First-Order Primal-Dual Repairs for Scalable Branch and Bound
Abstract
Mixed-integer programs have found applications in many real-world problems but remain difficult to solve at scale. To improve scalability, there has been increasing interest in incorporating first-order methods as subroutines for solving node relaxations, with success primarily on problems involving specific types of constraints. However, when more general constraints are considered, first-order methods cannot be directly applied because, at finite iterations, they typically produce infeasible primal and dual solutions, preventing their direct integration into branch-and-bound. To overcome this limitation, we introduce repair procedures that extract valid primal and dual bounds from infeasible first-order iterates. These bounds enable node pruning, incumbent updates, and early stopping at any iteration. Empirically, our approach achieves one to two orders of magnitude speedups when solving real-world instances (with another order of magnitude speedup when running on GPUs) and certifies optimality on instances that are difficult for commercial solvers.
Related
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- Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
- Real radical branch recovery for second order semialgebraic optimization
- Finding Simple Proofs for First-Order Optimization
BibTeX
@misc{pivi2026firstorder,
title = {{First-Order Primal-Dual Repairs for Scalable Branch and Bound}},
author = {Francesco Pivi and Jiachang Liu and Andrea Lodi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26509},
eprint = {2610.26509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26509},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - First-Order Primal-Dual Repairs for Scalable Branch and Bound
AU - Francesco Pivi
AU - Jiachang Liu
AU - Andrea Lodi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26509
UR - https://arxiv.org/abs/2610.26509
AB - Mixed-integer programs have found applications in many real-world problems but remain difficult to solve at scale. To improve scalability, there has been increasing interest in incorporating first-order methods as subroutines for solving node relaxations, with success primarily on problems involving specific types of constraints. However, when more general constraints are considered, first-order methods cannot be directly applied because, at finite iterations, they typically produce infeasible primal and dual solutions, preventing their direct integration into branch-and-bound. To overcome this limitation, we introduce repair procedures that extract valid primal and dual bounds from infeasible first-order iterates. These bounds enable node pruning, incumbent updates, and early stopping at any iteration. Empirically, our approach achieves one to two orders of magnitude speedups when solving real-world instances (with another order of magnitude speedup when running on GPUs) and certifies optimality on instances that are difficult for commercial solvers.
ER -