The preceding mean-variance frontier example illustrates the in-sample, backward-looking optimization. In practice, portfolio optimization requires forward-looking input. Expected returns are notoriously difficult to estimate accurately.

The covariance matrix can be estimated somewhat more reliably, which has given rise to several alternative approaches. However, covariance matrices with correlated assets pose computational challenges since the optimization problem requires inverting the matrix. The high condition number induces numerical instability, which in turn gives rise to Markovitz curse: the more diversification is required (by correlated investment opportunities), the more unreliable the weights produced by the algorithm. 

Many investors prefer to use portfolio-optimization techniques with less onerous input requirements. We now introduce several alternatives that aim to address these shortcomings, including more recent approaches based on machine learning.