Combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods

In this work, the optimal adjustment algorithm for p coordinates, which arose from a generalization of the optimal pair adjustment algorithm is used to accelerate the convergence of interior point methods using a hybrid iterative approach for solving the linear systems of the interior point method. Its main advantages are simplicity and fast initial convergence. At each interior point iteration, the preconditioned conjugate gradient method is used in order to solve the normal equation system. The controlled Cholesky factorization is adopted as the preconditioner in the first outer iterations and the splitting preconditioner is adopted in the final outer iterations. The optimal adjustment algorithm is applied in the preconditioner transition in order to improve both speed and robustness. Numerical experiments on a set of linear programming problems showed that this approach reduces the total number of interior point iterations and running time for some classes of problems. Furthermore, some problems were solved only when the optimal adjustment algorithm for p coordinates was used in the change of preconditioners.