Hristo S. Sendov

Hyperbolic Polynomials and Convex Analysis

A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the uni- variate polynomial tp(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gu proved in 1951 that the largest such root is a convex function of x ,a nd showed var- ious ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gu result to arbitrary symmetric functions of the roots. Many classi- cal and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

Twice Differentiable Spectral Functions

A function F on the space of n × n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument. Spectral functions are just symmetric functions of the eigenvalues. We show that a spectral function is twice (continuously) differentiable at a matrix if and only if the corresponding symmetric function is twice (continuously) differentiable at the vector of eigenvalues. We give a concise and usable formula for the Hessian.

Quadratic expansions of spectral functions

A function, F, on the space of n × n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F( A)= F( UAU T ) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on R n : those that are invariant under arbitrary swapping of their arguments. In this paper, we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessian’ of the spectral function. In the case of convex functions we show that a positive definite ‘Hessian’ of f implies positive definiteness of the ‘Hessian’ of F.

Generalized Hadamard Product and the Derivatives of Spectral Functions

Real valued functions,

Loci of complex polynomials, part I

F(X)

Self-concordant barriers for hyperbolic means

, on a symmetric matrix argument are called spectral if

Loci of complex polynomials, part II: polar derivatives

F(U^TXU) = F(X)

The Shape of the Borwein–Affleck–Girgensohn Function Generated by Completely Monotone and Bernstein Functions

for every orthogonal matrix

Two Walsh-Type Theorems for the Solutions of Multi-Affine Symmetric Polynomials

U

Separable self-concordant spectral functions and a conjecture of Tunçel

and