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

The Kantorovich function ( x T A x ) ( x T A − 1 x ) , 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 + 2 2 . 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 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 + 2 6 , 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.