Given a matrix $M$, and it is perturbed by a matrix $E$.
Denote the new perturbed matrix by $M'=M+\epsilon E$. I wish to find out the optimal $\epsilon^*$ that minimizes the spectral radius of $M'$. Any suggestion to solve this problem (either numerical or analytic methods)?
Note that the task is to minimize $\rho(M')$, not the 2-norm $\|M'\|_2$. I tried to use Linear Matrix Inequality (LMI) to minimize $\|M'\|_2$, but corresponding $\epsilon$ does not minimize $\rho(M')$.