Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the diagonal matrices have in their diagonal exactly one $1$, one $-1$ and rest $0$s and $(2)$ all the matrices are of the same even dimension say $2n$ and (3) $k=n$)
Now I start with $B$ and among $D_{11}$ and $D_{12}$, say I add to $B$ the one which causes the minimum upshift in the spectral radius of $B$. Thus I get the matrix $B_1$ (which is either $B + D_{11}$ or $B+D_{12}$). Similarly I add to $B_1$ either $D_{21}$ or $D_{22}$ depending on the same criteria. And I continue so on.
Say that the final objective is to find the matrix $B+D_{1*}+D_{2*}+..+D_{k*}$ such that it has the minimum spectral radius among all the $2^k$ options that are possible corresponding to choosing $* = 1$ or $*=2$ for each of the $k$ $D$s.
- Will this global optimization be achieved by doing the afore mentioned sequence of local optimizations?
If it helps (unlikely to!) one can replace the above $2k$ diagonal matrices with $2k$ other off-diagonal symmetric matrices of dimension $2n$ say $M_{1+}, M_{1-},..,M_{k+}, M_{k-}$ such that $M_{i\pm}$ has all zeros everywhere except $2$ symmetric entries (at the same location) which are both $1$ or both $-1$ and if $a \neq b$ then $M_{a\pm}$ and $M_{b\pm}$ have their $4$ non-zero entries in different array locations for all $a$ and $b$.