Timeline for A (linear) optimization problem subject to (linear) matrix inequality constraints
Current License: CC BY-SA 4.0
16 events
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| May 1, 2020 at 14:26 | comment | added | DSM | Any luck using the explicit solution for a Lyapunov Equation? For $AX+XA^\top=-Q$, $Q\succeq 0$, the unique solution is $X = \int_{0}^{\infty}e^{A^\top t}Qe^{At}dt$. If one chooses $Q=\mbox{diag}(-\mbox{real}(\lambda_i))$, $tr(AX)=-tr(Q)/2=-1/2$. But then I am only able to show $X\leq \beta^2 ||F||||F^{-1}|| I$, where $A=FJF^{-1}$ and $\beta$ is such that $||e^{Jt}||\leq \beta e^{\sigma t}$. | |
| Oct 1, 2018 at 2:00 | comment | added | Toni Mhax | ok then so if we take $A$ in this form can a diagonal $X$ do the constraint $AX+XA^T$ it got to have a $1/2$ term on its diagonal we are in dimension $2$. | |
| Sep 30, 2018 at 21:04 | comment | added | Ludwig | In the $2\times 2$ case I don't think it is possible to find such a matrix $A$. Indeed, in case $A+A^\top\not<0$, by virtue of the Schur-Horn Theorem, there always exists an orthogonal matrix $U$ such that $U^\top A U$ has one diagonal entry equal to $\mathrm{tr}(A)$ and the other entry equal to zero. | |
| Sep 30, 2018 at 20:47 | comment | added | Toni Mhax | $U^*=U^T$ real matrice | |
| Sep 30, 2018 at 20:40 | comment | added | Toni Mhax | Try to find $A+A^T$ a $2\times 2$ matrix having one positive eignevalue at least and such that $U^*AU$ has strictly negative diagonal for all unitary $U$, the counter example is equivalent to that. | |
| Sep 30, 2018 at 20:22 | comment | added | Ludwig | Yes, I think it is more complicated than that. However, if you manage to find an explicit counterexample, please let me know. | |
| Sep 30, 2018 at 20:20 | comment | added | Toni Mhax | Or it is more complicated | |
| Sep 30, 2018 at 20:09 | comment | added | Toni Mhax | Pick $A$ with strictly negative diagonal and $A+A^T$ having some positive eigenvalue, just that. | |
| Sep 30, 2018 at 18:12 | comment | added | Ludwig | Thanks for your comment. I see your point, however I couldn't find any numerical counterexample yet (I've run an extensive number of random numerical simulation for $n=2,3,\dots,10$). If you have some ideas about how to construct such a counterexample, please let me know. | |
| Sep 30, 2018 at 9:10 | history | edited | Toni Mhax | CC BY-SA 4.0 |
correct synthaxe
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| Sep 30, 2018 at 6:38 | history | edited | Toni Mhax | CC BY-SA 4.0 |
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| Sep 30, 2018 at 6:27 | history | edited | Toni Mhax | CC BY-SA 4.0 |
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| Sep 30, 2018 at 6:20 | history | edited | Toni Mhax | CC BY-SA 4.0 |
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| Sep 30, 2018 at 6:14 | history | edited | Toni Mhax | CC BY-SA 4.0 |
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| Sep 30, 2018 at 6:10 | review | First posts | |||
| Sep 30, 2018 at 7:02 | |||||
| Sep 30, 2018 at 6:08 | history | answered | Toni Mhax | CC BY-SA 4.0 |