% Section 8.4.1, Boyd & Vandenberghe "Convex Optimization"
% Original version by Lieven Vandenberghe
% Updated for CVX by Almir Mutapcic - Jan 2006
% (a figure is generated)
%
% We find the ellipsoid E of maximum volume that lies inside of
% a polyhedra C described by a set of linear inequalities.
%
% C = { x | a_i^T x <= b_i, i = 1,...,m } (polyhedra)
% E = { Bu + d | || u || <= 1 } (ellipsoid)
%
% This problem can be formulated as a log det maximization
% which can then be computed using the det_rootn function, ie,
%     maximize     log det B
%     subject to   || B a_i || + a_i^T d <= b,  for i = 1,...,m

% problem data
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)];

% generate A,b
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

% formulate and solve the problem
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

% make the plots
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 4.0: 34 variables, 15 equality constraints
   For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

 num. of constraints = 15
 dim. of sdp    var  =  6,   num. of sdp  blk  =  2
 dim. of socp   var  = 15,   num. of socp blk  =  5
 dim. of linear var  =  6
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
   HKM      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|1.8e+01|9.4e+00|1.3e+03| 1.441021e+01  0.000000e+00| 0:0:00| chol  1  1 
 1|1.000|0.628|8.3e-06|3.6e+00|5.4e+02| 5.502362e+01 -5.884367e+00| 0:0:00| chol  1  1 
 2|1.000|0.991|5.8e-06|4.2e-02|4.4e+01| 3.953225e+01 -8.633868e-02| 0:0:00| chol  1  1 
 3|0.915|1.000|7.6e-07|1.0e-03|3.9e+00| 3.947314e+00  4.254554e-02| 0:0:00| chol  1  1 
 4|0.691|1.000|2.8e-07|1.0e-04|1.9e+00| 2.071907e+00  1.972134e-01| 0:0:00| chol  1  1 
 5|0.972|1.000|1.1e-08|1.0e-05|5.5e-01| 1.147511e+00  5.991432e-01| 0:0:00| chol  1  1 
 6|1.000|0.939|1.6e-09|1.6e-06|7.2e-02| 9.841548e-01  9.117756e-01| 0:0:00| chol  1  1 
 7|0.951|0.954|5.3e-10|1.7e-07|3.6e-03| 9.536201e-01  9.500190e-01| 0:0:00| chol  1  1 
 8|0.978|0.975|2.5e-10|1.4e-08|8.8e-05| 9.523368e-01  9.522492e-01| 0:0:00| chol  1  1 
 9|0.955|0.977|4.9e-11|3.7e-10|2.7e-06| 9.523088e-01  9.523061e-01| 0:0:00| chol  1  1 
10|0.987|1.000|6.5e-13|9.8e-12|2.4e-07| 9.523076e-01  9.523074e-01| 0:0:00| chol  1  1 
11|0.993|1.000|5.2e-15|1.0e-12|1.3e-08| 9.523075e-01  9.523075e-01| 0:0:00|
  stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
 number of iterations   = 11
 primal objective value =  9.52307513e-01
 dual   objective value =  9.52307500e-01
 gap := trace(XZ)       = 1.33e-08
 relative gap           = 4.59e-09
 actual relative gap    = 4.59e-09
 rel. primal infeas (scaled problem)   = 5.20e-15
 rel. dual     "        "       "      = 1.00e-12
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 2.2e+00, 2.6e+00, 4.6e+00
 norm(A), norm(b), norm(C) = 7.6e+00, 2.0e+00, 3.7e+00
 Total CPU time (secs)  = 0.23  
 CPU time per iteration = 0.02  
 termination code       =  0
 DIMACS: 5.2e-15  0.0e+00  1.1e-12  0.0e+00  4.6e-09  4.6e-09
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.952308