Convexity Conditions of Kantorovich Function and Related Semi-infinite Linear Matrix Inequalities

TL;DR

This paper characterizes the convexity of the Kantorovich function based on the condition number of its matrix, providing exact bounds in 2D and sufficient conditions in higher dimensions, linking it to semi-infinite linear matrix inequalities.

Contribution

It establishes precise convexity conditions for the Kantorovich function in 2D and 3D, connecting matrix condition numbers with convexity and semi-infinite linear matrix inequalities.

Findings

Convexity in 2D occurs if and only if the condition number ≤ 3+2√2.

Condition number ≤ √(5+2√6) guarantees convexity in any finite dimension.

Explicit solutions to semi-infinite linear matrix inequalities underpin the convexity analysis.

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+2\sqrt{2}, $ and 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 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+2\sqrt{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.