Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

DCC: Differentiable Cardinality Constraints for Partial Index Tracking

Wooyeon Jo, Hyunsouk Cho

TL;DR

The paper addresses the NP-hard challenge of partial replication in index tracking by introducing the Differentiable Cardinality Constraint (DCC) and its floating-point precision-aware variant, DCC_fpp. By replacing the non-differentiable cardinality indicator with differentiable surrogates such as $\widetilde{b}(w_i)=1-\frac{1}{a w_i+1}$ and $\tilde{b}_{fpp}(w_i)=\frac{1}{1+e^{-a(w_i-\epsilon)}}$, the authors integrate the constraint into standard differentiable optimization frameworks, notably SLSQP, while proving conditions under which actual cardinality is enforced. They establish accuracy and assurance criteria (C0–C2) and prove that, with a sufficiently large $a$ (e.g., $a \ge 1.38\times10^5$ in 64-bit precision), the surrogate preserves correct cardinality and constraint satisfaction, achieving polynomial-time complexity. Empirical results across S&P 100/500 and KOSPI 100 show that DCC_fpp yields competitive tracking performance—often outperforming heuristic baselines—while offering improved efficiency and interpretability. The work provides practical hyperparameter guidance and demonstrates the approach’s viability for real-world portfolio optimization under cardinality constraints.

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 $a$.

DCC: Differentiable Cardinality Constraints for Partial Index Tracking

TL;DR

The paper addresses the NP-hard challenge of partial replication in index tracking by introducing the Differentiable Cardinality Constraint (DCC) and its floating-point precision-aware variant, DCC_fpp. By replacing the non-differentiable cardinality indicator with differentiable surrogates such as and , the authors integrate the constraint into standard differentiable optimization frameworks, notably SLSQP, while proving conditions under which actual cardinality is enforced. They establish accuracy and assurance criteria (C0–C2) and prove that, with a sufficiently large (e.g., in 64-bit precision), the surrogate preserves correct cardinality and constraint satisfaction, achieving polynomial-time complexity. Empirical results across S&P 100/500 and KOSPI 100 show that DCC_fpp yields competitive tracking performance—often outperforming heuristic baselines—while offering improved efficiency and interpretability. The work provides practical hyperparameter guidance and demonstrates the approach’s viability for real-world portfolio optimization under cardinality constraints.

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 () for index tracking and introduce a floating-point precision-aware method () 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 ensures that 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 .

Paper Structure

This paper contains 26 sections, 5 theorems, 16 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Full Replication
  5. Partial Replication
  6. Differentiable Cardinality Constraint
  7. Rational Function Approach
  8. Sigmoid Function Approach ($\textbf{DCC}_{fpp}$)
  9. Cardinality Constraint with Cutoff Threshold Consideration
  10. Conditions for Accurate Cardinality Calculation
  11. Conditions for Assurance of the $\textbf{DCC}_{fpp}$
  12. Complex Analysis
  13. Experiments
  14. Experimental Settings
  15. Data
  16. ...and 11 more sections

Key Result

Theorem 1

The Assurance of the cardinality constraint $N$ is the number of stocks, $e$ is constant in Lemma 1 (You can see in Appendix), and $K$ is integer such that $K<N$.

Figures (6)

  • Figure 1: (a) red: binary function, $b(w_i)$. green lines: rational approximated function of $b(w_i)$. (b) red: binary function with cutoff threshold, $b_{fpp}(w_i)$. blue lines: sigmoid approximated function of $b_{fpp}(w_i)$.
  • Figure 2: Comparison of the tracking indices (orange, green, red, blue) with the S&P 100 target index (grey) over time. Each row indicates SLSQP with forward selection, SLSQP with backward selection, SNN, and SLSQP with $\textbf{DCC}_{fpp}$, respectively. Additionally, the graph below shows the absolute error between the tracking index and the target index (full cardainlity).
  • Figure 3: Comparison of secondary evaluations between portfolio of $\textbf{DCC}_{fpp}$ at different cardinality $K=20, 25$ and $30$ and the portfolio of full replication.
  • Figure 4: Comparison of the tracking indices (orange, green, red, blue) on KOSPI 100 and S&P 500. Each rows represents forward, backward, SNN, and $\textbf{DCC}_{fpp}$, respectively.
  • Figure 5: Time comparison of each baseline. Each color indicates method: forward selection (orange), backward selection (green), SNN (red), and $\textbf{DCC}_{fpp}$ (blue), respectively. The $x$-axis and $y$-axis represent the cardinality $K$ and the average time (sec) taken to optimize.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof