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 $a$.

翻译：指数跟踪是一种旨在优化投资组合的流行被动投资策略，但完全复制指数可能导致高昂的交易成本。为解决此问题，部分复制方法被提出。然而，基数约束使得该问题非凸、不可微且通常为NP难问题，导致采用启发式或基于神经网络的方法，这些方法可能缺乏可解释性或具有NP难复杂度。为克服这些限制，我们提出了一种用于指数跟踪的可微基数约束（$\textbf{DCC}$），并引入了一种浮点精度感知方法（$\textbf{DCC}_{fpp}$）以解决实现问题。我们从理论上证明了我们的方法能够准确计算基数，并以多项式时间复杂度强制实现实际基数。基于理论证明和实验，我们提出了超参数$a$的取值范围，以确保$\textbf{DCC}_{fpp}$在实际实现中无误差。我们的方法应用于数学方法，在多个数据集上均优于基线方法，证明了所识别超参数$a$的有效性。