Some extensions of the Kantorovich inequality and statistical applications

Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A⁻¹X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A⁻¹X| where X is n × k matrix such that X′X = I_k. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A⁻¹X, X′B²X(X′BCX)⁻¹ X′C²X(X′BCX)⁻¹ where X and Y are n × k matrices such that X′X = Y′Y = I_k and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.     

MATRIX NORM VERSIONS OF THE KANTOROVICH INEQUALITY AND ITS APPLICATIONS

In this paper we discuss the generalizations of the Kantorovich inequality and obtain some generalized Kantorovich inequalities in the sense of matrix norm. We further illustrate how to use these inequalities to determine the lower bound of relative efficiency of the parameter estimate in linear model.     

Kantorovich’s phenomenon

This is a short tribute to Leonid Kantorovich (1912–1986).     

A Cauchy-Khinchin matrix inequality

We derive a matrix inequality, which generalizes the Cauchy-Schwarz inequality for vectors, and Khinchin's inequality for zero–one matrices. Furthermore, we pose a related problem on the maximum irregularity of a directed graph with prescribed number of vertices and arcs, and make some remarks on this problem.     

The projection Kantorovich method for eigenvalue problems

A modified projection method for eigenvalues and eigenvectors of a compact operator T on a Banach space is defined and analyzed. The method is derived from the Kantorovich regularization for second-kind equations involving the operator T. It is shown that when T is a positive self-adjoint operator on a Hilbert space and the projections are orthogonal, the modified method always gives eigenvalue approximations which are at least as accurate as those obtained from the projection method. For self-adjoint operators, the required computation is essentially the same for both methods. Numerical computations for two integral operators are presented. One has T positive self-adjoint, while in the other T is not self-adjoint. In both cases the eigenvalue approximations from the modified method are more accurate than those from the projection method.     

Matrix versions of the Cauchy and Kantorovich inequalities

Summary A version of Cauchy's inequality is obtained which relates two matrices by an inequality in the sense of the Loewner ordering. In that ordering a symmetric idempotent matrix is dominated by the identity matrix and this fact yields a simple proof.A consequence of this matrix Cauchy inequality leads to a matrix version of the Kantorovich inequality, again in the sense of Loewner.     

Several Generalized Matrix Versions of Kantorovich Inequalities

Kantorovich inequalities are old results. In this paper we give several Kantorovich-type matrix inequalities.     

带约束的Kantorovich和Wielandt不等式的矩阵形式

本文利用矩阵的奇异值分解给出了带约束的Ｋａｎｔｏｒｏｖｉｃｈ不等式的矩阵形式，从而推广了王松桂和邵军１９９２年　　［１］　　的结果．并利用此结论得到了一般形式的带约束的Ｗｉｅｌａｎｄｔ不等式的矩阵形式．     

Recent developments of the operator Kantorovich inequality

First, we take a historical glimpse at some significant refinements and extensions of the Kantorovich inequality. Second, we present some operator Kantorovich inequalities involving unital positive linear mappings and the operator geometric mean in the framework of semi-inner product C^∗$C^{\ast }$-modules and give some new and classical results in a unified approach.     

Construction of a new family of Bernstein‐Kantorovich operators

In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first‐ and second‐order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein‐Kantorovich operators and their approximation behaviors.     

Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities

Summary A recent note by Marshall and Olkin (1990), in which the Cauchy-Schwarz and Kantorovich inequalities are considered in matrix versions expressed in terms of the Loewner partial ordering, is extended to cover positive semidefinite matrices in addition to positive definite ones.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Kantorovich算子在Ba空间中的逼近

利用Ditzian-Totik光滑模,研究了Kantorovich算子在Ba空间中的逼近.得到逼近的正定理与等价定理.所得结果改进,推广和统一了一些作者的结果.     

多元Kantorovich算子的最优逼近类

本文研究定义在单纯形上的多元Kantorovich算子逼近的正逆不等式与饱和定理,给出该算子在Lp(1≤p≤∞)空间的最优逼近类,即利用K-泛函的特征刻画分别满足‖Knf-f‖p=O(n-1) 与‖Knf-f‖p=o(n-1)的函数类．     

The Kantorovich Theorem and interior point methods

The Kantorovich Theorem is a fundamental tool in nonlinear analysis which has been extensively used in classical numerical analysis. In this paper we show that it can also be used in analyzing interior point methods. We obtain optimal bounds for Newtons method when relied upon in a path following algorithm for linear complementarity problems.Given a point z that approximates a point z() on the central path with complementarity gap , a parameter (0,1) is determined for which the point z satisfies the hypothesis of the affine invariant form of the Kantorovich Theorem, with regards to the equation defining z((1–)). The resulting iterative algorithm produces a point with complementarity gap less than in at most Newton steps, or simplified Newton steps, where ₀ is the complementarity gap of the starting point and n is the dimension of the problem. Thus we recover the best complexity results known to date and, in addition, we obtain the best bounds for Newtons method in this context.Mathematics Subject Classification (2000): 49M15, 65H10, 65K05, 90C33Work supported by the National Science Foundation under Grants No. 9996154, 0139701Acknowledgement The original version of this paper [38] was written when the author was visiting the Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB). The author would like to thank Peter Deuflhard, President of ZIB, for the excellent working conditions at ZIB, and for very interesting discussions on various mathematical problems related to the subject of the present paper. The author would also like to thank two anonymous referees whose comments and suggestions led to a better presentation of our results.     

Asymptotic expansion of multivariate Kantorovich type operators

In this paper we study the asymptotic expansion of sequences of multivariate Kantorovich type operators and their partial derivatives. In particular, we obtain the complete expansion for the Kantorovich Bernstein operators on the simplex and for two Kantorovich type modifications of the Bleimann, Butzer and Hahn operators that we introduce in the paper. AMS subject classification 41A36     

A Cauchy-Schwarz type inequality for fuzzy integrals

In this paper we prove a Cauchy-Schwarz type inequality for fuzzy integrals.