  In optimization routines used for on-line Model Predictive Control (MPC),
linear systems of equations are usually solved in each iteration. This is true
both for Active Set (AS) methods as well as for Interior Point (IP) methods,
and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main
computational effort is spent while solving these linear systems of equations,
and hence, it is of greatest interest to solve them efficiently. Classically,
the optimization problem has been formulated in either of two different ways.
One of them leading to a sparse linear system of equations involving relatively
many variables to solve in each iteration and the other one leading to a dense
linear system of equations involving relatively few variables. In this work, it
is shown that it is possible not only to consider these two distinct choices of
formulations. Instead it is shown that it is possible to create an entire
family of formulations with different levels of sparsity and number of
variables, and that this extra degree of freedom can be exploited to get even
better performance with the software and hardware at hand. This result also
provides a better answer to an often discussed question in MPC; should the
sparse or dense formulation be used. In this work, it is shown that the answer
to this question is that often none of these classical choices is the best
choice, and that a better choice with a different level of sparsity actually
can be found.
