Central Swaths (A Generalization of the Central Path)

TL;DR

This paper introduces a generalized concept of the central path in convex optimization using derivative cones of hyperbolicity cones, potentially enabling faster convergence to optimal solutions.

Contribution

It extends the central path notion via derivative cones, providing a new framework that could improve optimization efficiency across various conic problems.

Findings

Paths generated by derivative cones always lead to optimality

Special case of quadratic derivative cones recovers the classical central path

Higher-degree derivative cones may accelerate convergence to optimal solutions

Abstract

We develop a natural generalization to the notion of the central path -- a notion that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the "derivative cones" of a "hyperbolicity cone," the derivatives being direct and mathematically-appealing relaxations of the underlying (hyperbolic) conic constraint, be it the non-negative orthant, the cone of positive semidefinite matrices, or other. We prove that a dynamics inherent to the derivative cones generates paths always leading to optimality, the central path arising from a special case in which the derivative cones are quadratic. Derivative cones of higher degree better fit the underlying conic constraint, raising the prospect that the paths they generate lead to optimality quicker than the central path.