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The Differentiable Feasibility Pump

Matteo Cacciola, Alexandre Forel, Antonio Frangioni, Andrea Lodi

TL;DR

It is shown that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters and this reinterpretation opens many opportunities for improving the performance of the original algorithm.

Abstract

Although nearly 20 years have passed since its conception, the feasibility pump algorithm remains a widely used heuristic to find feasible primal solutions to mixed-integer linear problems. Many extensions of the initial algorithm have been proposed. Yet, its core algorithm remains centered around two key steps: solving the linear relaxation of the original problem to obtain a solution that respects the constraints, and rounding it to obtain an integer solution. This paper shows that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters. A central aspect of this reinterpretation is observing that the traditional algorithm differentiates the solution of the linear relaxation with respect to its cost. This reinterpretation opens many opportunities for improving the performance of the original algorithm. We study how to modify the gradient-update step as well as extending its loss function. We perform extensive experiments on MIPLIB instances and show that these modifications can substantially reduce the number of iterations needed to find a solution.

The Differentiable Feasibility Pump

TL;DR

It is shown that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters and this reinterpretation opens many opportunities for improving the performance of the original algorithm.

Abstract

Although nearly 20 years have passed since its conception, the feasibility pump algorithm remains a widely used heuristic to find feasible primal solutions to mixed-integer linear problems. Many extensions of the initial algorithm have been proposed. Yet, its core algorithm remains centered around two key steps: solving the linear relaxation of the original problem to obtain a solution that respects the constraints, and rounding it to obtain an integer solution. This paper shows that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters. A central aspect of this reinterpretation is observing that the traditional algorithm differentiates the solution of the linear relaxation with respect to its cost. This reinterpretation opens many opportunities for improving the performance of the original algorithm. We study how to modify the gradient-update step as well as extending its loss function. We perform extensive experiments on MIPLIB instances and show that these modifications can substantially reduce the number of iterations needed to find a solution.

Paper Structure

This paper contains 32 sections, 1 theorem, 23 equations, 4 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works.
  3. Contributions.
  4. Outline.
  5. Background
  6. Feasibility pump.
  7. Algorithm.
  8. Distance function.
  9. Restarts.
  10. Differentiable optimization.
  11. Differentiable Feasibility Pump
  12. Regularized gradient-descent interpretation.
  13. Integrality loss.
  14. Regularized loss.
  15. Gradient update.
  16. ...and 17 more sections

Key Result

Proposition 3.2

When $f(x) = \sum_{i=1}^n \min\{x_i, 1-x_i\}$, $\eta = 1$, $\gamma = 1$, and the Jacobian of the $\operatornamewithlimits{argmin}$ is computed as $J_{\theta} \hat{x}(\theta) = - I_n$, Algorithm alg:gen_fp generate the same sequence of solutions $\{ \hat{x}^{(k)} \}$ as the original feasibility pump

Figures (4)

  • Figure 1: Integrality loss as a function of $x$ for several loss orders: (a) $p=0.5$, (b) $p=1$, and (c) $p=2$.
  • Figure 2: Computational graph of the differentiable feasibility pump algorithm.
  • Figure 3: Soft rounding and its gradient for varying $\varepsilon$.
  • Figure 4: Number of instances that cannot be solved by DP1 for varying values of $\eta$ and $\gamma$.

Theorems & Definitions (5)

  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 4.1