Single- and Multi-period Portfolio Optimization with Cone Constraints and Discrete Decisions

Portfolio optimization literature has come quite far in the decades since the first publication, and many modern models are formulated using second-order cone constraints and take discrete decisions into consideration. In this study we consider both single-period and multi-period portfolio optimization problems based on the Markowitz (1952) mean/variance framework, where there is a trade-off between expected return and the risk that the investor may be willing to take on. Our model is aggregated from current literature. In this model, we have included transaction costs, conditional value-at-risk (CVaR) constraints, diversification-by-sector constraints and buy-in thresholds. Our numerical experiments are conducted on portfolios drawn from 20 to 400 different stocks available from the S&P 500 for the single period-model. The multi-period portfolio optimization model is obtained using a binary scenario tree that is constructed with monthly returns of the closing price of the stocks from the S&P 500. We solve these models with a MATLAB based Mixed Integer Linear and Nonlinear Optimizer (MILANO). We provide substantial improvement in runtimes using warmstarts in both branch-and-bound and outer approximation algorithms.