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- W4239722651 abstract "The Kantorovich function (xTAx)(xTA−1x), where A is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: when is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only if the condition number of its matrix is less than or equal to 3+22. Thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound ‘3+22’ is turned out to be a necessary condition for the convexity of the Kantorovich function 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+26, the Kantorovich function is convex. Furthermore, we prove that this general sufficient convexity condition can be improved to 2+3 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.
- W4239722651 created "2022-05-12" @default.
- W4239722651 date "1963-01-01" @default.
- W4239722651 modified "2023-09-26" @default.
- W4239722651 title "Miniaturization of inductance component parts" @default.
- W4239722651 doi "https://doi.org/10.1016/0026-2714(63)90181-1" @default.
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