Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A First Order Method for Linear Programming Parameterized by Circuit Imbalance

Richard Cole, Christoph Hertrich, Yixin Tao, László A. Végh

TL;DR

This work develops a first-order optimization framework for linear programs that ties convergence to circuit-imbalance measures rather than traditional Hoffman constants. By embedding a discretized cost and employing a two-loop scheme—an inner fast gradient method (R-FGM) with a binary search over a potential $F_{\tau}$ and an outer loop that fixes variables and reduces the problem—the authors obtain δ-feasible and δ-optimal solutions with polynomial dependence on the circuit-imbalance parameter $\bar{\kappa}(\mathcal{X}_A)$ and logarithmic dependence on $1/δ$ and on problem data. The methodology yields dual certificates and provable proximity results, and it recovers polynomial-time guarantees for totally unimodular constraint matrices, addressing cases where primal-dual methods may converge slowly. The approach integrates ideas from Hoffman proximity theory, quadratic-growth relaxations, and advanced first-order methods to provide robust, scalable LP solvers with strong theoretical guarantees and practical certifiability. Overall, the paper contributes a principled, data-aware framework that leverages circuit structure to achieve efficient convergence for large-scale LPs.

Abstract

Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have been established for first order LP algorithms, despite the fact that the underlying formulations are not strongly convex. However, the convergence rate typically depends on the Hoffman constant of a large matrix that contains the constraint matrix, as well as the right hand side, cost, and capacity vectors. We introduce a first order approach for LP optimization with a convergence rate depending polynomially on the circuit imbalance measure, which is a geometric parameter of the constraint matrix, and depending logarithmically on the right hand side, capacity, and cost vectors. This provides much stronger convergence guarantees. For example, if the constraint matrix is totally unimodular, we obtain polynomial-time algorithms, whereas the convergence guarantees for approaches based on primal-dual formulations may have arbitrarily slow convergence rates for this class. Our approach is based on a fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019); this algorithm is called repeatedly in a framework that gradually fixes variables to the boundary. This technique is based on a new approximate version of Tardos's method, that was used to obtain a strongly polynomial algorithm for combinatorial LPs (Oper. Res. 1986).

A First Order Method for Linear Programming Parameterized by Circuit Imbalance

TL;DR

This work develops a first-order optimization framework for linear programs that ties convergence to circuit-imbalance measures rather than traditional Hoffman constants. By embedding a discretized cost and employing a two-loop scheme—an inner fast gradient method (R-FGM) with a binary search over a potential and an outer loop that fixes variables and reduces the problem—the authors obtain δ-feasible and δ-optimal solutions with polynomial dependence on the circuit-imbalance parameter and logarithmic dependence on and on problem data. The methodology yields dual certificates and provable proximity results, and it recovers polynomial-time guarantees for totally unimodular constraint matrices, addressing cases where primal-dual methods may converge slowly. The approach integrates ideas from Hoffman proximity theory, quadratic-growth relaxations, and advanced first-order methods to provide robust, scalable LP solvers with strong theoretical guarantees and practical certifiability. Overall, the paper contributes a principled, data-aware framework that leverages circuit structure to achieve efficient convergence for large-scale LPs.

Abstract

Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have been established for first order LP algorithms, despite the fact that the underlying formulations are not strongly convex. However, the convergence rate typically depends on the Hoffman constant of a large matrix that contains the constraint matrix, as well as the right hand side, cost, and capacity vectors. We introduce a first order approach for LP optimization with a convergence rate depending polynomially on the circuit imbalance measure, which is a geometric parameter of the constraint matrix, and depending logarithmically on the right hand side, capacity, and cost vectors. This provides much stronger convergence guarantees. For example, if the constraint matrix is totally unimodular, we obtain polynomial-time algorithms, whereas the convergence guarantees for approaches based on primal-dual formulations may have arbitrarily slow convergence rates for this class. Our approach is based on a fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019); this algorithm is called repeatedly in a framework that gradually fixes variables to the boundary. This technique is based on a new approximate version of Tardos's method, that was used to obtain a strongly polynomial algorithm for combinatorial LPs (Oper. Res. 1986).
Paper Structure (34 sections, 33 theorems, 84 equations, 2 algorithms)

This paper contains 34 sections, 33 theorems, 84 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hoffman bounds and quadratic function growth.
  3. Our approach
  4. Circuit imbalance measures and proximity
  5. Guessing the condition numbers.
  6. Related work
  7. Notation.
  8. Overview.
  9. Linear convergence for functions with quadratic growth
  10. Circuit imbalances and proximity
  11. Conformal circuit decompositions
  12. Guessing the circuit imbalances
  13. Main ideas and key statements
  14. The feasibility algorithm
  15. The outer loop
  16. ...and 19 more sections

Key Result

Theorem 1.1

There is an FOM-based algorithm for eq:LP that obtains a solution $x$ that is $\delta$-feasible and $\delta$-optimal, or concludes that no feasible solution exists, and whose runtime is dominated by the cost of performing $O( n^{1.5} m^2 \|A\|_1^2 \cdot\bar{\kappa}^3(\mathcal{X}_{A})$$\log^3 \left(

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4: necoara2019
  • Theorem 2.5
  • Lemma 3.1: DadushHNV20
  • Lemma 3.2
  • Lemma 3.3
  • ...and 57 more