Constructing Factor Mimicking Portfolios

I'm working with some factor data from a third party company. Their factor model is estimated on a broad universe. I'm trying to re-estimate the model on a smaller subset (my own universe) to construct factor mimicking portfolios. Essentially, I want to find portfolios from within my universe that track their factor most closely. Let's begin with a factor model: $r_{i,t} = X_{i, t-1}^{'} F_t + \eta_{i, t}$ with a $k \times 1$ vector of factor returns $F_t$ , or in matrix form: $\mathbf{r}_t = \m