A new smoothing spectral conjugate gradient method for solving tensor complementarity problems

<p style='text-indent:20px;'>In recent years, the tensor complementarity problem has attracted widespread attention and has been extensively studied. The research work of tensor complementarity problem mainly focused on theory, solution methods and applications. In this paper, we study the solution method of tensor complementarity problem. Based on the equivalence relation of the tensor complementarity problem and unconstrained optimization problem, we propose a new smoothing spectral conjugate gradient method with Armijo line search. Under mild conditions, we establish the global convergence of the proposed method. Finally, some numerical results are given to show the effectiveness of the proposed method and verify our theoretical results.</p>

Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

In this paper, some properties of structured rectangular tensors are presented, and the relationship among these structured rectangular tensors is also given. It is shown that all the V-singular values of rectangular P-tensors are positive. Some necessary and/or sufficient conditions for a rectangular tensor to be a rectangular P-tensor are also obtained. A new subclass of rectangular tensors, which is called rectangular S-tensors, is introduced and it is proved that rectangular S-tensors can be defined by the feasible vectors of the corresponding rectangular tensor complementarity problem.

Generalized Tensor Complementarity Problems Over a Polyhedral Cone

This paper addresses a class of generalized tensor complementarity problems (GTCPs) over a polyhedral cone. As a new generalization of the well-studied tensor complementarity problems (TCPs) in the literature, we first show the nonemptiness of the solution set of GTCPs when the involved tensor is cone ER. Then, we study bounds of solutions, and in addition to deriving a Hölderian local error bound of the problem under consideration. Finally, we reformulate GTCPs over a polyhedral cone as a system of nonlinear equations, which is helpful to employ the Levenberg–Marquardt algorithm for finding a solution of the problem. Some preliminary numerical results show that such an algorithm is efficient for GTCPs.