I have a variable matrix $W$ that is Hermitian and is used in two SDP problems.
- Problem 1 has constraints that depend on the real diagonal elements of $W$.
Example of the constraint is $W_{ii}+x_{ij}+y_{ij}=0$ Where $W_{ii}$ is a real diagonal element of the Hermitian $W$ matrix.
- Problem 2 has constraints that depend on the complex off-diagonal elements of $W$.
Examples of these constraints are $real(W_{ij})+x_{ij}+y_{ij}=0$ and $imag(W_{ij})+x_{ij}+y_{ij}=0$ Where $W_{ij}$ is a complex off-diagonal element of the Hermitian $W$ matrix.
Where $i,j \in l, \forall l \in NT$ where $NT$ is the number of branches (Branch $l$ connects nodes $i$ and $j$). Note that in both cases, $x$ and $y$ are decision variables that aren't elements of a matrix variable.
For a large scale problem with thousands of such constraints (e.g $NT$=5000), Which problem would be easier to solve and why? Would Cholesky decomposition be cheaper for the first problem due to the dependency on the real diagonal elements of $W$?