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Smoothed Analysis of Interior-Point Algorithms: Condition Number

John Dunagan, Daniel A. Spielman, Shang-Hua Teng

TL;DR

The paper analyzes Renegar's condition number for linear programming under Gaussian perturbations (smoothed analysis) and links it to interior-point algorithm performance. By developing primal and dual geometric analyses and proving that the logarithm of the condition number is $O(\log (n d / \sigma))$, it derives smoothed complexity bounds of order $O(n^{3} \log (n d / \sigma))$ for Renegar's method and related IPMs. The results illuminate why interior-point methods perform well in practice despite worst-case polylogarithmic guarantees, and they establish a framework for studying condition-number-driven complexity in conic LPs. The work also discusses perturbation-model limitations (e.g., zero-preserving perturbations) and highlights open questions for extending smoothed analyses to broader perturbation schemes and related optimization problems.

Abstract

We show that the smoothed complexity of the logarithm of Renegar's condition number is O(log (n/sigma)).

Smoothed Analysis of Interior-Point Algorithms: Condition Number

TL;DR

The paper analyzes Renegar's condition number for linear programming under Gaussian perturbations (smoothed analysis) and links it to interior-point algorithm performance. By developing primal and dual geometric analyses and proving that the logarithm of the condition number is , it derives smoothed complexity bounds of order for Renegar's method and related IPMs. The results illuminate why interior-point methods perform well in practice despite worst-case polylogarithmic guarantees, and they establish a framework for studying condition-number-driven complexity in conic LPs. The work also discusses perturbation-model limitations (e.g., zero-preserving perturbations) and highlights open questions for extending smoothed analyses to broader perturbation schemes and related optimization problems.

Abstract

We show that the smoothed complexity of the logarithm of Renegar's condition number is O(log (n/sigma)).

Paper Structure

This paper contains 23 sections, 32 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. The Complexity of Linear Programming Algorithms
  3. Renegar's Condition Number
  4. Smoothed Analysis of Condition Number: Our Results
  5. Organization of the Paper
  6. Prior Smoothed Analyses of Linear Programming Algorithms
  7. Prior Average-Case Analyses of Interior Point Algorithms
  8. Lower Bounds for Interior Point Algorithms
  9. Notation and Basic Geometric Definitions
  10. Primal Condition Number
  11. Linear Programs in Conic Form and Basic Convex Probability Theory
  12. Primal condition number, feasible case
  13. Primal number, infeasible case
  14. Primal condition number, putting the feasible and infeasible cases together
  15. Log of the Primal Condition Number
  16. ...and 8 more sections

Key Result

Theorem 1.2.2

For any linear program of form (2) specified by $(A,\boldsymbol{\mathit{b}} ,\boldsymbol{\mathit{c}} )$ and parameter $\epsilon$, Renegar's interior-point algorithm, in $O (n^{3} \log (nC (A,\boldsymbol{\mathit{b}} ,\boldsymbol{\mathit{c}} )/\epsilon ))$ operations, finds a feasible solution $\bolds

Theorems & Definitions (70)

  • Definition 1.2.1: Primal Condition Number
  • Theorem 1.2.2: Renegar
  • Theorem 1.3.1: Smoothed Complexity of Renegar's Condition Number
  • Theorem 1.3.2: Smoothed Complexity of IPM
  • Proposition 1.8.1: Choice of norm
  • Definition 1.8.2: Ray
  • Definition 1.8.3: Non-pointed convex cone
  • Definition 1.8.4: Positive half-space
  • Definition 2.1.1: Generalized distance to ill-posedness
  • Lemma 2.1.2: Preserving the condition number
  • ...and 60 more