap = 1.25e-08
actual relative gap = 1.24e-08
rel. primal infeas = 5.43e-11
rel. dual infeas = 6.72e-11
norm(X), norm(y), norm(Z) = 1.9e+00, 2.6e+00, 6.6e+00
norm(A), norm(b), norm(C) = 5.1e+00, 2.0e+00, 7.1e+00
Total CPU time (secs) = 0.1
CPU time per iteration = 0.0
termination code = 0
DIMACS: 5.4e-11 0.0e+00 1.2e-10 0.0e+00 1.2e-08 1.3e-08
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +2.12132
Done!
------------------------------------------------------------------
The distance between the 2 polyhedra C and D is:
dist(C,D) = 2.1213