Watson and Woodruff (2011) developed a heuristic for computing
variable-dependent values of the penalty parameter $\rho$ from the model itself.
We combine this heuristic with a gradient-based method, in order to
obtain a new method for calculating $\rho$ values.
We then introduce a method for iteratively computing variable-dependent $\rho$ values.
This method is based on a first-order condition, and can be implemented with criteria
that allow the parameter to be updated as the algorithm progresses.
These new approaches for selecting and updating $\rho$ can have an important impact on overall convergence,
and on the behavior of decision variables and dual weights.