Mathematical Optimization for Portfolio Selection Under Uncertainty: Robust, Distributionally Robust, and Machine Learning-Enhanced Approaches to Modern Portfolio Theory

Abstract

Modern portfolio theory, originating with Markowitz's mean-variance framework in 1952, has evolved substantially through incorporation of robust optimization, distributionally robust optimization, and machine learning-enhanced approaches that address the practical challenges of parameter uncertainty, non-Gaussian returns, and non-stationarity. This paper presents a comprehensive mathematical treatment and empirical evaluation of three principal families of portfolio optimization methods applied to European equity markets during 2005-2024. The methods evaluated include: (1) classical mean-variance optimization with shrinkage estimation of the covariance matrix; (2) robust optimization with box and ellipsoidal uncertainty sets for expected returns; (3) distributionally robust optimization with Wasserstein-ball ambiguity sets; and (4) machine learning-enhanced approaches combining reinforcement learning with classical optimization. Empirical evaluation on European equity data covering 640 constituent stocks demonstrates that robust and distributionally robust methods produce portfolios with substantially improved out-of-sample Sharpe ratios (0.68 versus 0.42 for classical mean-variance) and lower maximum drawdowns (18.4 per cent versus 32.7 per cent) at modest cost in expected returns. Machine learning-enhanced methods achieve intermediate risk-adjusted performance. The paper contributes mathematical exposition, comparative empirical evaluation, and practical guidance for portfolio managers and asset allocation researchers navigating the modern portfolio optimization landscape.