matrix analysis

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

  • Yun-Bin Zhao
    • Categories Convex Optimization, Robust Optimization, Semi-infinite Programming Tags , , , ,

    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 2-dimensional Kantorovich function is convex if and only … Read more

    Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

  • Yun-Bin Zhao
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: \emph{When is the product of finitely many positive … Read more