Smoothing-type Newton Methods To Solve Tensor Related Problems

In the era of data information explosion,tensor,as a high-dimensional matrix,is widely used in signal and image processing,nuclear magnetic resonance imaging,wireless communication and other practical problems.In this paper,smoothing-type Newton methods are mainly used to solve tensor related problems,such as tensor complementarity problem and tensor absolute value equations.Tensor complementarity problem（TCP）is a kind of complementarity prob-lem proposed by Song and Qi in recent years.TCP,as a kind of nonlinear complemen-tarity problem defined by tensor,is a generalization of linear complementarity problem（LCP）and a special nonlinear complementarity problem（NCP）.Some theoretical prop-erties and algorithms of tensor complementarity problems have been widely studied and have made great progress.In the smoothing-type Newton method for TCP,when the tensor is symmetric S₀tensor and the problem satisfies q≤0,the solution of TCP can be obtained only by judging whether the accumulation point is feasible.Under this con-dition,the smoothing-type Newton method obtains global convergence and superlinear local convergence.The method combines with the structural characteristics of tensors and is different from the general nonlinear complementarity problem,which provides a new research idea for solving TCP.As a special kind of absolute value equations,tensor absolute value equations play an important role in scientific calculation and engineering application.In this paper,we study the smoothing-type Newton method for solving the tensor absolute value equations.Under the condition that it is weak and easily checkable,we prove that the smoothing-type Newton method is globally convergent.This paper gives numerical experiments to show the effectiveness of the algorithms.