Portfolio Optimization

Portfolio optimization is a core problem at the intersection of finance and OR. Given a candidate asset set (stocks, bonds, FX, commodities, etc.), expected returns, the variance-covariance matrix and risk-return constraints, it determines the weights. The classical formulation was set up by Markowitz (1952) as a quadratic-programming problem — the weight vector that maximizes expected return while minimizing variance. The output is either a single point (tangent portfolio, max Sharpe) or a curve (efficient frontier). Modern extensions: Black-Litterman (Bayesian, with expert views), Risk Parity (equal risk contribution), Robust Optimization (parameter uncertainty), CVaR optimization (tail risk), Mean-Semivariance (downside risk). In 2024 it is still the core module of every robo-advisor and institutional investment software.