Maximum volume inscribed ellipsoid in a polyhedron with YALMIP

[Raffaele Romagnoli]

Hi,

I'm computing the maximum volume inscribed ellipsoid in a polyhedron, following the theory developed in Boyd & Vandenberghe "Convex Optimization"
. After computing the polyhedron, using cvx I have no problems, instead using the same structure of the polyhedron with YALMIP I obtain an ellipsoid out of the polyhedron that in my case is a triangle:

%-----------Polyhedron-----------------------------

x1=0:0.1:5;
y1=0.2*x1; y2=x1; y3=-1*x1+4;
figure
plot(x1,y1,x1,y2,x1,y3)
%---------------------------------------------------------

I am wondering if I am wrong in the description of the polyhedron using YALMIP. Following the code I use

%-------------------------- Code ------------------------------------------------------

A=[-1 1;1 1;0.2 -1]; b=[0;4;0];   % it's a triangle

P = sdpvar(n,n);
x = sdpvar(2,1);

F = [P>=0, A(1,:)*P*A(1,:)'+A(1,:)*x<=b(1),...
       A(2,:)*P*A(2,:)'+A(2,:)*x <=b(2),...
       A(3,:)*P*A(3,:)'+A(3,:)*x<=b(3)];

optimize(F,-logdet(P),sdpsettings('solver','sdpt3'));

% Plot of the ellipsoid see  http://web.cvxr.com/cvx/examples/cvxbook/Ch08_geometric_probs/html/max_vol_ellip_in_polyhedra.html

noangles = 500;
angles   = linspace( 0, 2 * pi, noangles );
ellipse_inner  = value(P) * [ cos(angles) ; sin(angles) ] + value(x) * ones( 1, noangles );

%------------------------------------------------------------------------------------------------------

Thanks for your attention

Raffaele

[Johan Löfberg]

You're not solving the same problem, so of course you get something weird. Correct code would be

F = [P>=0, 

    norm( P*A(1,:)', 2 ) + A(1,:)*x <= b(1),

    norm( P*A(2,:)', 2 ) + A(2,:)*x <= b(2),

    norm( P*A(3,:)', 2 ) + A(3,:)*x <= b(3)];

optimize(F,-logdet(P),sdpsettings('solver','sdpt3'));

z = sdpvar(2,1);

plot([A*z <= b]);hold on

plot((value(x) - z)'*value(P)^-2*(value(x)-z) <= 1)