# Cardinality Minimization, Constraints, and Regularization: A Survey

arXiv: https://arxiv.org/abs/2106.09606 · HTML: https://mlsanity.com/paper/2106.09606/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2021.

## Built on by (most cited first, 12 of 12)

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- [Cost-Sensitive Best Subset Selection for Logistic Regression: A Mixed-Integer Conic Optimization Perspective](https://mlsanity.com/paper/2310.05464/) (2023)
- [A GPU-Accelerated Bi-linear ADMM Algorithm for Distributed Sparse Machine Learning](https://mlsanity.com/paper/2405.16267/) (2024)
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- [A Generic Branch-and-Bound Algorithm for ℓ0-Penalized Problems with Supplementary Material](https://mlsanity.com/paper/2506.03974/) (2025)
- [El0ps: An Exact L0-regularized Problems Solver](https://mlsanity.com/paper/2506.06373/) (2025)
- [Data-Driven Discovery of Mobility Periodicity for Understanding Urban Systems](https://mlsanity.com/paper/2508.03747/) (2025)
- [Local Duality for Sparse Support Vector Machines](https://mlsanity.com/paper/2601.20170/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and broadly applicable as well as problem-specific exact and heuristic solution approaches. While our perspective is that of mathematical optimization, a main goal of this work is to reach out to and build bridges between the different communities in which cardinality optimization problems are frequently encountered. In particular, we highlight that modern mixed-integer programming, which is often regarded as impractical due to commonly unsatisfactory behavior of black-box solvers applied to generic problem formulations, can in fact produce provably high-quality or even optimal solutions for cardinality optimization problems, even in large-scale real-world settings. Achieving such performance typically draws on the merits of problem-specific knowledge that may stem from different fields of application and, e.g., shed light on structural properties of a model or its solutions, or lead to the development of efficient heuristics; we also provide some illustrative examples.