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Autonomous Sparse Mean-CVaR Portfolio Optimization

Yizun Lin, Yangyu Zhang, Zhao-Rong Lai, Cheng Li

TL;DR

This work tackles the NP-hard problem of $\\\\ell_0$-constrained mean-CVaR portfolio optimization by substituting the hard sparsity constraint with a tailed indicator approximation and solving the resulting surrogate via a convergent Proximal Alternating Linearized Minimization (PALM) method with a nested Fixed-Point Proximity Algorithm (FPPA). The ASMCVaR framework achieves autonomous sparsity, maintaining substantial asset inclusion while adjusting pool size, and provides a theoretically grounded approximation to the original model as the approximation parameter $$ tends to zero. Empirically, ASMCVaR consistently outperforms state-of-the-art sparse and dense portfolio methods across six real-world datasets in cumulative wealth, alpha, and Sharpe ratio, and remains robust to transaction costs. These results offer a scalable, practical route for risk-controlled sparse portfolio optimization with broad potential applicability beyond finance.

Abstract

The $\ell_0$-constrained mean-CVaR model poses a significant challenge due to its NP-hard nature, typically tackled through combinatorial methods characterized by high computational demands. From a markedly different perspective, we propose an innovative autonomous sparse mean-CVaR portfolio model, capable of approximating the original $\ell_0$-constrained mean-CVaR model with arbitrary accuracy. The core idea is to convert the $\ell_0$ constraint into an indicator function and subsequently handle it through a tailed approximation. We then propose a proximal alternating linearized minimization algorithm, coupled with a nested fixed-point proximity algorithm (both convergent), to iteratively solve the model. Autonomy in sparsity refers to retaining a significant portion of assets within the selected asset pool during adjustments in pool size. Consequently, our framework offers a theoretically guaranteed approximation of the $\ell_0$-constrained mean-CVaR model, improving computational efficiency while providing a robust asset selection scheme.

Autonomous Sparse Mean-CVaR Portfolio Optimization

TL;DR

This work tackles the NP-hard problem of -constrained mean-CVaR portfolio optimization by substituting the hard sparsity constraint with a tailed indicator approximation and solving the resulting surrogate via a convergent Proximal Alternating Linearized Minimization (PALM) method with a nested Fixed-Point Proximity Algorithm (FPPA). The ASMCVaR framework achieves autonomous sparsity, maintaining substantial asset inclusion while adjusting pool size, and provides a theoretically grounded approximation to the original model as the approximation parameter tends to zero. Empirically, ASMCVaR consistently outperforms state-of-the-art sparse and dense portfolio methods across six real-world datasets in cumulative wealth, alpha, and Sharpe ratio, and remains robust to transaction costs. These results offer a scalable, practical route for risk-controlled sparse portfolio optimization with broad potential applicability beyond finance.

Abstract

The -constrained mean-CVaR model poses a significant challenge due to its NP-hard nature, typically tackled through combinatorial methods characterized by high computational demands. From a markedly different perspective, we propose an innovative autonomous sparse mean-CVaR portfolio model, capable of approximating the original -constrained mean-CVaR model with arbitrary accuracy. The core idea is to convert the constraint into an indicator function and subsequently handle it through a tailed approximation. We then propose a proximal alternating linearized minimization algorithm, coupled with a nested fixed-point proximity algorithm (both convergent), to iteratively solve the model. Autonomy in sparsity refers to retaining a significant portion of assets within the selected asset pool during adjustments in pool size. Consequently, our framework offers a theoretically guaranteed approximation of the -constrained mean-CVaR model, improving computational efficiency while providing a robust asset selection scheme.
Paper Structure (27 sections, 6 theorems, 66 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 27 sections, 6 theorems, 66 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Works
  3. VaR and CVaR
  4. Original Mean-CVaR Model
  5. Mean-CVaR Model with $\ell_1$ Regularization
  6. $\ell_0$-constrained Mean-CVaR Model
  7. Autonomous Sparse Mean-CVaR Portfolio Optimization
  8. Autonomous Sparse Mean-CVaR Model
  9. Proximal Alternating Linearized Minimization
  10. Experimental Results
  11. Parameter Setting and Autonomous Sparsity
  12. Cumulative Wealth
  13. $\alpha$ Factor
  14. Sharpe Ratio
  15. Transaction Cost
  16. ...and 12 more sections

Key Result

Lemma 3.2

Let $\bv^{*}$ be a solution of model model:ASMCVaR and $\bw^*:=\tilde{\bI}\bv^*$. Then there exists a constant $\tilde{L}>0$ such that

Figures (3)

  • Figure 1: Diagram for tailed approximation of $\ell_0$ constraint: $\tilde{\iota}_{m,\gamma}(\bw)$ with $N=10$ and $m=5$. As $\gamma \searrow 0$, $\frac{1}{2\gamma}\sum_{k=6}^{10}w_{j_k}^2 \nearrow +\infty$ and $\tilde{\iota}_{m,\gamma}(\bw)\nearrow \iota_{m}(\bw)$ for any fixed $\bw$ ($\|\bw\|_0>5$).
  • Figure 2: Cumulative wealths of different portfolio optimization models along with trade time on 6 benchmark data sets.
  • Figure 3: Final cumulative wealths of different portfolio optimization models w.r.t. transaction cost rate $\nu$ on $6$ benchmark data sets.

Theorems & Definitions (13)

  • Definition 3.1: Tailed Approximation of $\ell_0$ Constraint
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Proposition 3.7
  • proof
  • proof
  • proof
  • ...and 3 more