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An inexact infeasible arc-search interior-point method for linear optimization problems

Einosuke Iida, Makoto Yamashita

TL;DR

The paper introduces an inexact infeasible arc-search interior-point method (II-arc) for linear optimization, combining an ellipsoidal arc trajectory with inexact Newton solves to relax step-size restrictions and achieve a polynomial-time guarantee. Theoretical analysis yields an improved iteration bound of $O\left(n^{1.5}L\right)$, reflecting a gain over traditional inexact line-search IPMs, while numerical experiments on NETLIB problems confirm reduced iterations and faster runtimes. The method leverages an arc-based central-path approximation, inexact NES/MNES formulations for derivatives, and practical heuristics to balance convergence theory with numerical stability. Overall, II-arc demonstrates both stronger worst-case guarantees and tangible performance gains, suggesting significant practical impact for large-scale LP solvers.

Abstract

We propose an inexact infeasible arc-search interior-point method for solving linear optimization problems. The method combines an arc-search strategy with inexact solutions to Newton systems and admits a polynomial iteration complexity bound. In existing inexact infeasible interior-point methods, both the linearization error of the central path and the inexactness of the Newton system accumulate along the search direction, which forces the algorithm to take very small steps. The proposed method mitigates this effect by using an arc-search strategy: the curved search path provides a more accurate approximation of the central path, so the step size can remain larger even when the Newton system is solved inexactly. As a result, the proposed method achieves a provably tighter worst-case iteration bound than existing inexact infeasible line-search methods. Numerical experiments on NETLIB benchmark problems demonstrate that the proposed method reduces both the number of iterations and the computation time.

An inexact infeasible arc-search interior-point method for linear optimization problems

TL;DR

The paper introduces an inexact infeasible arc-search interior-point method (II-arc) for linear optimization, combining an ellipsoidal arc trajectory with inexact Newton solves to relax step-size restrictions and achieve a polynomial-time guarantee. Theoretical analysis yields an improved iteration bound of , reflecting a gain over traditional inexact line-search IPMs, while numerical experiments on NETLIB problems confirm reduced iterations and faster runtimes. The method leverages an arc-based central-path approximation, inexact NES/MNES formulations for derivatives, and practical heuristics to balance convergence theory with numerical stability. Overall, II-arc demonstrates both stronger worst-case guarantees and tangible performance gains, suggesting significant practical impact for large-scale LP solvers.

Abstract

We propose an inexact infeasible arc-search interior-point method for solving linear optimization problems. The method combines an arc-search strategy with inexact solutions to Newton systems and admits a polynomial iteration complexity bound. In existing inexact infeasible interior-point methods, both the linearization error of the central path and the inexactness of the Newton system accumulate along the search direction, which forces the algorithm to take very small steps. The proposed method mitigates this effect by using an arc-search strategy: the curved search path provides a more accurate approximation of the central path, so the step size can remain larger even when the Newton system is solved inexactly. As a result, the proposed method achieves a provably tighter worst-case iteration bound than existing inexact infeasible line-search methods. Numerical experiments on NETLIB benchmark problems demonstrate that the proposed method reduces both the number of iterations and the computation time.
Paper Structure (22 sections, 12 theorems, 121 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 22 sections, 12 theorems, 121 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. The proposed method
  5. Theoretical proof
  6. Numerical experiments
  7. Inexact infeasible line-search IPM
  8. Test problems
  9. Implementation details
  10. Parameter settings
  11. Solving systems of linear equations
  12. The modification of the second order derivatives
  13. Step size
  14. Stopping criteria
  15. Initial points
  16. ...and 7 more sections

Key Result

Lemma 3.1

Assume a step length $\alpha \in (0, \pi / 2]$ satisfies $G_i^k(\alpha) \ge 0$ and $g^k(\alpha) \ge 0$. Then, $(x^k(\alpha), y^k(\alpha), s^k(\alpha)) \in \mathcal{N}(\gamma_1, \gamma_2)$.

Figures (5)

  • Figure 1: Performance profile of the number of iterations with II-arc-Yang and II-arc-Constant
  • Figure 2: Performance profile of the computation time with II-arc-Yang and II-arc-Constant
  • Figure 3: Performance profile of the number of iterations with II-arc-Yang and II-line-Yang
  • Figure 4: Performance profile of the computation time with II-arc-Yang and II-line-Yang
  • Figure 5: Computation time to obtain the search directions for solving PDS-10

Theorems & Definitions (17)

  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2: yang2020arc
  • Proposition 4.1
  • Lemma 4.3
  • proof
  • Lemma 4.4: mohammadisiahroudi2024efficient
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • ...and 7 more