Symbolic computation in hyperbolic programming

TL;DR

This paper introduces a symbolic computation approach for hyperbolic programming, leveraging algebraic boundary structures to develop exact algorithms that certify solutions without requiring determinantal representations.

Contribution

It presents a novel symbolic method for hyperbolic programming that does not depend on determinantal representations, enabling certification of solutions and optimal values.

Findings

Developed exact algorithms for hyperbolic programming

Certified multiplicity of solutions and optimal values

Operates without determinantal representability

Abstract

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming. We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function.