Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the sys...Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.展开更多
Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors t...Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.展开更多
The tensor complementarity problem is a special instance in the class of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems and traffic...The tensor complementarity problem is a special instance in the class of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems and traffic equilibrium problems. Two most important research issues are how to identify the solvability and how to solve such a problem via analyzing the structure of the involved tensor. In this paper, based on the concept of monotone mappings, we introduce a new class of structured tensors and the corresponding monotone tensor complementarity problem. We show that the solution set of the monotone tensor complementarity problem is nonempty and compact under the feasibility assumption. Moreover, a necessary and sufficient condition for ensuring the feasibility is given via analyzing the structure of the involved tensor. Based on the Huber function,we propose a regularized smoothing Newton method to solve the monotone tensor complementarity problem and establish its global convergence. Under some mild assumptions, we show that the proposed algorithm is superlinearly convergent. Preliminary numerical results indicate that the proposed algorithm is very promising.展开更多
This paper considers semidefinite relaxation for linear and nonlinear complementarity problems.For some particular copositive matrices and tensors,the existence of a solution for the corresponding complementarity prob...This paper considers semidefinite relaxation for linear and nonlinear complementarity problems.For some particular copositive matrices and tensors,the existence of a solution for the corresponding complementarity problems is studied.Under a general assumption,we show that if the solution set of a complementarity problem is nonempty,then we can get a solution by the semidefinite relaxation method;while if it does not have a solution,we can obtain a certificate for the infeasibility.Some numerical examples are given.展开更多
In this paper,we consider the second-order cone tensor eigenvalue complementarity problem(SOCTEiCP)and present three different reformulations to the model under consideration.Specifically,for the general SOCTEiCP,we ...In this paper,we consider the second-order cone tensor eigenvalue complementarity problem(SOCTEiCP)and present three different reformulations to the model under consideration.Specifically,for the general SOCTEiCP,we first show its equivalence to a particular variational inequality under reasonable conditions.A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem.Then,for the symmetric and sub-symmetric SOCTEiCPs,we reformulate them as appropriate nonlinear programming problems,which are extremely beneficial for designing reliable solvers to find solutions of the considered problem.Finally,we report some preliminary numerical results to verify our theoretical results.展开更多
This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems ...This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.展开更多
文摘Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11601261, 11571095, 11601134）, the Hong Kong Research Grant Council （Grant No .PolyU 502111, 501212, 501913, 15302114）, the Natural Science Foundation of Shandong Province （No. ZR2016AQ12）, and the China Postdoctoral Science Foundation （Grant No. 2017M622163）.
文摘Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.
基金supported by National Natural Science Foundation of China(Grant No.12171271)。
文摘The tensor complementarity problem is a special instance in the class of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems and traffic equilibrium problems. Two most important research issues are how to identify the solvability and how to solve such a problem via analyzing the structure of the involved tensor. In this paper, based on the concept of monotone mappings, we introduce a new class of structured tensors and the corresponding monotone tensor complementarity problem. We show that the solution set of the monotone tensor complementarity problem is nonempty and compact under the feasibility assumption. Moreover, a necessary and sufficient condition for ensuring the feasibility is given via analyzing the structure of the involved tensor. Based on the Huber function,we propose a regularized smoothing Newton method to solve the monotone tensor complementarity problem and establish its global convergence. Under some mild assumptions, we show that the proposed algorithm is superlinearly convergent. Preliminary numerical results indicate that the proposed algorithm is very promising.
基金supported by the National Natural Science Foundation of China(Nos.12171105,11271206)the Fundamental Research Funds for the Central Universities(No.FRF-DF-19-004).
文摘This paper considers semidefinite relaxation for linear and nonlinear complementarity problems.For some particular copositive matrices and tensors,the existence of a solution for the corresponding complementarity problems is studied.Under a general assumption,we show that if the solution set of a complementarity problem is nonempty,then we can get a solution by the semidefinite relaxation method;while if it does not have a solution,we can obtain a certificate for the infeasibility.Some numerical examples are given.
基金the National Natural Science Foundation of China(Nos.11171083,11301123,and 11571087)the Natural Science Foundation of Zhejiang Province(Nos.LZ14A010003 and LY17A010028).
文摘In this paper,we consider the second-order cone tensor eigenvalue complementarity problem(SOCTEiCP)and present three different reformulations to the model under consideration.Specifically,for the general SOCTEiCP,we first show its equivalence to a particular variational inequality under reasonable conditions.A notable benefit is that such a reformulation possibly provides an efficient way for the study of properties of the problem.Then,for the symmetric and sub-symmetric SOCTEiCPs,we reformulate them as appropriate nonlinear programming problems,which are extremely beneficial for designing reliable solvers to find solutions of the considered problem.Finally,we report some preliminary numerical results to verify our theoretical results.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671220, 11401331, 11771244 and 11271221)the Nature Science Foundation of Shandong Province (Grant Nos. ZR2015AQ013 and ZR2016AM29)the Hong Kong Research Grant Council (Grant Nos. PolyU 501913,15302114, 15300715 and 15301716)
文摘This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.
