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Sparse Extended Mean-Variance-CVaR Portfolios with Short-selling

Ahmad Mousavi, Maziar Salahi, Zois Boukouvalas

TL;DR

The paper tackles sparse portfolio optimization under an extended mean-variance-CVaR objective that incorporates short-selling and a cardinality constraint. It introduces a customized penalty decomposition algorithm that decouples nonconvex and nondifferentiable terms by using auxiliary variables and a penalty subproblem solved via block coordinate descent, with several subproblems admitting closed-form updates. The authors establish convergence to a Lu–Zhang sparse minimizer under Robinson's constraint qualification and demonstrate strong numerical performance on real data, outperforming direct solvers and a prior penalty method in CPU time while maintaining competitive risk-return profiles. This work provides a theoretically grounded and computationally efficient framework for sparse, robust portfolio optimization with practical relevance for financial decision-making under uncertainty.

Abstract

This paper introduces a novel penalty decomposition algorithm customized for addressing the non-differentiable and nonconvex problem of extended mean-variance-CVaR portfolio optimization with short-selling and cardinality constraints. The proposed algorithm solves a sequence of penalty subproblems using a block coordinate descent (BCD) method while striving to fully exploit each component of the objective function or constraints. Through rigorous analysis, the well-posed nature of each subproblem of the BCD method is established, and closed-form solutions are derived where possible. A comprehensive theoretical convergence analysis is provided to confirm the efficacy of the introduced algorithm in reaching a local minimizer of this intractable optimization problem in finance, whereas generic optimization techniques either only capture a partial minimum or are not efficient. Numerical experiments conducted on real-world datasets validate the practical applicability, effectiveness, and robustness of the introduced algorithm across various criteria. Notably, the existence of closed-form solutions within the BCD subproblems prominently underscores the efficiency of our algorithm when compared to state-of-the-art methods.

Sparse Extended Mean-Variance-CVaR Portfolios with Short-selling

TL;DR

The paper tackles sparse portfolio optimization under an extended mean-variance-CVaR objective that incorporates short-selling and a cardinality constraint. It introduces a customized penalty decomposition algorithm that decouples nonconvex and nondifferentiable terms by using auxiliary variables and a penalty subproblem solved via block coordinate descent, with several subproblems admitting closed-form updates. The authors establish convergence to a Lu–Zhang sparse minimizer under Robinson's constraint qualification and demonstrate strong numerical performance on real data, outperforming direct solvers and a prior penalty method in CPU time while maintaining competitive risk-return profiles. This work provides a theoretically grounded and computationally efficient framework for sparse, robust portfolio optimization with practical relevance for financial decision-making under uncertainty.

Abstract

This paper introduces a novel penalty decomposition algorithm customized for addressing the non-differentiable and nonconvex problem of extended mean-variance-CVaR portfolio optimization with short-selling and cardinality constraints. The proposed algorithm solves a sequence of penalty subproblems using a block coordinate descent (BCD) method while striving to fully exploit each component of the objective function or constraints. Through rigorous analysis, the well-posed nature of each subproblem of the BCD method is established, and closed-form solutions are derived where possible. A comprehensive theoretical convergence analysis is provided to confirm the efficacy of the introduced algorithm in reaching a local minimizer of this intractable optimization problem in finance, whereas generic optimization techniques either only capture a partial minimum or are not efficient. Numerical experiments conducted on real-world datasets validate the practical applicability, effectiveness, and robustness of the introduced algorithm across various criteria. Notably, the existence of closed-form solutions within the BCD subproblems prominently underscores the efficiency of our algorithm when compared to state-of-the-art methods.
Paper Structure (19 sections, 7 theorems, 73 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 19 sections, 7 theorems, 73 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. An Efficient Customized Penalty Decomposition Algorithm
  3. Methodology
  4. Subproblems of Algorithm \ref{['algo: BCD']}
  5. Subproblem of $x$
  6. Subproblem of $y$
  7. Subproblem of $z$
  8. Subproblem of $w$ and $\gamma$
  9. Convergence Analysis
  10. Analysis of Algorithm \ref{['algo: BCD']}
  11. Analysis of Algorithm \ref{['algo: PD_method_p']}
  12. Numerical Results
  13. Conclusion
  14. Appendix
  15. Proof of Lemma \ref{['lem: solution_p_x']}
  16. ...and 4 more sections

Key Result

Lemma 2.1

The solution to the problem (pr: px) is where

Figures (1)

  • Figure 1: Comparison of returns, risks, Sharpe ratios, and CVaR for different $k$ values for 130 stocks of the S&P index.

Theorems & Definitions (15)

  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 5 more