Suppose I have an optimization model (P1) which its decision variable is a symmetric matrix $W$ (but not necessarily negative/positive semidefinite). In my case, this model can be converted to a vector-based model using half vectorizing $W$; let this new equivalent model be (P2). I want to only work with (P2) and add a constraint to it or add a term to its objective function that promotes the low-rankness of the original matrix $W$ in (P1) when I go back. In other words, I want to revise the equivalent half-vectorized based problem (P2) in a way that once I go back to the original matrix $W$, it is low-rank.
Another way to think of this question is can I use the components of a symmetric matrix and promote low-rankness using them? I think one way to do this is using the trace of $W$ (which is solely based on components of $W$ so it can be used in (P2)) but I am not sure if minimizing the trace of a symmetric matrix (not necessarily negative/positive semidefinite) promotes its low-rankness! Any hint is appreciated!