A1 Refereed original research article in a scientific journal
Boundary problem and overfitting reduction in convex regression
Authors: Liao, Zhiqiang; Dai, Sheng; Lim, Eunji; Kuosmanen, Timo
Publisher: Elsevier BV
Publication year: 2026
Journal: European Journal of Operational Research
Volume: 333
Issue: 2
First page : 555
Last page: 566
ISSN: 0377-2217
eISSN: 1872-6860
DOI: https://doi.org/10.1016/j.ejor.2026.04.009
Publication's open availability at the time of reporting: No Open Access
Publication channel's open availability : Partially Open Access publication channel
Web address : https://doi.org/10.1016/j.ejor.2026.04.009
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/523239733
Self-archived copy's licence: CC BY NC ND
Self-archived copy's version: Final draft
Convex regression is a nonparametric approach for estimating a convex or concave function from observed data. It is widely used in operations research, economics, machine learning, and related fields. However, empirical evidence has shown that convex regression can yield excessively large subgradients on the boundary. In this paper, we provide theoretical evidence of this boundary problem. To address such a problem, we propose two new estimators by placing a bound on the subgradients of the convex function. We further prove that they converge to the underlying true convex function and that their subgradients converge to the gradient of the underlying function, both uniformly over the domain with probability one as the sample size increases to infinity. The proposed methods also help to reduce overfitting in finite samples: Monte Carlo simulations and empirical illustrations with large-scale datasets confirm the superior performance of the proposed estimators in predictive power over the existing methods.
Funding information in the publication:
Zhiqiang Liao gratefully acknowledges financial support from the BNBU Start-up Research Fund [grant no. UICR0700139-26]. Sheng Dai gratefully acknowledges financial support from the National Natural Science Foundation of China [grant no. 72501303].
