Robust Portfolio Optimization under Computational Complexity: A P-vs-NP-Inspired Markowitz-CAPM Framework with Cardinality Constraints and a Black-Scholes Derivative-Pricing Overlay

his study develops a robust computational-finance framework for cardinality-constrained portfolio optimization by integrating Markowitz mean-variance logic, CAPM expected-return calibration, a P-vs-NP-inspired support-selection layer, stochastic and metaheuristic search, exact reduced benchmarking, robustness diagnostics, and a Black-Scholes derivative-pricing overlay. The empirical universe is a fixed Damodaran January 2026 U.S. industry dataset containing 94 industry portfolios after excluding aggregate market rows. Expected returns are constructed as CAPM-implied priors using a risk-free rate of 3.97% and an equity risk premium of 4.23%, while portfolio risk is evaluated through single-index covariance geometry, correlation diagnostics, and eigenvalue concentration tests. The baseline K = 10 sparse-selection problem contains C(94,10) = 9,041,256,841,903 possible supports before continuous weights are optimized, motivating a layered comparison of greedy search, Monte Carlo sampling, genetic algorithms, and GA-plus-continuous reoptimization. Exact evidence is supplied through a reduced n = 20, K = 6 benchmark in which all 38,760 supports are enumerated. The study reports best-found full-universe incumbents under documented search coverage rather than claiming certified global optimality. Robustness layers examine covariance alternatives, Rf/ERP perturbations, weight caps, transaction costs, beta drift, risk contributions, stress scenarios, and falsification gates. The Black-Scholes overlay demonstrates how option value, delta exposure, leverage-adjusted beta, and volatility can be incorporated without treating derivative leverage as free performance. The contribution is methodological and empirical: it shows how computational complexity, asset-pricing priors, covariance dependence, algorithmic search, and derivative realism can be combined in a transparent, reproducible, and claim-bounded portfolio-optimization architecture. The study does not prove P ≠ NP, does not certify full n = 94 global optimality, and does not establish realized historical investment dominance.