The Differentiable Feasibility Pump

TL;DR

This paper reinterprets the feasibility pump heuristic for mixed-integer linear problems as a differentiable, gradient-based algorithm, enabling new modifications that significantly improve its efficiency on benchmark instances.

Contribution

It introduces a differentiable perspective to the feasibility pump, allowing for gradient-based improvements and extensions to enhance performance.

Findings

Modifications reduce the number of iterations needed to find solutions.

Reinterpretation as gradient descent opens new avenues for algorithm enhancement.

Experimental results show substantial performance improvements.

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.