# Coordinate Optimality Reformulation for Mixed-Integer Convex Programs with Indicators

arXiv: https://arxiv.org/abs/2608.01385 · HTML: https://mlsanity.com/paper/2608.01385/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 69 references that are in the map.
- First posted on arXiv 2 Aug 2026.

## Builds on (most cited first, 28 of 28)

- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms](https://mlsanity.com/paper/1803.01454/) (2018)
- [Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization](https://mlsanity.com/paper/2004.06152/) (2020)
- [Learning Sparse Classifiers: Continuous and Mixed Integer Optimization Perspectives](https://mlsanity.com/paper/2001.06471/) (2020)
- [Integer Programming for Learning Directed Acyclic Graphs from Continuous Data](https://mlsanity.com/paper/1904.10574/) (2019)
- [Subset Selection with Shrinkage: Sparse Linear Modeling when the SNR is low](https://mlsanity.com/paper/1708.03288/) (2017)
- [Bayesian Network Learning via Topological Order](https://mlsanity.com/paper/1701.05654/) (2017)
- [Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection](https://mlsanity.com/paper/1510.06083/) (2015)
- [Rank-one Convexification for Sparse Regression](https://mlsanity.com/paper/1901.10334/) (2019)
- [Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks](https://mlsanity.com/paper/2005.14346/) (2020)
- [Ideal formulations for constrained convex optimization problems with indicator variables](https://mlsanity.com/paper/2007.00107/) (2020)
- [A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators](https://mlsanity.com/paper/2110.12547/) (2021)
- [Outer Approximation With Conic Certificates For Mixed-Integer Convex Problems](https://mlsanity.com/paper/1808.05290/) (2018)
- [Integer Programming for Learning Directed Acyclic Graphs from Non-identifiable Gaussian Models](https://mlsanity.com/paper/2404.12592/) (2024)
- [A geometric way to build strong mixed-integer programming formulations](https://mlsanity.com/paper/1811.10409/) (2018)
- [Outlier detection in regression: conic quadratic formulations](https://mlsanity.com/paper/2307.05975/) (2023)
- [Polyhedral approximation in mixed-integer convex optimization](https://mlsanity.com/paper/1607.03566/) (2016)
- [Two-Term Disjunctions on the Second-Order Cone](https://mlsanity.com/paper/1404.7813/) (2014)
- [A combinatorial approach for small and strong formulations of disjunctive constraints](https://mlsanity.com/paper/1607.04803/) (2016)
- [Subset selection in sparse matrices](https://mlsanity.com/paper/1810.02757/) (2018)
- [Solution Path of Time-varying Markov Random Fields with Discrete Regularization](https://mlsanity.com/paper/2307.13750/) (2023)
- [Polyhedral Analysis of Quadratic Optimization Problems with Stieltjes Matrices and Indicators](https://mlsanity.com/paper/2404.04236/) (2024)
- [A Parametric Approach for Solving Convex Quadratic Optimization with Indicators Over Trees](https://mlsanity.com/paper/2404.08178/) (2024)
- [Real-time solution of quadratic optimization problems with banded matrices and indicator variables](https://mlsanity.com/paper/2405.03051/) (2024)
- [An Asymptotically Optimal Coordinate Descent Algorithm for Learning Bayesian Networks from Gaussian Models](https://mlsanity.com/paper/2408.11977/) (2024)
- [Solving convex QPs with structured sparsity under indicator conditions](https://mlsanity.com/paper/2411.11722/) (2024)
- [Solving Convex Quadratic Optimization with Indicators Over Structured Graphs](https://mlsanity.com/paper/2603.02103/) (2026)
- [Computation of Least Trimmed Squares: A Branch-and-Bound framework with Hyperplane Arrangement Enhancements](https://mlsanity.com/paper/2604.11584/) (2026)

## Abstract (arXiv)

We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the Coordinate Optimality Reformulation (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable problem structure. We first develop the main components of CORe, including coordinate-wise optimality conditions, closed-form characterizations, and disjunctive reformulations. We then demonstrate the framework across multiple problem families, including quadratic problems and robust single-index models. Computational experiments show that CORe can substantially improve solver performance compared with standard big-M formulations.