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

The Kantorovich function (xT Ax)(xT A−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 2dimensional 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.