# Can you maximize the spectral norm of a matrix in a semidefinite program?

Consider the following optimization problem: Maximize $||X||$, subject to $X$ being Hermitian (or symmetric, if you prefer) and a bunch of semidefinite constraints on $X$. I want to know if this can be written as a semidefinite program (SDP). Here, $||X||$ is the spectral norm of $X$, which is just the largest eigenvalue of $X$ by magnitude since $X$ is Hermitian.

Instead of maximize $||X||$, if we had minimize $||X||$, then this would be easy. We could add a new variable $t$ and minimize $t$ subject to $||X|| \leq t$ and the semidefinite constraints on $X$. Finally, $||X|| \leq t$ can be written as the constraint $-tI \preceq X \preceq tI$, which makes this a valid SDP.

My question is whether this can be done for max $||X||$.

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The following slides might be of interest to you: leo.technion.ac.il/DelRob11/talks/Henrion.pdf –  Markus Schweighofer Oct 24 '12 at 11:10

For example, suppose linear constraints leave you a one-dimensional feasible region, e.g. $\pmatrix{1 & t\cr t & 1\cr}$ which is positive semidefinite for $-1 \le t \le 1$. The maximum norm is $2$, attained at $t=-1$ and $t=1$. No one-to-one change of variables will make a two-point set into a convex set. –  Robert Israel Oct 25 '12 at 6:48