Cardinality Minimization, Constraints, and Regularization: A Survey
Abstract
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 generalpurpose modeling techniques and broadly applicable as well as problemspecific 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 mixedinteger programming, which is often regarded as impractical due to commonly unsatisfactory behavior of blackbox solvers applied to generic problem formulations, can in fact produce provably highquality or even optimal solutions for cardinality optimization problems, even in largescale realworld settings. Achieving such performance typically draws on the merits of problemspecific 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.09606
 Bibcode:
 2021arXiv210609606T
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Information Theory