PGMO days 2020: Hyperbolic polynomials : algorithms and implementations скачать в хорошем качестве

PGMO days 2020: Hyperbolic polynomials : algorithms and implementations

Title : Hyperbolic polynomials : algorithms and implementations Author: Simone Naldi Abstract: A real univariate polynomial is hyperbolic whenever all its roots are real or, in other words, if it equals the characteristic polynomial of a real symmetric matrix. This property can be extended to the multivariate case via the classical algebraic tool of symmetric determinantal representations. By the way, not every multivariate hyperbolic polynomial admits such a representation/certificate. Hyperbolic Programming (HP) is the natural convex optimization problem asking to minimize a linear function over the hyperbolicity cone of a multivariate hyperbolic polynomial. HP generalizes Linear (LP) and Semidefinite Programming (SDP), central problems in mathematics and its applications. The goal of this project is to contribute to the following two research directions: (1) the development of symbolic-numerical algorithms and implementations for the general HP problem, and (2) efficient computation of hyperbolicity certificates such as symmetric determinantal representations. The two questions above are highly related since when a polynomial has a symmetric determinantal representation or, more generally, when its hyperbolicity cone is a section of the cone of positive semidefinite matrices, then the associated HP problem reduces to a SDP problem.