# Maximum volume inscribed ellipsoid in a polyhedron

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    % 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 sedumi: 34 variables, 15 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 = 15, order n = 23, dim = 42, blocks = 8
    nnz(A) = 53 + 0, nnz(ADA) = 119, nnz(L) = 67
     it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
      0 :            4.08E+00 0.000
      1 :   1.21E+00 1.71E+00 0.000 0.4194 0.9000 0.9000   2.95  1  1  1.6E+00
      2 :   6.35E-01 5.08E-01 0.000 0.2970 0.9000 0.9000   2.13  1  1  3.8E-01
      3 :   8.84E-01 9.53E-02 0.000 0.1874 0.9000 0.9000   0.94  1  1  8.0E-02
      4 :   9.48E-01 5.96E-03 0.000 0.0625 0.9900 0.9900   0.95  1  1  5.1E-03
      5 :   9.52E-01 1.60E-04 0.000 0.0268 0.9900 0.9900   1.00  1  1  1.4E-04
      6 :   9.52E-01 8.02E-06 0.372 0.0502 0.9904 0.9900   1.00  1  1  5.3E-06
      7 :   9.52E-01 1.58E-06 0.000 0.1969 0.9120 0.9000   1.00  1  1  1.0E-06
      8 :   9.52E-01 3.36E-07 0.121 0.2127 0.9168 0.9000   1.00  1  1  2.3E-07
      9 :   9.52E-01 5.63E-08 0.000 0.1675 0.9096 0.9000   1.00  1  1  4.4E-08
     10 :   9.52E-01 8.02E-09 0.000 0.1425 0.9101 0.9000   1.00  1  1  7.2E-09

    iter seconds digits       c*x               b*y
     10      0.1   8.1  9.5230751775e-01  9.5230750982e-01
    |Ax-b| =   1.3e-10, [Ay-c]_+ =   4.2E-09, |x|=  2.2e+00, |y|=  2.6e+00

    Detailed timing (sec)
       Pre          IPM          Post
    1.000E-02    9.000E-02    0.000E+00
    Max-norms: ||b||=1, ||c|| = 2.474874e+00,
    Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.03991.
    ------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +0.952308