A General Preconditioner for Tensor Complementarity Problems

Preconditioning techniques have been focused on multi-linear systems or tensor equations. However, to our knowledge, there is no relevant research on tensor complementarity problems. In this paper, we present a general preconditioner P for solving the tensor complementarity problem (TCP) involved with an L -tensor A and a vector q, and we prove that the TCP (A, q) is equivalent to the TCP (PA, Pq) under the assumption that PA is an L -tensor. Based on this equivalence, a preconditioned fixed point iteration method is proposed for solving the tensor complementarity problem and its convergence analysis is given. For actual computations, we provide a concrete choice for the preconditioner P associated with a parameter satisfying above-mentioned hypothesis. In addition, it is proved theoretically that the convergence rate of the fixed point method with the chosen preconditioner P is at least as fast as that of the corresponding method without preprocessing. Meanwhile, we also obtain the monotony of the parameter on the performance of the preconditioned iterative method. Lastly, numerical examples are used to demonstrate the theoretical results.