# Outlier detection in regression: conic quadratic formulations

arXiv: https://arxiv.org/abs/2307.05975 · HTML: https://mlsanity.com/paper/2307.05975/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 62 references that are in the map.
- First posted on arXiv 12 Jul 2023.

## Builds on (most cited first, 20 of 20)

- [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)
- [Sparse High-Dimensional Regression: Exact Scalable Algorithms and Phase Transitions](https://mlsanity.com/paper/1709.10029/) (2017)
- [Robust Regression via Hard Thresholding](https://mlsanity.com/paper/1506.02428/) (2015)
- [Integer Programming for Learning Directed Acyclic Graphs from Continuous Data](https://mlsanity.com/paper/1904.10574/) (2019)
- [The ALAMO approach to machine learning](https://mlsanity.com/paper/1705.10918/) (2017)
- [Subset Selection with Shrinkage: Sparse Linear Modeling when the SNR is low](https://mlsanity.com/paper/1708.03288/) (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)
- [Iterative Least Trimmed Squares for Mixed Linear Regression](https://mlsanity.com/paper/1902.03653/) (2019)
- [L0Learn: A Scalable Package for Sparse Learning using L0 Regularization](https://mlsanity.com/paper/2202.04820/) (2022)
- [Sparse and Smooth Signal Estimation: Convexification of L0 Formulations](https://mlsanity.com/paper/1811.02655/) (2018)
- [P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints](https://mlsanity.com/paper/2202.05198/) (2022)
- [On the convex hull of convex quadratic optimization problems with indicators](https://mlsanity.com/paper/2201.00387/) (2022)
- [Subset Selection for Multiple Linear Regression via Optimization](https://mlsanity.com/paper/1701.07920/) (2017)
- [A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators](https://mlsanity.com/paper/2110.12547/) (2021)
- [Simultaneous Feature Selection and Outlier Detection with Optimality Guarantees](https://mlsanity.com/paper/2007.06114/) (2020)
- [Linear regression with partially mismatched data: local search with theoretical guarantees](https://mlsanity.com/paper/2106.02175/) (2021)
- [Minimization of Akaike's Information Criterion in Linear Regression Analysis via Mixed Integer Nonlinear Program](https://mlsanity.com/paper/1606.05030/) (2016)

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

- [Scalable First-order Method for Certifying Optimal k-Sparse GLMs](https://mlsanity.com/paper/2502.09502/) (2025)
- [Computation of Least Trimmed Squares: A Branch-and-Bound framework with Hyperplane Arrangement Enhancements](https://mlsanity.com/paper/2604.11584/) (2026)
- [Coordinate Optimality Reformulation for Mixed-Integer Convex Programs with Indicators](https://mlsanity.com/paper/2608.01385/) (2026)

## Abstract (arXiv)

In many applications, when building linear regression models, it is important to account for the presence of outliers, i.e., corrupted input data points. Such problems can be formulated as mixed-integer optimization problems involving cubic terms, each given by the product of a binary variable and a quadratic term of the continuous variables. Existing approaches in the literature, typically relying on the linearization of the cubic terms using big-M constraints, suffer from weak relaxation and poor performance in practice. In this work we derive stronger second-order conic relaxations that do not involve big-M constraints. Our computational experiments indicate that the proposed formulations are several orders-of-magnitude faster than existing big-M formulations in the literature for this problem.