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On the Curvature of the Central Path of Linear Programming Theory

Jean-Pierre Dedieu, Gregorio Malajovich, Mike Shub

TL;DR

Addressing the curvature of the central path in linear programming, the paper derives a bound on the total curvature when summing over all sign conditions and shows that the average curvature over natural probability measures scales with the primal variable count $n$ but is largely independent of the number of constraints $m$. The main technique combines integral geometry (Gauss map and Grassmannian intersections) with multi-homogeneous Bézout bounds for the associated polynomial systems describing central paths. The authors prove a general bound on the total curvature and its average, and supply a detailed, technically involved proof of a key proposition via complexification, continuation of nondegenerate roots, elimination of variables, and spurious-root analysis. This work provides theoretical support for the efficiency of long-step interior-point methods by indicating small curvature on average.

Abstract

We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of independent variables of the primal problem, and independent on the number of constraints.

On the Curvature of the Central Path of Linear Programming Theory

TL;DR

Addressing the curvature of the central path in linear programming, the paper derives a bound on the total curvature when summing over all sign conditions and shows that the average curvature over natural probability measures scales with the primal variable count but is largely independent of the number of constraints . The main technique combines integral geometry (Gauss map and Grassmannian intersections) with multi-homogeneous Bézout bounds for the associated polynomial systems describing central paths. The authors prove a general bound on the total curvature and its average, and supply a detailed, technically involved proof of a key proposition via complexification, continuation of nondegenerate roots, elimination of variables, and spurious-root analysis. This work provides theoretical support for the efficiency of long-step interior-point methods by indicating small curvature on average.

Abstract

We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of independent variables of the primal problem, and independent on the number of constraints.

Paper Structure

This paper contains 16 sections, 25 theorems, 74 equations.

Table of Contents

  1. Introduction.
  2. Description of the central path.
  3. Curvature.
  4. An integral geometry formula.
  5. A Bézout bound for multi-homogeneous systems.
  6. The total curvature of the central path on the average.
  7. Proof of Proposition \ref{['prop-mu']}
  8. Complexification of the equations
  9. Continuation of non-degenerate roots
  10. Non-degeneracy at the maximum
  11. Genericity
  12. Simplification of the equations
  13. Elimination of variables
  14. The Bézout bound
  15. The spurious roots
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $m > n \ge 1$. Let $A$ be an $m \times n$ matrix of rank $n$, and let $b \in {\mathbb R}^m$ and $c \in {\mathbb R}^n$, $c$ non-zero. The sum over all $2^m$ sign conditions of the total curvature of the primal/dual central paths (resp. primal central paths, dual central paths) is less than or equ

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 4.1
  • Definition 4.2
  • Corollary 4.3
  • Lemma 4.4
  • proof
  • Theorem 5.1
  • Lemma 6.1
  • ...and 32 more