# Convex mixed-integer optimization with Frank-Wolfe methods

arXiv: https://arxiv.org/abs/2208.11010 · HTML: https://mlsanity.com/paper/2208.11010/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 66 references that are in the map.
- First posted on arXiv 23 Aug 2022.

## Builds on (most cited first, 17 of 17)

- [Solving Mixed Integer Programs Using Neural Networks](https://mlsanity.com/paper/2012.13349/) (2020)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Parallelizing the dual revised simplex method](https://mlsanity.com/paper/1503.01889/) (2015)
- [On the Global Linear Convergence of Frank-Wolfe Optimization Variants](https://mlsanity.com/paper/1511.05932/) (2015)
- [Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms](https://mlsanity.com/paper/1803.01454/) (2018)
- [A unified approach to mixed-integer optimization problems with logical constraints](https://mlsanity.com/paper/1907.02109/) (2019)
- [MathOptInterface: a data structure for mathematical optimization problems](https://mlsanity.com/paper/2002.03447/) (2020)
- [Lazifying Conditional Gradient Algorithms](https://mlsanity.com/paper/1610.05120/) (2016)
- [FrankWolfe.jl: a high-performance and flexible toolbox for Frank-Wolfe algorithms and Conditional Gradients](https://mlsanity.com/paper/2104.06675/) (2021)
- [Restarting Frank-Wolfe: Faster Rates Under H"olderian Error Bounds](https://mlsanity.com/paper/1810.02429/) (2018)
- [Birkhoff's Decomposition Revisited: Sparse Scheduling for High-Speed Circuit Switches](https://mlsanity.com/paper/2011.02752/) (2020)
- [Sparser Kernel Herding with Pairwise Conditional Gradients without Swap Steps](https://mlsanity.com/paper/2110.12650/) (2021)
- [Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes](https://mlsanity.com/paper/2106.11943/) (2021)
- [A Frank-Wolfe Based Branch-and-Bound Algorithm for Mean-Risk Optimization](https://mlsanity.com/paper/1507.05914/) (2015)
- [A Feasible Active Set Method with Reoptimization for Convex Quadratic Mixed-Integer Programming](https://mlsanity.com/paper/1512.02369/) (2015)
- [Dual Prices for Frank--Wolfe Algorithms](https://mlsanity.com/paper/2101.02087/) (2021)
- [Scalable Frank-Wolfe on Generalized Self-concordant Functions via Simple Steps](https://mlsanity.com/paper/2105.13913/) (2021)

## Built on by (most cited first, 3 of 3)

- [The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning](https://mlsanity.com/paper/2603.15914/) (2026)
- [Scalable DC Optimization via Adaptive Frank-Wolfe Algorithms](https://mlsanity.com/paper/2507.17545/) (2025)
- [Adaptive Conditional Gradient Descent](https://mlsanity.com/paper/2510.11440/) (2025)

## Abstract (arXiv)

Mixed-integer nonlinear optimization encompasses a broad class of problems that present both theoretical and computational challenges. We propose a new type of method to solve these problems based on a branch-and-bound algorithm with convex node relaxations. These relaxations are solved with a Frank-Wolfe algorithm over the convex hull of mixed-integer feasible points instead of the continuous relaxation via calls to a mixed-integer linear solver as the linear minimization oracle. The proposed method computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of mixed-integer linear solvers without an outer approximation scheme and can exploit inexact solutions of node subproblems.