Shrink-Wrapping trajectories for Linear Programming

TL;DR

This paper explores the geometry of Shrink-Wrapping trajectories in Linear Programming, extending interior-point methods and analyzing their behavior near the central line, with implications for solving LP more effectively.

Contribution

It introduces a geometric analysis of Shrink-Wrapping trajectories for LP and compares their behavior to the central path, providing new insights into hyperbolic relaxations.

Findings

Shrink-Wrapping trajectories generalize the central path.

Analysis of trajectory behavior near the central line.

Elementary proof of convexity of hyperbolicity cones.

Abstract

Hyperbolic Programming (HP) --minimizing a linear functional over an affine subspace of a finite-dimensional real vector space intersected with the so-called hyperbolicity cone-- is a class of convex optimization problems that contains well-known Linear Programming (LP). In particular, for any LP one can readily provide a sequence of HP relaxations. Based on these hyperbolic relaxations, a new Shrink-Wrapping approach to solve LP has been proposed by Renegar. The resulting Shrink-Wrapping trajectories, in a sense, generalize the notion of central path in interior-point methods. We study the geometry of Shrink-Wrapping trajectories for Linear Programming. In particular, we analyze the geometry of these trajectories in the proximity of the so-called central line, and contrast the behavior of these trajectories with that of the central path for some pathological LP instances. In addition, we provide an elementary proof of convexity of hyperbolicity cones over reals.