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A first order method for linear programming parameterized by circuit imbalance.

Richard Cole1, Christoph Hertrich2, Yixin Tao3

  • 1Courant Institute, New York University, NY, 10012 USA.

Mathematical Programming
|May 1, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a novel first-order method for Linear Programming (LP) optimization, offering improved convergence rates. The new approach provides stronger theoretical guarantees, especially for specific problem classes like totally unimodular matrices.

Keywords:
Circuit ImbalancesFirst Order MethodsHoffman ProximityLinear Programming

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Area of Science:

  • Optimization Theory
  • Computational Mathematics
  • Operations Research

Background:

  • First-order methods are widely used for solving large-scale Linear Programming (LP) problems.
  • Existing theoretical analyses show linear convergence rates for these methods, but these rates depend on the Hoffman constant, which can be problematic.
  • There is a need for LP algorithms with stronger convergence guarantees and better performance on specific matrix classes.

Purpose of the Study:

  • To develop a new first-order approach for Linear Programming (LP) optimization.
  • To establish stronger theoretical convergence guarantees for LP algorithms.
  • To improve LP solver performance, particularly for matrices like totally unimodular ones.

Main Methods:

  • A novel first-order LP optimization approach is introduced.
  • The convergence rate is analyzed based on the circuit imbalance measure and logarithmic dependence on problem vectors.
  • The method utilizes a repeated application of a fast gradient method within a framework that fixes variables to the boundary, inspired by Tardos's method.

Main Results:

  • The proposed method exhibits a convergence rate that depends polynomially on the circuit imbalance measure and logarithmically on the right-hand side, capacity, and cost vectors.
  • This leads to significantly stronger convergence guarantees compared to existing methods.
  • For totally unimodular matrices, the approach yields polynomial-time algorithms, outperforming primal-dual methods for this class.

Conclusions:

  • The new first-order LP optimization method offers enhanced theoretical convergence guarantees.
  • The approach demonstrates particular effectiveness for specific classes of matrices, such as totally unimodular matrices.
  • This work advances the theoretical understanding and practical potential of first-order methods in Linear Programming.