Parallel primal-dual active-set algorithm with nonlinear and linear preconditioners

Abstract

The primal-dual active-set (PDAS) algorithm is a well-established and efficient method for addressing complementarity problems. However, the majority of existing approaches primarily concentrate on solving this non-smooth system with linear cases, and the straightforward extension of the primal-dual active-set method for solving nonlinear large-scale engineering problems does not work as well as expected, due to the unbalanced nonlinearities that bring about the difficulty of the slow convergence or stagnation. In the paper, we present the primal-dual active-set method with backtracking on the parallel computing framework for solving the nonlinear complementarity problem (NCP) arising from the discretization of partial differential equations. Some adaptive nonlinear preconditioning strategies based on nonlinear elimination are presented to handle the high nonlinearity of the nonsmooth system, and a family of linear preconditioners based on domain decomposition is developed to enhance the efficiency and scalability of this Newton-type method. Moreover, rigorous proof to establish both the monotone and superlinear convergence of the primal-dual active-set algorithm is also provided for the theoretical analysis. A series of numerical experiments for a family of multiphase reservoir problems, i.e., the CO₂ injection model, are carried out to demonstrate the robustness and efficiency of the proposed parallel algorithm.