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YunBin Zhao(y.zhao.2bham.ac.uk) Abstract: The Kantorovich function $(x^TAx)( x^T A^{1} x)$, 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 2dimensional Kantorovich function is convex if and only if the condition number of its matrix is less than or equal to $3+2\sqrt{2}. $ Thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound `$3+2\sqrt{2} $' is turned out to be a necessary condition for the convexity of the Kantorovich function in any finitedimensional 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+2\sqrt{6}}, $ the Kantorovich function is convex. Furthermore, we prove that this general sufficient convexity condition can be improved to $2+\sqrt{3} $ in 3dimensional space. Our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semiinfinite linear matrix inequality or `robust positive semidefiniteness' of symmetric matrices. In fact, our main result for 3dimensional cases has been proved by finding an explicit solution range to some semiinfinite linear matrix inequalities. Keywords: Matrix analysis, condition number, Kantorovich function, convex analysis, positive definite matrix. Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Robust Optimization Category 3: Infinite Dimensional Optimization (Semiinfinite Programming ) Citation: Download: [PDF] Entry Submitted: 01/29/2011 Modify/Update this entry  
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