The article examines cardinality-constrained portfolio optimization by integrating the P versus NP computational-complexity problem with the Markowitz–Capital Asset Pricing Model (CAPM) framework, augmented by Black–Scholes derivative pricing. The aim of the study is to develop an integrated economic-computational framework that operationalizes combinatorial hardness within a realistic, auditable portfolio-selection setting under a hard cardinality constraint. The empirical analysis uses cross-sectional data for approximately 94 U.S. industry portfolios, including levered betas, annualized volatilities, firm-count information, and CAPM-implied expected returns, with a fully constructible covariance matrix derived through a single-index market model. The study employs mixed-integer quadratic programming (MIQP) logic alongside greedy selection, Monte Carlo sampling, and genetic-algorithm search, implemented through a reproducibility protocol with random-seed control, distributional performance evaluation, convergence diagnostics, and runtime profiling. The results demonstrate that under a cardinality constraint of K = 10, the maximum Sharpe ratio reaches 0.0880, with expected return μₚ = 0.0868 and risk σₚ = 0.5356, confirming that combinatorial restrictions materially reshape the efficient frontier. Dependence diagnostics reveal strong systemic correlation, with median pairwise correlation of 0.7692 and the leading eigenvalue explaining 76.10% of total variance, indicating structurally limited diversification potential. Sensitivity analysis shows that increasing the equity risk premium by 100 basis points raises the median Sharpe ratio from 0.0866 to 0.1070, while variations in the risk-free rate leave Sharpe unchanged under the CAPM structure. A reduced-universe exact benchmark (n = 20, K = 6) confirms that heuristic methods recover the same optimal Sharpe ratio as full enumeration, validating the reliability of the approximation schemes. The Black–Scholes overlay demonstrates that embedding a European call option via delta-based linearization sharply raises expected return and risk in tandem, confirming that the derivative mainly introduces leverage rather than genuine risk-adjusted improvement. The findings suggest that improved computational approaches to constrained portfolio optimization can enhance capital-allocation efficiency, strengthen portfolio-level risk diagnostics, and reduce systemic vulnerability in financially constrained and computationally complex decision-making environments. The article highlights the value of combining rigorous complexity theory with transparent, reproducible empirical practice to address socioeconomic challenges in institutional capital allocation.

Keywords: socioeconomic challenges, portfolio optimization, P versus NP, cardinality constraints, Markowitz-CAPM, mixed-integer quadratic programming, Black–Scholes model, systemic risk.

JEL Classification: G11, G12, C61, C63, C15, G13.

Gondauri, D. (2026). Addressing socioeconomic challenges through robust portfolio optimization: Integrating the P versus NP problem, the Markowitz–capital asset pricing model framework, cardinality constraints, and Black–Scholes derivative pricing. SocioEconomic Challenges, 10(2), 37-70. https://doi.org/10.61093/sec.10(2).37-70.2026