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An infeasible interior-point arc-search method with Nesterov's restarting strategy for linear programming problems

Einosuke Iida, Makoto Yamashita

TL;DR

The paper addresses efficient LP solving by fusing arc-search interior-point methods with Nesterov’s restarting strategy to inject momentum and accelerate convergence. It proposes two variants, including a Mehrotra-type arc-search, and proves convergence to an optimal solution with a polynomial-time bound ${O}(nL)$ in the central-path neighborhood ${\mathcal N}(\theta)$; a practical variant further improves execution time. The numerical results on Netlib LP instances show reduced iteration counts and CPU time, with the momentum-based variant outperforming the baseline arc-search and a line-search method. The approach offers a provably efficient and practically faster LP solver that can be extended to other conic problems and further refined with enhanced centrality strategies. Overall, the work advances interior-point methods by integrating momentum into the arc-based central-path tracing for robust, scalable optimization.

Abstract

An arc-search interior-point method is a type of interior-point methods that approximates the central path by an ellipsoidal arc, and it can often reduce the number of iterations. In this work, to further reduce the number of iterations and computation time for solving linear programming problems, we propose two arc-search interior-point methods using Nesterov's restarting strategy that is well-known method to accelerate the gradient method with a momentum term. The first one generates a sequence of iterations in the neighborhood, and we prove that the convergence of the generated sequence to an optimal solution and the computation complexity is polynomial time. The second one incorporates the concept of the Mehrotra-type interior-point method to improve numerical performance. The numerical experiments demonstrate that the second one reduced the number of iterations and computational time. In particular, the average number of iterations was reduced compared to existing interior-point methods due to the momentum term.

An infeasible interior-point arc-search method with Nesterov's restarting strategy for linear programming problems

TL;DR

The paper addresses efficient LP solving by fusing arc-search interior-point methods with Nesterov’s restarting strategy to inject momentum and accelerate convergence. It proposes two variants, including a Mehrotra-type arc-search, and proves convergence to an optimal solution with a polynomial-time bound in the central-path neighborhood ; a practical variant further improves execution time. The numerical results on Netlib LP instances show reduced iteration counts and CPU time, with the momentum-based variant outperforming the baseline arc-search and a line-search method. The approach offers a provably efficient and practically faster LP solver that can be extended to other conic problems and further refined with enhanced centrality strategies. Overall, the work advances interior-point methods by integrating momentum into the arc-based central-path tracing for robust, scalable optimization.

Abstract

An arc-search interior-point method is a type of interior-point methods that approximates the central path by an ellipsoidal arc, and it can often reduce the number of iterations. In this work, to further reduce the number of iterations and computation time for solving linear programming problems, we propose two arc-search interior-point methods using Nesterov's restarting strategy that is well-known method to accelerate the gradient method with a momentum term. The first one generates a sequence of iterations in the neighborhood, and we prove that the convergence of the generated sequence to an optimal solution and the computation complexity is polynomial time. The second one incorporates the concept of the Mehrotra-type interior-point method to improve numerical performance. The numerical experiments demonstrate that the second one reduced the number of iterations and computational time. In particular, the average number of iterations was reduced compared to existing interior-point methods due to the momentum term.
Paper Structure (21 sections, 17 theorems, 96 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 21 sections, 17 theorems, 96 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Arc-search interior-point method
  5. The proposed method
  6. Theoretical proof
  7. Convergence of Algorithm \ref{['algo_arc_restarting_for_proof']}
  8. Proof for Proposition \ref{['proposition_s_bounded_below_away_from_zero']}
  9. Numerical experiments
  10. Test problems
  11. The modified algorithm
  12. Implementation details
  13. Initial points
  14. Step size
  15. Stopping criteria
  16. ...and 6 more sections

Key Result

Proposition 4.1

There exists $L_0 > 0$ such that $s_i^k \ge L_0 \mu_k$ for each $i=1,\dots,n$.

Figures (6)

  • Figure 1: Performance profile of the number of iterations with different restarting parameters in Algorithm \ref{['algo_arc_restarting_for_proof']}
  • Figure 2: Performance profile of the number of iterations in Algorithms \ref{['algo_arc_restarting_for_proof']} and \ref{['algo_arc_restarting_for_calculation']} with $\beta = 0.001$
  • Figure 3: Trajectories for $\left\|r_b(x^k)\right\|_\infty$. The left is the result with the setting of \ref{['def_beta_for_proof']}, and the right is with \ref{['def_beta_for_calculation']}.
  • Figure 4: Performance profile of the number of iterations with different setting of $\beta_k$
  • Figure 5: Performance profile of the number of iterations with different restarting parameters in Algorithm \ref{['algo_arc_restarting_for_calculation']}
  • ...and 1 more figures

Theorems & Definitions (27)

  • Proposition 4.1
  • Lemma 4.1
  • proof
  • Corollary 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.3
  • proof
  • ...and 17 more