[Paper] Covariance-Aware Simplex Projection for Cardinality-Constrained Portfolio Optimization

Source: arXiv - 2512.19986v1

Overview

The paper proposes Covariance‑Aware Simplex Projection (CASP), a new “repair” step for metaheuristic portfolio optimizers that must respect a cardinality limit (i.e., a maximum number of assets). Unlike the usual Euclidean projection that treats assets as independent, CASP incorporates the assets’ covariance matrix, yielding portfolios that are noticeably less risky while still satisfying the cardinality constraint.

Key Contributions

• Two‑stage repair operator
  1. Volatility‑normalized asset selection – picks the target set of assets based on risk‑adjusted scores.
  2. Covariance‑aware simplex projection – maps the raw weight vector onto the feasible simplex using a distance metric derived from the covariance matrix (tracking‑error geometry).
• Theoretical grounding – shows that a covariance‑induced distance is a natural choice from a portfolio‑theoretic perspective.
• Empirical validation – on S&P 500 data (2020‑2024), the CASP‑Basic variant cuts portfolio variance by a statistically significant margin compared with standard Euclidean repair, without needing expected‑return forecasts.
• Ablation study – demonstrates that most of the variance reduction stems from the volatility‑normalized selection, while the covariance‑aware projection adds a consistent secondary boost.
• Optional return‑aware extension – by feeding in return estimates, CASP can also improve Sharpe ratios, and out‑of‑sample tests confirm that the gains translate into realized performance.
• Drop‑in compatibility – CASP can replace Euclidean projection in any existing metaheuristic optimizer (genetic algorithms, particle swarm, etc.) with minimal code changes.

Methodology

1. Problem setting – The optimizer searches over weight vectors w that must (a) lie on the simplex (weights sum to 1, non‑negative) and (b) contain at most k non‑zero entries (cardinality constraint).
2. Standard repair – When a candidate violates the cardinality limit, Euclidean projection simply truncates the smallest weights and rescales the rest, ignoring how assets co‑move.
3. CASP – Stage 1 (Selection)
   • Compute a volatility‑normalized score for each asset:
     $$ s_i = \frac{|w_i|}{\sigma_i} $$
     where $\sigma_i$ is the asset’s standard deviation.
   • Pick the top‑k assets by score; this favors assets that contribute weight relative to their own risk.
4. CASP – Stage 2 (Projection)
   • Define a covariance‑induced norm:
     $$ |x|_{\Sigma} = \sqrt{x^\top \Sigma x} $$
     where $\Sigma$ is the asset covariance matrix.
   • Solve a quadratic program that finds the closest feasible weight vector (in the $\Sigma$‑norm) to the original candidate, restricted to the selected assets.
   • The solution has a closed‑form “water‑filling” style expression, making it fast enough for iterative metaheuristics.
5. Extensions – A return‑aware variant adds a linear term $-\mu^\top w$ to the objective, steering the projection toward higher expected returns when reliable forecasts are available.

Results & Findings

• Variance reduction is robust across different cardinalities (k = 10, 20, 30) and across market regimes within the 2020‑2024 window.
• Ablation shows that using volatility‑normalized selection alone already cuts variance by ~22 %; adding the covariance‑aware projection contributes an extra ~5 % improvement.
• Return‑aware extension modestly lifts Sharpe ratios, confirming that the framework can be tuned for risk‑only or risk‑return objectives.
• Runtime impact is negligible: the projection solves in $O(k^{2})$ time, adding < 1 ms per iteration on a typical desktop CPU.

Practical Implications

• Quantitative developers building evolutionary or swarm‑based portfolio optimizers can swap in CASP to respect cardinality while honoring the true risk geometry of the market.
• Risk‑focused robo‑advisors can adopt CASP to generate tighter‑risk portfolios without needing sophisticated return forecasts, valuable when forward‑looking estimates are noisy.
• ETF or index‑fund construction teams that must limit the number of constituents (e.g., for transaction‑cost or regulatory reasons) can use CASP to keep the resulting basket well‑diversified.
• Open‑source libraries (e.g., PyPortfolioOpt, DEAP) can expose CASP as an optional projector, giving practitioners a ready‑made, theoretically justified alternative to the default Euclidean projection.
• Performance monitoring: because CASP works directly with the covariance matrix, any improvements in covariance estimation (shrinkage, factor models, machine‑learning‑based forecasts) translate immediately into better repair outcomes.

Limitations & Future Work

• Covariance estimation – CASP’s benefits hinge on a reasonably accurate $\Sigma$. In highly volatile or regime‑changing periods, estimation error could erode the variance gains.
• Scalability to ultra‑large universes – while $O(k^{2})$ is fine for typical cardinalities (≤ 50), projecting in universes with thousands of assets may require sparse or low‑rank approximations of $\Sigma$.
• Return‑aware extension assumes reliable expected‑return inputs; noisy forecasts can negate the Sharpe‑ratio boost.
• Integration with transaction‑cost models – the current formulation ignores turnover; extending CASP to jointly consider cost‑aware constraints is an open avenue.
• Dynamic cardinality – future research could explore adaptive k (varying the number of assets over time) within the CASP framework, possibly guided by market‑state detectors.

Bottom line: CASP offers a principled, easy‑to‑plug‑in improvement for any cardinality‑constrained portfolio optimizer, delivering lower risk without sacrificing computational efficiency—a win for developers who need robust, real‑world‑ready solutions.

Authors

• Nikolaos Iliopoulos

Paper Information

• arXiv ID: 2512.19986v1
• Categories: q-fin.PM, cs.LG, cs.NE, q-fin.CP
• Published: December 23, 2025
• PDF: Download PDF