A direct proof and a generalization for a Kantorovich type inequality

Huang, Jianguo; Zhou, Jieyong

doi:10.1016/j.laa.2004.10.009

Cited by 5 publications

(3 citation statements)

References 6 publications

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“…To prove that the lower bound on the α-weight angle exists, we make use of the result derived by Bauer and Householder [3] who extended the Kantorovich inequality described in Theorem 1. The proof for this is investigated by Huang and Zhou [25].…”

Section: Upper Bound On the α-Weight Anglementioning

confidence: 99%

Geometric insights into robust portfolio construction with gearing

Dalmeyer¹,

Gebbie²

2021

Preprint

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We investigate and extend the results of Golts and Jones [18] that an alpha-weight angle resulting from unconstrained quadratic portfolio optimisations has an upper bound dependent on the condition number of the covariance matrix. This implies that better conditioned covariance matrices produce weights from unconstrained mean-variance optimisations that are better aligned with each assets expected return. We provide further clarity on the mathematical insights that relate the inequality between the α-weight angle and the condition number and extend the result to include portfolio optimisations with gearing constraints. We provide an extended family of robust optimisations that include the gearing constraints, and discuss their interpretation.

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Section: Upper Bound On the α-Weight Anglementioning

confidence: 99%

Geometric insights into robust portfolio construction with gearing

Dalmeyer¹,

Gebbie²

2021

Preprint

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“…A particular instance of the problem is when B = A −1 . Then the Legendre-Fenchel transform of f (x) = q A (x) q A −1 (x) would allow us to recover inequalities like that of Kantorovich [Huang05]:…”

Section: Problem 11 the Legendre-fenchel Transform Of The Product Ofmentioning

confidence: 99%

Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization

Hiriart-Urruty¹

2007

SIAM Rev.

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We present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization. These problems can be classified into three groups: problems of pure mathematical interest, problems motivated by scientific computing and applications, and problems whose solutions are known but for which we would like to know better proofs. For each problem we provide a succinct presentation, a list of appropriate references, and a view of the state of the art of the subject.

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“…This inequality and its variants have many applications in matrix analysis, statistics, numerical algebra, and optimization (see e.g. [7,11,15,16,22,23,25,26,30,32,33,34]). In this paper, K(x) is referred to as the 'Kantorovich function'.…”

Section: Introductionmentioning

confidence: 99%

Convexity conditions of Kantorovich function and related semi-infinite linear matrix inequalities

Zhao¹

2011

Journal of Computational and Applied Mathematics

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The Kantorovich function (x T Ax)(x T A −1 x), where A is a positive definite matrix, is not convex in general. From matrix/convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we investigate the convexity of this function by the condition number of its matrix. In 2-dimensional space, we prove that the Kantorovich function is convex if and only if the condition number of its matrix is bounded above by 3 + 2 √ 2, and thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound '3+2 √ 2' is turned out to be a necessary condition for the convexity of Kantorovich functions in any finite-dimensional spaces. We also point out that when the condition number of the matrix (which can be any dimensional) is less than or equal to 5 + 2 √ 6, the Kantorovich function is convex. Furthermore, we prove that this general sufficient convexity condition can be remarkably improved in 3-dimensional space. Our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semi-infinite linear matrix inequality or 'robust positive semi-definiteness' of symmetric matrices. In fact, our main result for 3-dimensional cases has been proved by finding an explicit solution range to some semi-infinite linear matrix inequalities.

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A direct proof and a generalization for a Kantorovich type inequality

Cited by 5 publications

References 6 publications

Geometric insights into robust portfolio construction with gearing

Geometric insights into robust portfolio construction with gearing

Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization

Convexity conditions of Kantorovich function and related semi-infinite linear matrix inequalities

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