Maximum volume inscribed ellipsoid in a polyhedron
n = 2;
px = [0 .5 2 3 1];
py = [0 1 1.5 .5 -.5];
m = size(px,2);
pxint = sum(px)/m; pyint = sum(py)/m;
px = [px px(1)];
py = [py py(1)];
A = zeros(m,n); b = zeros(m,1);
for i=1:m
A(i,:) = null([px(i+1)-px(i) py(i+1)-py(i)])';
b(i) = A(i,:)*.5*[px(i+1)+px(i); py(i+1)+py(i)];
if A(i,:)*[pxint; pyint]-b(i)>0
A(i,:) = -A(i,:);
b(i) = -b(i);
end
end
cvx_begin
variable B(n,n) symmetric
variable d(n)
maximize( det_rootn( B ) )
subject to
for i = 1:m
norm( B*A(i,:)', 2 ) + A(i,:)*d <= b(i);
end
cvx_end
noangles = 200;
angles = linspace( 0, 2 * pi, noangles );
ellipse_inner = B * [ cos(angles) ; sin(angles) ] + d * ones( 1, noangles );
ellipse_outer = 2*B * [ cos(angles) ; sin(angles) ] + d * ones( 1, noangles );
clf
plot(px,py)
hold on
plot( ellipse_inner(1,:), ellipse_inner(2,:), 'r--' );
plot( ellipse_outer(1,:), ellipse_outer(2,:), 'r--' );
axis square
axis off
hold off
Calling SDPT3: 28 variables, 9 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 9
dim. of sdp var = 6, num. of sdp blk = 2
dim. of socp var = 15, num. of socp blk = 5
*******************************************************************
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|9.1e+00|2.5e+00|1.6e+02| 7.205107e+00| 0:0:00| chol 1 1
1|1.000|0.871|2.8e-06|4.0e-01|2.4e+01| 5.831517e+00| 0:0:00| chol 1 1
2|1.000|1.000|4.6e-07|8.8e-03|2.7e+00| 1.671491e+00| 0:0:00| chol 1 1
3|0.830|1.000|8.5e-08|8.9e-04|4.8e-01| 8.822259e-01| 0:0:00| chol 1 1
4|1.000|0.901|2.8e-07|1.7e-04|5.4e-02| 9.487353e-01| 0:0:00| chol 1 1
5|0.967|0.974|1.0e-08|1.3e-05|1.6e-03| 9.522458e-01| 0:0:00| chol 1 1
6|0.969|0.976|6.5e-10|1.2e-06|4.4e-05| 9.523109e-01| 0:0:00| chol 1 1
7|0.941|0.978|8.0e-11|2.6e-08|2.1e-06| 9.523080e-01| 0:0:00| chol 1 1
8|1.000|1.000|5.4e-13|1.6e-11|2.3e-07| 9.523075e-01| 0:0:00| chol 1 1
9|0.994|1.000|7.5e-14|1.0e-12|6.7e-09| 9.523075e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 9
primal objective value = 9.52307512e-01
dual objective value = 9.52307506e-01
gap := trace(XZ) = 6.74e-09
relative gap = 2.32e-09
actual relative gap = 2.32e-09
rel. primal infeas = 7.55e-14
rel. dual infeas = 1.00e-12
norm(X), norm(y), norm(Z) = 1.9e+00, 2.6e+00, 4.6e+00
norm(A), norm(b), norm(C) = 6.6e+00, 2.0e+00, 3.7e+00
Total CPU time (secs) = 0.2
CPU time per iteration = 0.0
termination code = 0
DIMACS: 7.5e-14 0.0e+00 1.1e-12 0.0e+00 2.3e-09 2.3e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.952308