# The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization

arXiv: https://arxiv.org/abs/1508.01922 · HTML: https://mlsanity.com/paper/1508.01922/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Aug 2015.

## Built on by (most cited first, 5 of 5)

- [Grouped Variable Selection with Discrete Optimization: Computational and Statistical Perspectives](https://mlsanity.com/paper/2104.07084/) (2021)
- [Predicting Census Survey Response Rates With Parsimonious Additive Models and Structured Interactions](https://mlsanity.com/paper/2108.11328/) (2021)
- [Best Subset Selection with Efficient Primal-Dual Algorithm](https://mlsanity.com/paper/2207.02058/) (2022)
- [Dynamic Incremental Optimization for Best Subset Selection](https://mlsanity.com/paper/2402.02322/) (2024)
- [Buffered AUC maximization for scoring systems via mixed-integer optimization](https://mlsanity.com/paper/2601.05544/) (2026)

## Abstract (arXiv)

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a Mixed Integer Linear Optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed in [8] and the continuous nonconvex quadratic optimization framework of [33]. We propose new discrete first-order methods, which when paired with state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular ℓ1-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling.