SIAM Review, Volume 66, Issue 3, Page 403-477, May 2024.
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 we give concrete examples from diverse application fields such as signal and image processing, portfolio selection, and 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 the 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 involves drawing on the merits of problem-specific knowledge that may stem from different fields of application and, e.g., can shed light on structural properties of a model or its solutions, or can lead to the development of efficient heuristics. We also provide some illustrative examples.

中文翻译：

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基数最小化、约束和正则化：一项调查


《SIAM 评论》，第 66 卷，第 3 期，第 403-477 页，2024 年 5 月。

我们调查涉及约束或目标函数中变量向量基数的优化问题。我们提供了关于一般问题类别和模型的统一观点，并给出了信号和图像处理、投资组合选择和机器学习等不同应用领域的具体示例。本文讨论了通用建模技术和广泛适用的以及特定问题的精确和启发式解决方法。虽然我们的观点是数学优化，但这项工作的主要目标是在经常遇到基数优化问题的不同社区之间建立桥梁。我们特别强调，现代混合整数规划通常被认为是不切实际的，因为黑盒求解器应用于通用问题公式时通常表现不佳，但实际上可以为基数优化产生可证明的高质量甚至最优解决方案即使在大规模的现实环境中也会遇到问题。实现这样的性能通常涉及利用可能源自不同应用领域的特定问题知识的优点，例如，可以阐明模型或其解决方案的结构特性，或者可以导致有效启发法的发展。我们还提供了一些说明性示例。