Global uniqueness and solvability of tensor complementarity problems for ℋ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docume

Inthis paper, we study the global uniqueness and solvability of tensor complementarity problems for $\mathcal {H}_{+}$-tensors. We obtain a sufficient condition of the global uniqueness and solvability of tensor complementarity problems for $\mathcal {H}_{+}$-tensors. We present nonlinear dynamical system models for solving the tensor complementarity problem (TCP). We prove that the presented dynamical system models are stable in the sense of Lyapunov stability theory for considering three classes of structured tensors. The computer simulation results further substantiate that the considered dynamical system can be used to solve the TCP.

[1]  Stefen Hui,et al.  Solving linear programming problems with neural networks: a comparative study , 1995, IEEE Trans. Neural Networks.

[2]  Ke Chen,et al.  Global exponential convergence and stability of gradient-based neural network for online matrix inversion , 2009, Appl. Math. Comput..

[3]  Shuai Li,et al.  Hybrid GNN-ZNN models for solving linear matrix equations , 2018, Neurocomputing.

[4]  Katta G. Murty,et al.  Linear complementarity, linear and nonlinear programming , 1988 .

[5]  Yimin Wei,et al.  Positive-Definite Tensors to Nonlinear Complementarity Problems , 2015, J. Optim. Theory Appl..

[6]  G. Lin.  Nonlinear Programming without Computation , 2022 .

[7]  Liao Li-Zhi,et al.  A neural network for the linear complementarity problem , 1999 .

[8]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[9]  Liping Zhang,et al.  A mixed integer programming approach to the tensor complementarity problem , 2018, Journal of Global Optimization.

[10]  Seakweng Vong,et al.  Tensor complementarity problems: the GUS-property and an algorithm , 2018 .

[11]  Liqun Qi,et al.  Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors , 2014, 1412.0113.

[12]  Edgar Sanchez-Sinencio,et al.  Nonlinear switched capacitor 'neural' networks for optimization problems , 1990 .

[13]  L. Qi,et al.  M-tensors and nonsingular M-tensors , 2013, 1307.7333.

[14]  Yunong Zhang,et al.  A set of nonlinear equations and inequalities arising in robotics and its online solution via a primal neural network , 2006, Neurocomputing.

[15]  Predrag S. Stanimirovic,et al.  Recurrent Neural Network for Computing the Drazin Inverse , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[16]  Predrag S. Stanimirovic,et al.  Complex ZFs for computing time-varying complex outer inverses , 2018, Neurocomputing.

[17]  Yong Wang,et al.  Exceptionally regular tensors and tensor complementarity problems , 2015, Optim. Methods Softw..

[18]  Yimin Wei,et al.  Bounds for eigenvalues of nonsingular H-tensor , 2015 .

[19]  Shuai Li,et al.  Accelerating a Recurrent Neural Network to Finite-Time Convergence for Solving Time-Varying Sylvester Equation by Using a Sign-Bi-power Activation Function , 2012, Neural Processing Letters.

[20]  Liqun Qi,et al.  P-tensors, P0-tensors, and their applications , 2018, Linear Algebra and its Applications.

[21]  Yimin Wei,et al.  Solving Multi-linear Systems with $$\mathcal {M}$$M-Tensors , 2016, J. Sci. Comput..

[22]  Feng Ding,et al.  Gradient-Based Identification Methods for Hammerstein Nonlinear ARMAX Models , 2006 .

[23]  Naihua Xiu,et al.  The sparsest solutions to Z-tensor complementarity problems , 2015, Optim. Lett..

[24]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

[25]  Yi-min Wei,et al.  ℋ-tensors and nonsingular ℋ-tensors , 2016 .

[26]  Francisco Facchinei,et al.  A semismooth equation approach to the solution of nonlinear complementarity problems , 1996, Math. Program..

[27]  Jorge J. Moré,et al.  Global Methods for Nonlinear Complementarity Problems , 1994, Math. Oper. Res..

[28]  Liqun Qi,et al.  Properties of Some Classes of Structured Tensors , 2014, J. Optim. Theory Appl..

[29]  Yimin Wei,et al.  Neural networks based approach solving multi-linear systems with M-tensors , 2019, Neurocomputing.

[30]  Yong Wang,et al.  Global Uniqueness and Solvability for Tensor Complementarity Problems , 2015, J. Optim. Theory Appl..

[31]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[32]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[33]  A. Berman,et al.  Some properties of strong H-tensors and general H-tensors , 2015 .

[34]  Yisheng Song,et al.  Properties of Solution Set of Tensor Complementarity Problem , 2015, J. Optim. Theory Appl..

[35]  Dong-Hui Li,et al.  An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem , 2017, J. Optim. Theory Appl..

[36]  Yunong Zhang,et al.  Convergence Properties Analysis of Gradient Neural Network for Solving Online Linear Equations , 2009 .

[37]  Zheng-Hai Huang,et al.  Formulating an n-person noncooperative game as a tensor complementarity problem , 2016, Comput. Optim. Appl..

[38]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[39]  Liqun Qi,et al.  Tensor absolute value equations , 2017, Science China Mathematics.

[40]  Liqun Qi,et al.  A Strongly Semismooth Integral Function and Its Application , 2003, Comput. Optim. Appl..

[41]  Li-Zhi Liao,et al.  Solving nonlinear complementarity problems with neural networks: a reformulation method approach , 2001 .

[42]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[43]  G. Isac.  Complementarity Problems , 1992 .

[44]  Lixing Han,et al.  A homotopy method for solving multilinear systems with M-tensors , 2017, Appl. Math. Lett..

[45]  Liqun Qi,et al.  P-Tensors, P$_0$-Tensors, and Tensor Complementarity Problem , 2015, 1507.06731.

[46]  Yimin Wei,et al.  Stochastic $$R_0$$R0 tensors to stochastic tensor complementarity problems , 2018, Optim. Lett..

[47]  Yimin Wei,et al.  Modified gradient dynamic approach to the tensor complementarity problem , 2018, Optim. Methods Softw..

[48]  Abdesselam Bouzerdoum,et al.  Neural network for quadratic optimization with bound constraints , 1993, IEEE Trans. Neural Networks.

[49]  Yimin Wei,et al.  Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse , 2017 .

[50]  Alexander Shapiro,et al.  Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets , 1999, SIAM J. Optim..

[51]  Yimin Wei,et al.  Existence and uniqueness of positive solution for H+-tensor equations , 2019, Appl. Math. Lett..

[52]  Lixing Han,et al.  A Continuation Method for Tensor Complementarity Problems , 2018, J. Optim. Theory Appl..

[53]  Predrag S. Stanimirovic,et al.  Gradient Neural Network with Nonlinear Activation for Computing Inner Inverses and the Drazin Inverse , 2017, Neural Processing Letters.

[54]  Predrag S. Stanimirovic,et al.  Nonlinearly Activated Recurrent Neural Network for Computing the Drazin Inverse , 2017, Neural Processing Letters.

[55]  Jerzy Zabczyk,et al.  Mathematical control theory - an introduction , 1992, Systems & Control: Foundations & Applications.

[56]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[57]  Liqun Qi,et al.  Tensor Complementarity Problem and Semi-positive Tensors , 2015, J. Optim. Theory Appl..

[58]  Yiju Wang,et al.  Nonsingular $H$-tensor and its criteria , 2016 .

[59]  S. Ramezani,et al.  Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory , 2013 .

[60]  A. Fischer.  A special newton-type optimization method , 1992 .

[61]  Dong-Hui Li,et al.  An equivalent tensor equation to the tensor complementarity problem with positive semi-definite Z-tensor , 2019, Optim. Lett..