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I am interested in the following optimization problem: $$ \min_{\lambda > 0} \| \lambda A - I \|_{2}, $$ where $A$ is an arbitrary matrix and $\| \cdot \|_2$ is the spectral norm.

As suggested by Spectral norm minimization via semidefinite programming, we could write the above problem as the following semidefinite program (SPD): \begin{align} &\min_{s,\lambda} s \\ &s.t.\begin{pmatrix} s I & \lambda A - I \\ \lambda A^{\top} - I & s I \end{pmatrix} \succcurlyeq O. \end{align} I wonder what next I could do in order to find an analytic solution to the above problem.

What's more, any other method that can solve the original matrix norm minimization problem is appreciated.

Zhang Xinyu
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  • It seems likely that your problem does not have a nice "analytic solution". Do you have any reason to believe that a closed form for the solution exists? – Ben Grossmann May 14 '21 at 15:11
  • I just think this problem is a univariate one, and thus may have an analytic solution of some form. Do you have any idea on solving the aforementioned SPD problem? Non-analytic solver is also appreciated. – Zhang Xinyu May 15 '21 at 06:46

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