Outlier Detection in Regression: Conic Quadratic Formulations

In many applications, when building linear regression models, it is important to account for the presence of outliers, that is, 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. In addition, we verify that exact solution of the mixed-integer optimization problems can lead to substantially better-quality solutions than simpler relaxations or heuristics commonly used in practice. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by a public grant as part of the Investissement d’avenir project [Grant ANR-11-LABX-0056-LMH, LabEx LMH]. A. Gómez is supported by the US Air Force Office of Scientific Research [Grant FA9550-22-1-0369]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1215 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1215 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .