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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

Convex_guy
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  • Is this a full information mixed $H_2 - H_{\infty}$ problem or something like that? If so, check results by Carsten Scherer, he has a lot of work in this direction. Actually, check them regardless... – Johan Löfberg May 16 '22 at 10:53
  • I do not think this is full information mixed $H_2-H_\infty$ problem. Can you please let me know exactly what ref/references I must look for by Carsten Scherer? – Convex_guy May 16 '22 at 13:29
  • Nothing specific, but he worked on very similar LMIs and relaxations of XY=I and tight cases etc. – Johan Löfberg May 16 '22 at 17:12
  • Can you please suggest a reference where he has relaxations like $XY=I$. I am kind of struggling to find those works since he has a lot of papers on LMIs. I would be really grateful – Convex_guy May 16 '22 at 17:21

1 Answers1

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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
Johan Löfberg
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  • I feel $BB^\mathrm{T}$ is close to singular up to numerical accuracy (one of the eigenvalues of $BB^\mathrm{T}$ is 0.0014). Please can you confirm? However, in the question, I had mentioned that $BB^\mathrm{T}$ is nonsingular – Convex_guy May 16 '22 at 22:50
  • I think you are right in generating the right counterexample. Thanks a lot, Sir for your time for this problem – Convex_guy May 17 '22 at 02:07