Consider the following optimization problem: 


Maximize $\|X\|\_2$, subject to $X$ being Hermitian (or symmetric) and a bunch of semidefinite constraints on $X$. Here, $\|X\|\_2$ is the spectral norm of $X$, i.e., the largest eigenvalue of $X$ by magnitude (since $X$ is Hermitian).


Can this be written as a semidefinite program (SDP)?


Instead of maximizing $\|X\|\_2$, if we minimized $\|X\|\_2$, then this would be easy. We could add a new variable $t$ and minimize $t$ subject to $\|X\|\_2 \leq t$ and the semidefinite constraints on $X$. Finally, $\|X\|\_2 \leq t$ 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 when maximizing $\|X\|\_2$.



Can one maximize the spectral norm of a matrix via semidefinite programming?
No: maximizing the norm makes it a non-convex problem.


