Survey and Review

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.