Covariance-Aware Simplex Projection for Cardinality-Constrained Portfolio Optimization

ArXiv ID: 2512.19986 “View on arXiv”

Authors: Nikolaos Iliopoulos

Abstract

Metaheuristic algorithms for cardinality-constrained portfolio optimization require repair operators to map infeasible candidates onto the feasible region. Standard Euclidean projection treats assets as independent and can ignore the covariance structure that governs portfolio risk, potentially producing less diversified portfolios. This paper introduces Covariance-Aware Simplex Projection (CASP), a two-stage repair operator that (i) selects a target number of assets using volatility-normalized scores and (ii) projects the candidate weights using a covariance-aware geometry aligned with tracking-error risk. This provides a portfolio-theoretic foundation for using a covariance-induced distance in repair operators. On S&P 500 data (2020-2024), CASP-Basic delivers materially lower portfolio variance than standard Euclidean repair without relying on return estimates, with improvements that are robust across assets and statistically significant. Ablation results indicate that volatility-normalized selection drives most of the variance reduction, while the covariance-aware projection provides an additional, consistent improvement. We further show that optional return-aware extensions can improve Sharpe ratios, and out-of-sample tests confirm that gains transfer to realized performance. CASP integrates as a drop-in replacement for Euclidean projection in metaheuristic portfolio optimizers.

Keywords: portfolio optimization, covariance structure, metaheuristic algorithms, repair operators, tracking error

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail
• Why: The paper introduces a novel, covariance-aware projection operator with explicit mathematical derivation and financial interpretation (tracking error), demonstrating high mathematical sophistication. It is backed by extensive empirical validation on S&P 500 data (2020-2024), including statistical significance tests, ablation studies, out-of-sample tests, and clear integration guidance for practical implementation.

  flowchart TD
    A["Research Goal"] --> B["Data & Inputs"]
    B --> C["Standard Euclidean Projection<br/>Baseline Repair Operator"]
    B --> D["CASP Methodology<br/>Two-Stage Repair Operator"]
    D --> E["Covariance-Aware<br/>Projection"]
    D --> F["Volatility-Normalized<br/>Selection"]
    C & E --> G["Outcomes & Findings"]
    F --> G

    subgraph Key Findings
        G --> H["Lower Portfolio Variance<br/>Without Return Estimates"]
        G --> I["Volatility Selection<br/>Drives Main Reduction"]
        G --> J["Robust & Significant<br/>Improvements"]
    end

    style A fill:#e1f5fe
    style D fill:#f3e5f5
    style H fill:#e8f5e8
    style I fill:#e8f5e8
    style J fill:#e8f5e8