  Many techniques for real-time trajectory optimization and control require the
solution of optimization problems at high frequencies. However,
ill-conditioning in the optimization problem can significantly reduce the speed
of first-order primal-dual optimization algorithms. We introduce a
preconditioning technique and step-size heuristic for Proportional-Integral
Projected Gradient (PIPG), a first-order primal-dual algorithm. The
preconditioning technique, based on the QR factorization, aims to reduce the
condition number of the KKT matrix associated with the optimization problem.
Our step-size selection heuristic chooses step-sizes to minimize the upper
bound on the convergence of the primal-dual gap for the optimization problem.
These algorithms are tested on two model predictive control problem examples
and show a solve-time reduction of at least 3.6x.
