The central curve in linear programming

TL;DR

This paper investigates the algebraic and geometric properties of the central curve in linear programming, providing explicit formulas for its invariants and implications for interior point methods.

Contribution

It determines the degree, genus, and defining ideal of the central curve, linking these to the input matrix's matroid and extending previous curvature bounds.

Findings

Explicit formulas for the degree and genus of the central curve.

Connection of geometric invariants to the matroid of the input matrix.

Instance-specific bounds on the total curvature of the central path.

Abstract

The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior point methods. The global geometry of central curves is studied in detail.