Tensor Complementarity Problem Research Articles

The horizontal tensor complementarity problem

This article explores a new type of nonlinear complementarity problem, namely the horizontal tensor complementarity problem (HTCP), which is a natural extension of the horizontal linear complementarity problem studied in Sznajder [Degree-theoretic analysis of the vertical and horizontal linear complementarity problems [Ph.D. Thesis]. University of Maryland Baltimore County; 1994]. We extend the concepts of R 0 , R and P pairs from a pair of linear transformations given in Chi et al. [The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra. J Global Optim. 2019;73:153–169] to a pair of tensors. When a given pair of tensors has these properties, we use degree-theoretic tools to discuss the existence and boundedness of solutions to the HTCP. Finally, we study a uniqueness result of the solution of the HTCP.

An effective smoothing Newton projection algorithm for finding sparse solutions to NP-hard tensor complementarity problems

In this paper, we study the sparse solution of the tensor complementarity problem (TCP), which is an NP hard problem due to the nonconvexity and noncontinuity of the ℓ0 norm. We transform the complementarity constraints into a fixed point equation with projection type and use 0/1-loss function to relax ℓ0 norm. We develop a smoothing Newton projection (SNP) algorithm to solve the transformed problem. The subproblem with 0/1-loss function is solved by Newton method. Finally, numerical results show that the SNP algorithm can effectively get the sparse solution to a TCP.

Projected fixed-point method for vertical tensor complementarity problems

Projected fixed-point method for vertical tensor complementarity problems

A generalized alternating direction method of multipliers for tensor complementarity problems

A generalized alternating direction method of multipliers for tensor complementarity problems

The projected splitting iterative methods based on tensor splitting and its majorization matrix splitting for the tensor complementarity problem

The projected splitting iterative methods based on tensor splitting and its majorization matrix splitting for the tensor complementarity problem

Some Properties of the Solution of the Extended Vertical Tensor Complementarity Problem

Some Properties of the Solution of the Extended Vertical Tensor Complementarity Problem

Generalized Multilinear Games and Vertical Tensor Complementarity Problems

Generalized Multilinear Games and Vertical Tensor Complementarity Problems

Improved Fixed Point Iterative Methods for Tensor Complementarity Problem

Improved Fixed Point Iterative Methods for Tensor Complementarity Problem

Structure of the w-solution set of the tensor complementarity problem

Motivated by the study of column competent and column adequate matrices in the standard linear complementarity problems, we define column competent tensor, column adequate tensor and study the underlying properties of these tensors. We establish equivalent conditions for a column adequate tensor within the class of symmetric tensors and Z tensors. We then introduce the -solution set for the tensor complementarity problem and examine the uniqueness and finiteness of the -solution set. At last, we define column competent property and prove that under some additional condition, column competent property gives the finiteness of the -solution set of the corresponding tensor complementarity problem.

A linearized method for solving tensor complementarity problems with implicit Z-tensors

A linearized method for solving tensor complementarity problems with implicit Z-tensors

A projected splitting method for vertical tensor complementarity problems

A projected splitting method for vertical tensor complementarity problems

Finding the Least Element of a Nonnegative Solution Set of a Class of Polynomial Inequalities

In this paper, we consider the least element problem of a nonnegative solution set of a system of inequalities defined by a homogeneous polynomial mapping and a vector. In the set under consideration, the homogeneous polynomial mapping is defined by a tensor. When the tensor involved is square, the set under consideration is just the feasible region of the tensor complementarity problem (TCP). We first introduce the concept of the generalized -tensor, which is a natural generalization of the -tensor. Then, under the assumption that the considered set is nonempty and the tensor involved is a generalized -tensor, we propose an iterative method for finding the least element of the considered set. Specifically, by recognizing the position indices of positive components in the least element continuously, we solve a series of lower-dimensional system of tensor equations corresponding to these indices and prove that the least element of the set can be obtained within finite step iterations. When the tensor involved is square, the least element obtained is also a solution of the TCP. Compared with the existing methods for finding the least-element solution of the TCP with a -tensor, our method does not require any additional assumptions and has lower computational cost. Preliminary numerical experiments show that the proposed method is effective.

A smoothing projected HS method for solving stochastic tensor complementarity problem

A smoothing projected HS method for solving stochastic tensor complementarity problem

A smoothing Newton method preserving nonnegativity for solving tensor complementarity problems with <inline-formula><tex-math id="M1">$ P_0 $</tex-math></inline-formula> mappings

<p style='text-indent:20px;'>In this paper, we prove that the tensor complementarity problem with the <inline-formula><tex-math id="M2">\begin{document}$ P_0 $\end{document}</tex-math></inline-formula> mapping on the <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional nonnegative orthant is solvable and the solution set is nonempty and compact under mild assumptions. Since the involved homogeneous polynomial is a <inline-formula><tex-math id="M4">\begin{document}$ P_0 $\end{document}</tex-math></inline-formula> mapping on the <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional nonnegative orthant, the existing smoothing Newton methods are not directly used to solve this problem. So, we propose a smoothing Newton method preserving nonnegativity via a new one-dimensional line search rule for solving such problem. The global convergence is established and preliminary numerical results illustrate that the proposed algorithm is efficient and very promising.</p>

Some existence results for the generalized tensor absolute value equation

This paper introduces a new type of tensor absolute value equation, namely the generalized tensor absolute value equation (GTAVE), and studies its equivalence to the generalized tensor complementarity problem (GTCP).With the help of this equivalence and the degree theory approach, some existence results for the GTAVE are achieved.

Solving tensor complementarity problems with $ Z $-tensors via a weighted fixed point method

<p style='text-indent:20px;'>In this paper, we focus on solving the tensor complementarity problems with <inline-formula><tex-math id="M2">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-tensors. To this end, a weighted fixed point method is proposed for solving such tensor complementarity problems. Then, it is showed that the iterative sequence generated by the algorithm is monotonically decreasing with the help of <inline-formula><tex-math id="M3">\begin{document}$ Z $\end{document}</tex-math></inline-formula>-tensors. Moreover, the limit point of the iterative sequence is a solution of the corresponding tensor complementarity problem. Finally numerical experiments show the effectiveness of the algorithm and testify the theoretical conclusions.</p>

A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors

A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors

Neural network models for time-varying tensor complementarity problems

The existence and uniqueness of solutions and fast algorithms for tensor complementarity problems are hot topics in nowadays. We present a time-varying tensor complementarity problem (TVTCP) under tensor-tensor product (t-product). Theoretical analysis shows that the TVTCP is equivalent to a time-varying absolute value equation (TVAVE) under the mild conditions. Based on the absolute value equation, some neural networks for solving time-varying tensor inverse and TVTCP under the t-product are proposed and corresponding convergence are studied. Moreover, if the activation function (AF) of the neural networks is Mwsbp function, then we present the upper bound of the convergence time for the proposed neural networks. The numerical test results further illustrate that the proposed neural networks can solve time-varying tensor inverse and TVTCP effectively.

Solvability of monotone tensor complementarity problems

Solvability of monotone tensor complementarity problems

Copositivity for a Class of Fourth-Order Symmetric Tensors Given by Scalar Dark Matter

The mathematical model of general scalar potentials may be written as a fourth-order symmetric tensor with a particular structure in particle physics. In this paper, we mainly discuss the copositivity of a class of tensors defined by the scalar dark matter with the Higgs doublet and an inert doublet and a complex singlet. With the help of its structure, we obtain the necessary and sufficient conditions, which attains the analytic conditions required by the physical problems. At the same time, this work presents how to determine a unique solution of the tensor complementarity problem with a parameter.