Choosing Augmentation Parameters in OSQP-A New Approach based on Conjugate Directions

OSQP is a general purpose solver, based upon the alternating direction method of multipliers, for convex quadratic programs. Within this solver's algorithm, the idea of the augmented Lagrangian with a penalty parameter- a parameter which captures the relative weight-age on the objective function and the constraints of the problem in-hand- is utilized to develop an algorithm with so-called augmentation parameters. Although, the selection of these parameters is a crucial task, the optimal way to do the selection is not yet known. This work proposes a new method to select these parameters by utilizing the information of the conjugate directions of the coefficient matrix of a linear system of equations present in the algorithm. This selection makes it possible to cache these conjugate directions, instead of computing them at each iteration, resulting in a faster computation of the solution of the linear system, thus reducing the overall computation time. This reduction is demonstrated by a numerical example by comparing the total time it takes for the algorithms to converge sufficiently close to the optimal solution.