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- W2078857100 abstract "The Kantorovich function $(x^TAx)(x^T A^{-1} x)$, 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+2sqrt{2}, $ and thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound `$3+2sqrt{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 $sqrt{5+2sqrt{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." @default.
- W2078857100 created "2016-06-24" @default.
- W2078857100 creator A5050493484 @default.
- W2078857100 date "2011-06-01" @default.
- W2078857100 modified "2023-10-14" @default.
- W2078857100 title "Convexity conditions of Kantorovich function and related semi-infinite linear matrix inequalities" @default.
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- W2078857100 doi "https://doi.org/10.1016/j.cam.2011.03.034" @default.
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