The Legendre–Fenchel Conjugate of the Product of Two Positive Definite Quadratic Forms

It is well-known that the Legendre-Fenchel conjugate of a positive-denite quadratic form can be explicitly expressed as another positive-denite quadratic form, and that the conjugate of the sum of several positive-denite quadratic forms can be expressed via inf-convolution. However, the Legendre-Fenchel conjugate of the product of two positive-denite quadratic forms is not clear at present. Jean-Baptiste Hiriart-Urruty posted it as an open question in the eld of nonlinear analysis and optimization [`Question 11 ' in SIAM Review 49 (2007), 255-273]. From convex analysis point of view, it is interesting and important to address such a question. The purpose of this paper is to answer this question and to provide a formula for the conjugate of the product of two positive-denite quadratic forms. We prove that the computation of the conjugate can be implemented via nding a root to certain univariate polynomial equation. Furthermore, we show that the conjugate can be explicitly expressed as a single function in some situations. Our analysis shows that the relationship between the matrices of quadratic forms plays a vital role in determining whether or not the conjugate can be expressed explicitly, and our analysis also sheds some light on the computational complexity of the Legendre-Fenchel conjugate for the product of quadratic forms

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

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.Comment: 24 page

Angles, Majorization, Wielandt Inequality and Applications

In this thesis we revisit two classical definitions of angle in an inner product space: real-part angle and Hermitian angle. Special attention is paid to Krein’s inequality and its analogue. Some applications are given, leading to a simple proof of a basic lemma for a trace inequality of unitary matrices and also its extension. A brief survey on recent results of angles between subspaces is presented. This naturally brings us to the world of majorization. After introducing the notion of majorization, we present some classical as well as recent results on eigenvalue majorization. Several new norm inequalities are derived by making use of a powerful decomposition lemma for positive semidefinite matrices. We also consider coneigenvalue majorization. Some discussion on the possible generalization of the majorization bounds for Ritz values is presented. We then turn to a basic notion in convex analysis, the Legendre-Fenchel conjugate. The convexity of a function is important in finding the explicit expression of the transform for certain functions. A sufficient convexity condition is given for the product of positive definite quadratic forms. When the number of quadratic forms is two, the condition is also necessary. The condition is in terms of the condition number of the underlying matrices. The key lemma in our derivation is found to have some connection with the generalized Wielandt inequality. A new inequality between angles in inner product spaces is formulated and proved. This leads directly to a concise statement and proof of the generalized Wielandt inequality, including a simple description of all cases of equality. As a consequence, several recent results in matrix analysis and inner product spaces are improved