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
In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a ste...In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.展开更多
Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series o...Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series of linear programming. It is proved that a sparse solution can be found under the assumption that the connected matrixes have range space property(RSP). Numerical experiments are also conducted to verify the efficiency of the proposed algorithm.展开更多
On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations...On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.展开更多
In this paper,we consider the tensor absolute value equations(TAVEs),which is a newly introduced problem in the context of multilinear systems.Although the system of the TAVEs is an interesting generalization of matri...In this paper,we consider the tensor absolute value equations(TAVEs),which is a newly introduced problem in the context of multilinear systems.Although the system of the TAVEs is an interesting generalization of matrix absolute value equations(AVEs),the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity(or multilinearity)of the problem under consideration.Therefore,we first study the solutions existence of some classes of the TAVEs with the help of degree theory,in addition to showing,by fixed point theory,that the system of the TAVEs has at least one solution under some checkable conditions.Then,we give a bound of solutions of the TAVEs for some special cases.To find a solution to the TAVEs,we employ the generalized Newton method and report some preliminary results.展开更多
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.
文摘In this paper, we give a smoothing neural network algorithm for absolute value equations (AVE). By using smoothing function, we reformulate the AVE as a differentiable unconstrained optimization and we establish a steep descent method to solve it. We prove the stability and the equilibrium state of the neural network to be a solution of the AVE. The numerical tests show the efficient of the proposed algorithm.
文摘Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series of linear programming. It is proved that a sparse solution can be found under the assumption that the connected matrixes have range space property(RSP). Numerical experiments are also conducted to verify the efficiency of the proposed algorithm.
基金This work was supported in part by the National Natural Science Foundation of China(Nos.11171252,11201332 and 11431002).
文摘On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.
基金supported by National Natural Science Foundation of China(Grant Nos.11571087 and 11771113)Natural Science Foundation of Zhejiang Province(Grant No.LY17A010028)supported by the Hong Kong Research Grant Council(Grant Nos.PolyU 15302114,15300715,15301716 and 15300717)。
文摘In this paper,we consider the tensor absolute value equations(TAVEs),which is a newly introduced problem in the context of multilinear systems.Although the system of the TAVEs is an interesting generalization of matrix absolute value equations(AVEs),the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity(or multilinearity)of the problem under consideration.Therefore,we first study the solutions existence of some classes of the TAVEs with the help of degree theory,in addition to showing,by fixed point theory,that the system of the TAVEs has at least one solution under some checkable conditions.Then,we give a bound of solutions of the TAVEs for some special cases.To find a solution to the TAVEs,we employ the generalized Newton method and report some preliminary results.
