Let $v_1,\cdots,v_k \in \mathbb{C}^n$ with $||v_i||=1$ for all $1\leq i\leq k$ be a certain discretization of the n-sphere of $\mathbb{C}^n$.
In some optimization problem, I have to work with a complex semi-definite matrix variable $W$ of size $n$ for which I want to approximate linearly (by relying essentially on the trace operator) the maximum eigenvalue, noted $\lambda_{W,\text{max}}$ .
Since $\lambda_{W,\text{max}}=\max_{v\in\mathbb{C}^n\text{ s.t. }||v||=1} \ \ \ v^HWv$ (not linear in $W$), my idea (which may be not that new, although I have not find any article going in that direction so far) is to approximate this maximum eigenvalue of $W$ by
$\max_{1\leq i \leq k} \ \ \ v_i^HWv_i$
which I believe could be expressed linearly in semi-definite programming for instance.
My question is the following : since the $v_i$ are known, is there a way (either analytically or algorithmically) to find, before performing the optimization I am actually interested in, the complex SDP matrix $W$ that maximizes
$\dfrac{\lambda_{W,\text{max}}}{\max_{1\leq i \leq k} v_i^HWv_i}$
along with the value of that maximum ratio ?
Note 1 : in my particular case, it could be sufficient to determine that maximum ratio by considering only rank one $W$ matrices: I'm not sure it would simplify the analysis, but if it does, let's exploit that.
Note 2 : this ratio would be a way to know in advance by how much the n-sphere discretization can miss the actual value of the maximum eigenvalue (therefore the approximation used would be quantified which can be very valuable in my application)
For any matrix SDP matrix $W$ of rank 1 there exists a unitary matrix $U$ and a positive $\lambda$ such that $W = U^*diag(\lambda,0,...)U$, and conversely. Hence the ratio you want to maximise can be maximised for $U$ unitary and $\lambda$ positive. Now for any $U$ and $\lambda$ we have $$ v_i^*U^*diag(\lambda,0,...)Uv_i=\lambda |(Uv_i)_1|^2 $$ so your maximal ratio is $$ \max_{U^*U = Id, \lambda >0}\frac{\lambda}{\max_{1\leq i \leq k}\lambda |(Uv_i)_1|^2} = \left( \min_{U^*U=Id} \max_{1\leq i \leq k} |(Uv_i)_1|\right)^{-2}. $$ Also for any unity vector $u$, its image by the application $U \mapsto Uu$ is the unity sphere so you are left with $$ \left( \min_{u \in \mathbb{S}} \max_{1\leq i \leq k} |\langle u,v_i\rangle|\right)^{-2}. $$