A Fixed Point Iterative Method for Tensor Complementarity Problems
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A Fixed Point Iterative Method for Tensor Complementarity Problems Ping-Fan Dai1 Received: 24 October 2019 / Revised: 19 May 2020 / Accepted: 12 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Tensor complementarity problem (TCP) has attracted many attentions in recent years. In this paper, we equivalently reformulate tensor complementarity problem as a fixed point equation. Based on the fixed point equation, projected fixed point iterative methods are proposed and corresponding convergence proofs on the fixed point iterative methods for the tensor complementarity problem associated with a power Lipschitz tensor are investigated. Furthermore, the monotone convergence analysis of the fixed point iteration method for the tensor complementarity problem involving an L tensor is given. Numerical examples are tested to illustrate the given approach. Keywords Tensor complementarity problem · Fixed point method · Power Lipschitz tensor · L tensor Mathematics Subject Classification 90C33 · 90C30 · 65H10
1 Introduction Let R[m,n] and Rn be the set of all real m-order n-dimensional tensor and the set of all real n-dimensional vector, respectively. Given a tensor A ∈ R[m,n] and a vector f ∈ Rn , the tensor complementarity problem, abbreviated by TCP(A, f ), is to find a vector x ∈ Rn such that x ≥ 0, Ax m−1 − f ≥ 0, x T (Ax m−1 − f ) = 0, (1.1) in which Ax m−1 is a vector in Rn with its ith component as (Ax m−1 )i =
n
aii2 ...im xi2 . . . xim , ∀ i ∈ {1, 2, . . . , n}.
i 2 ,...,i m =1
This work was supported by the National Natural Science Foundation of China (11671318) and the Natural Science Foundations of Fujian Province of China (2016J01028, 2017N0029).
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Ping-Fan Dai [email protected] School of Information Engineering, Sanming University, Sanming 365004, Fujian, People’s Republic of China 0123456789().: V,-vol
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The tensor complementarity problem has growing applications, for instance, n-person noncooperative games, the hypergraph clustering problem and traffic equilibrium problems, for detail, see [1,2]. The tensor complementarity problem was firstly introduced by Song and Qi [3] and is a generalization of the linear complementarity problem. In the last few years, there were a lot of progress on the theoretical results of TCP, such as, the existence and global uniqueness of solutions [3–8], stability of solutions [9], error bound [10,11], and so on. A comprehensive summary for the theory and application of TCP can be found in [2,12]. It is well known that structured matrices play key roles in theory and methods of the linear complementary problem. Naturally, various structure tensors will also play important roles in the research of the TCP(A, f ). There are a number of structured tensors that are introduced, such as, strong strictly semi-positive tensor [13], exceptionally regular tensors [14], M-tensor [15], H-tensor [16], strong P -tensor [8], B-tensor [17], S -tensor [18], semi-positive tensor [
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