Projected fixed-point method for vertical tensor complementarity problems

It is well known that the standard complementarity problem can be equivalently reformulated as a projected fixed-point equation, and this reformulation plays an important role in the theoretical and algorithmic research of complementarity problems. The vertical linear complementarity problem proposed by Cottle and Dantzig is an important generalization of the standard linear complementarity problem, and recently it has been further generalized to the case of tensors, called the vertical tensor complementarity problem (VTCP). In this paper, we give a projected fixed-point reformulation of the VTCP, and then, we design a fixed-point iteration method for solving the VTCP with a vertical block implicit Z-tensor. When there are non-positive diagonal entries in the representation subtensors of the tensor involved in the VTCP, we can reduce the computational cost of our method by solving a lower dimensional VTCP. Under the assumption that the problem under consideration is feasible, we prove that the designed method converges monotonically to a solution of the problem. The numerical results show the effectiveness of the proposed method.