Counterexample for a convex problem

Question

The convex optimization problem is as follows: \begin{align} \underset{\mathbb{X},\mathbb{Y}\in\mathbb{S}_n^+}{\min}\quad &\operatorname{Tr}(X)+ \operatorname{Tr}\left(D Y \right)\nonumber\\ \text{s.t.}\;\; &AX+XA^T+BB^T\geq 0 \nonumber\\ &\begin{bmatrix} YA+A^\top Y -\gamma I & YB \\ B^\top Y & -I\end{bmatrix} \preceq 0\nonumber\\ &\begin{bmatrix} X&I\\I& Y \end{bmatrix}\geq 0\nonumber \end{align}

I feel at optimality $XY$ might not be equal to I. Any counterexamples

Answer 1

It holds on generic examples, but at the same time it is easy to find failures. Here shown via YALMIP in MATLAB

    A = [   0.1616    0.7101    0.3724
          -0.5743   -0.9687    1.0386
          -0.9781   -2.0428    0.1549];

    B = [ -0.8609    0.0831   -0.0633
           -0.3639   -0.6948    0.0915
           -0.6820   -1.4623    0.1106];
    a = 1;
    gamma = 1;
    D = a*B*B';
    n = 3;
    m = 3;
    X = sdpvar(n);
    Y = sdpvar(n);
    Model = [A*X+X*A' + B*B' >=0,
        [Y*A+A'*Y-gamma*eye(n) Y*B;B'*Y -eye(m)] <= 0,
        [X eye(n);eye(n) Y] >= 0];
    optimize(Model,trace(X) + trace(D*Y))
    value(X)
    inv(value(Y))
   
ans =

    1.1136    0.1004   -0.0039
    0.1004    0.7580    0.6263
   -0.0039    0.6263    1.1868


ans =

    1.1136    0.0991   -0.0042
    0.0991    0.5808    0.5944
   -0.0042    0.5944    1.1811