Sparse index tracking is a prominent passive portfolio management strategy that constructs a sparse portfolio to track a financial index. A sparse portfolio is preferable to a full portfolio in terms of reducing transaction costs and avoiding illiquid assets. To achieve portfolio sparsity, conventional studies have utilized $\ell_p$-norm regularizations as a continuous surrogate of the $\ell_0$-norm regularization. Although these formulations can construct sparse portfolios, their practical application is challenging due to the intricate and time-consuming process of tuning parameters to define the precise upper limit of assets in the portfolio. In this paper, we propose a new problem formulation of sparse index tracking using an $\ell_0$-norm constraint that enables easy control of the upper bound on the number of assets in the portfolio. Moreover, our approach offers a choice between constraints on portfolio and turnover sparsity, further reducing transaction costs by limiting asset updates at each rebalancing interval. Furthermore, we develop an efficient algorithm for solving this problem based on a primal-dual splitting method. Finally, we illustrate the effectiveness of the proposed method through experiments on the S&P500 and Russell3000 index datasets.

翻译：稀疏指数追踪是一种重要的被动型投资组合管理策略，通过构建稀疏投资组合来追踪金融指数。相较于全样本组合，稀疏组合在降低交易成本和规避非流动性资产方面具有显著优势。为实现组合稀疏性，现有研究普遍采用$\ell_p$范数正则化作为$\ell_0$范数正则化的连续松弛近似。尽管这类方法能够构建稀疏组合，但由于需要调整参数以精确界定组合中资产数量的上限，这一复杂且耗时的过程严重制约了其实际应用。本文提出一种基于$\ell_0$范数约束的稀疏指数追踪新问题范式，能够便捷地控制组合中资产数量的上限。此外，我们的方法可在组合稀疏性与换手率稀疏性约束之间自由选择，通过限制每个再平衡周期的资产更新次数进一步降低交易成本。我们基于原始-对偶分裂方法开发了求解该问题的高效算法。最后，通过标普500指数和罗素3000指数数据集的实验验证了所提方法的有效性。