DCC: Differentiable Cardinality Constraints for Partial Index Tracking

Abstract: Index tracking is a popular passive investment strategy aimed at optimizing portfolios, but fully replicating an index can lead to high transaction costs. To address this, partial replication have been proposed. However, the cardinality constraint renders the problem non-convex, non-differentiable, and often NP-hard, leading to the use of heuristic or neural network-based methods, which can be non-interpretable or have NP-hard complexity. To overcome these limitations, we propose a Differentiable Cardinality Constraint ($\textbf{DCC}$) for index tracking and introduce a floating-point precision-aware method ($\textbf{DCC}{fpp}$) to address implementation issues. We theoretically prove our methods calculate cardinality accurately and enforce actual cardinality with polynomial time complexity. We propose the range of the hyperparameter $a$ ensures that $\textbf{DCC}{fpp}$ has no error in real implementations, based on theoretical proof and experiment. Our method applied to mathematical method outperforms baseline methods across various datasets, demonstrating the effectiveness of the identified hyperparameter $a$.