In this paper the author writes a simple characterization for the best copositive approximation to elements of C(Q) by elements of finite dimensional strict Chebyshev subspaces of C(Q) in the case when Q is any compac...In this paper the author writes a simple characterization for the best copositive approximation to elements of C(Q) by elements of finite dimensional strict Chebyshev subspaces of C(Q) in the case when Q is any compact subset of real numbers. At the end of the paper the author applies this result for different classes of Q.展开更多
In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces, and shows that it is u-ni...In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces, and shows that it is u-nique.展开更多
The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Ja...The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Jackson type estimate ｜｜f-r｜｜p≤Cδl^2μωφ（f,1/n）p holds, where μ is a natural number ≥3/2＋1/p, and Cδ is a positive constant depending only on δ.展开更多
This paper is part II of ＂On Copositive Approximation in Spaces of Contin- uous Functions＂. In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Che...This paper is part II of ＂On Copositive Approximation in Spaces of Contin- uous Functions＂. In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C （Q）, then for any admissible function f ∈ C（Q）／M, the best copositive approximation to f from M is unique.展开更多
The present note investigates the divergence phenomenon of copositive approximation by poiynomials in L'spaces. We show that the Jackson type estimates cannot hold, in general, for moduli of smoothness of higher d...The present note investigates the divergence phenomenon of copositive approximation by poiynomials in L'spaces. We show that the Jackson type estimates cannot hold, in general, for moduli of smoothness of higher degree.展开更多
A real symmetric tensor A=(ai_(1…im))∈R^([m,n]) is copositive(resp.,strictly copositive)if Ax^(m)≥0(resp.,Ax^(m)>0)for any nonzero nonnegative vector x∈ℝ^(n).By using the associated hypergraph of A,we give nece...A real symmetric tensor A=(ai_(1…im))∈R^([m,n]) is copositive(resp.,strictly copositive)if Ax^(m)≥0(resp.,Ax^(m)>0)for any nonzero nonnegative vector x∈ℝ^(n).By using the associated hypergraph of A,we give necessary and sufficient conditions for the copositivity of A.For a real symmetric tensor A satisfying the associated negative hypergraph H−(A)and associated positive hypergraph H+(A)are edge disjoint subhypergraphs of a supertree or cored hypergraph,we derive criteria for the copositivity of A.We also use copositive tensors to study the positivity of tensor systems.展开更多
In this paper, we establish a new type of alternation theory for more general restricted ranges Chebyshev approximation with equalities. The uniqueness and strong uniqueness theorems are given. Applying the results, w...In this paper, we establish a new type of alternation theory for more general restricted ranges Chebyshev approximation with equalities. The uniqueness and strong uniqueness theorems are given. Applying the results, we obtain the alternation theorem and uniqueness theorem for best coposilive approximation.展开更多
A nonlinear optimization problem (P) with inequality constraints can be converted into a new optimization problem (PE) with equality constraints only. This is a Valentine method for finite dimensional optimization. We...A nonlinear optimization problem (P) with inequality constraints can be converted into a new optimization problem (PE) with equality constraints only. This is a Valentine method for finite dimensional optimization. We review second order optimality conditions for (PE) in connection with those of (P). A strictly complementary slackness condition can be made to get the property that sufficient optimality conditions for (P) imply the same property for (PE). We give some new results (see Theorems 3.1, 3.2 and 3.3) .Without any assumption, a counterexample is given to show that these conditions are not equivalent.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
文摘In this paper the author writes a simple characterization for the best copositive approximation to elements of C(Q) by elements of finite dimensional strict Chebyshev subspaces of C(Q) in the case when Q is any compact subset of real numbers. At the end of the paper the author applies this result for different classes of Q.
文摘In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces, and shows that it is u-nique.
基金supported by the National Natural Science Foundation of China (10901044)Research Project of Hangzhou Normal University (YS05203154)
文摘The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Jackson type estimate ｜｜f-r｜｜p≤Cδl^2μωφ（f,1/n）p holds, where μ is a natural number ≥3/2＋1/p, and Cδ is a positive constant depending only on δ.
文摘This paper is part II of ＂On Copositive Approximation in Spaces of Contin- uous Functions＂. In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C （Q）, then for any admissible function f ∈ C（Q）／M, the best copositive approximation to f from M is unique.
文摘The present note investigates the divergence phenomenon of copositive approximation by poiynomials in L'spaces. We show that the Jackson type estimates cannot hold, in general, for moduli of smoothness of higher degree.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11801115,12071097,12042103)the Natural Science Foundation of Heilongjiang Province(No.QC2018002)and the Fundamental Research Funds for the Central Universities.
文摘A real symmetric tensor A=(ai_(1…im))∈R^([m,n]) is copositive(resp.,strictly copositive)if Ax^(m)≥0(resp.,Ax^(m)>0)for any nonzero nonnegative vector x∈ℝ^(n).By using the associated hypergraph of A,we give necessary and sufficient conditions for the copositivity of A.For a real symmetric tensor A satisfying the associated negative hypergraph H−(A)and associated positive hypergraph H+(A)are edge disjoint subhypergraphs of a supertree or cored hypergraph,we derive criteria for the copositivity of A.We also use copositive tensors to study the positivity of tensor systems.
文摘In this paper, we establish a new type of alternation theory for more general restricted ranges Chebyshev approximation with equalities. The uniqueness and strong uniqueness theorems are given. Applying the results, we obtain the alternation theorem and uniqueness theorem for best coposilive approximation.
文摘A nonlinear optimization problem (P) with inequality constraints can be converted into a new optimization problem (PE) with equality constraints only. This is a Valentine method for finite dimensional optimization. We review second order optimality conditions for (PE) in connection with those of (P). A strictly complementary slackness condition can be made to get the property that sufficient optimality conditions for (P) imply the same property for (PE). We give some new results (see Theorems 3.1, 3.2 and 3.3) .Without any assumption, a counterexample is given to show that these conditions are not equivalent.
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.
