  Hyperbolic polynomials is a class of real-roots polynomials that has wide
range of applications in theoretical computer science. Each hyperbolic
polynomial also induces a hyperbolic cone that is of particular interest in
optimization due to its generality, as by choosing the polynomial properly, one
can easily recover the classic optimization problems such as linear programming
and semidefinite programming. In this work, we develop efficient algorithms for
hyperbolic programming, the problem in each one wants to minimize a linear
objective, under a system of linear constraints and the solution must be in the
hyperbolic cone induced by the hyperbolic polynomial. Our algorithm is an
instance of interior point method (IPM) that, instead of following the central
path, it follows the central Swath, which is a generalization of central path.
To implement the IPM efficiently, we utilize a relaxation of the hyperbolic
program to a quadratic program, coupled with the first four moments of the
hyperbolic eigenvalues that are crucial to update the optimization direction.
We further show that, given an evaluation oracle of the polynomial, our
algorithm only requires $O(n^2d^{2.5})$ oracle calls, where $n$ is the number
of variables and $d$ is the degree of the polynomial, with extra $O((n+m)^3
d^{0.5})$ arithmetic operations, where $m$ is the number of constraints.
