SIAM Review, Volume 66, Issue 3, Page 401-401, May 2024.
In “Cardinality Minimization, Constraints, and Regularization: A Survey," Andreas M. Tillmann, Daniel Bienstock, Andrea Lodi, and Alexandra Schwartz consider a class of optimization problems that involve the cardinality of variable vectors in constraints or in the objective function. Such problems have many important applications, e.g., medical imaging (like X-ray tomography), face recognition, wireless sensor network design, stock picking, crystallography, astronomy, computer vision, classification and regression, interpretable machine learning, and statistical data analysis. The emphasis in this paper is on continuous variables, which distinguishes it from a myriad of classical operation research or combinatorial optimization problems. Three general problem classes are studied in detail: cardinality minimization problems, cardinality-constrained problems, and regularized cardinality problems. The paper provides a road map connecting several disciplines and offers an overview of many different computational approaches that are available for cardinality optimization problems. Since such problems are of cross-disciplinary nature, the authors organized their review according to specific application areas and point out overlaps and differences. The paper starts with prominent cardinality optimization problems, namely, signal and image processing, portfolio optimization and management, high-dimensional statistics and machine learning, and some related problems from combinatorics, matrix sparsification, and group/block sparsity. It then continues with exact models and solution methods. The further sections are devoted to relaxations and heuristics, scalability of exact and heuristic algorithms. The authors made a strong effort regarding the organization of their quite long paper, meaning that tables and figures guide the reader to an application or result of interest. In addition, they provide an extensive overview on the literature with more than 400 references.

中文翻译：

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 调查与回顾


《SIAM 评论》，第 66 卷，第 3 期，第 401-401 页，2024 年 5 月。

在“基数最小化、约束和正则化：一项调查”中，Andreas M. Tillmann、Daniel Bienstock、Andrea Lodi 和 Alexandra Schwartz 考虑了一类优化问题，这些问题涉及约束或目标函数中变量向量的基数。问题有许多重要的应用，例如医学成像（如X射线断层扫描）、人脸识别、无线传感器网络设计、选股、晶体学、天文学、计算机视觉、分类和回归、可解释的机器学习和统计数据分析。本文的重点是连续变量，这将其与众多经典运筹学或组合优化问题区分开来，详细研究了三种常见问题：基数最小化问题、基数约束问题和正则化基数问题。连接多个学科的路线图，概述了可用于基数优化问题的许多不同的计算方法。由于此类问题具有跨学科性质，作者根据具体应用领域组织综述，并指出重叠和差异。本文从突出的基数优化问题开始，即信号和图像处理、投资组合优化和管理、高维统计和机器学习，以及组合学、矩阵稀疏化和组/块稀疏性的一些相关问题。然后继续使用精确模型和求解方法。接下来的部分专门讨论松弛和启发式算法、精确算法和启发式算法的可扩展性。 作者在组织这篇相当长的论文方面付出了巨大的努力，这意味着表格和图形引导读者找到感兴趣的应用程序或结果。此外，它们还提供了 400 多篇参考文献的广泛概述。