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A Primal-Dual Frank-Wolfe Algorithm for Linear Programming

Matthew Hough, Stephen A. Vavasis

TL;DR

It is shown that FWLP-P converges to a primal-dual solution with error $O(1/\sqrt{k})$ after $k$ iterations, while no convergence guarantees are provided for FWLP.

Abstract

We present two first-order primal-dual algorithms for solving saddle point formulations of linear programs, namely FWLP (Frank-Wolfe Linear Programming) and FWLP-P. The former iteratively applies the Frank-Wolfe algorithm to both the primal and dual of the saddle point formulation of a standard-form LP. The latter is a modification of FWLP in which regularizing perturbations are used in computing the iterates. We show that FWLP-P converges to a primal-dual solution with error $O(1/\sqrt{k})$ after $k$ iterations, while no convergence guarantees are provided for FWLP. We also discuss the advantages of using FWLP and FWLP-P for solving very large LPs. In particular, we argue that only part of the matrix $A$ is needed at each iteration, in contrast to other first-order methods.

A Primal-Dual Frank-Wolfe Algorithm for Linear Programming

TL;DR

It is shown that FWLP-P converges to a primal-dual solution with error after iterations, while no convergence guarantees are provided for FWLP.

Abstract

We present two first-order primal-dual algorithms for solving saddle point formulations of linear programs, namely FWLP (Frank-Wolfe Linear Programming) and FWLP-P. The former iteratively applies the Frank-Wolfe algorithm to both the primal and dual of the saddle point formulation of a standard-form LP. The latter is a modification of FWLP in which regularizing perturbations are used in computing the iterates. We show that FWLP-P converges to a primal-dual solution with error after iterations, while no convergence guarantees are provided for FWLP. We also discuss the advantages of using FWLP and FWLP-P for solving very large LPs. In particular, we argue that only part of the matrix is needed at each iteration, in contrast to other first-order methods.
Paper Structure (11 sections, 9 theorems, 77 equations, 3 algorithms)

This paper contains 11 sections, 9 theorems, 77 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. FWLP and FWLP-P
  3. Convergence analysis of FWLP-P
  4. Deriving the recursion and potential function
  5. Bounding the potential function
  6. Proving convergence of the potential function implies convergence of the iterates
  7. Relating $\bm{U_k}$ to the standard primal-dual gap
  8. Efficient implementation of FWLP and FWLP-P
  9. FWLP
  10. FWLP-P
  11. Conclusion

Key Result

Lemma 1

Recall $\epsilon_{k+1}$ defined in eq:epsk1. We have the bound

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 8 more