Portfolio Optimization with a Mean Absolute Deviation Criterion

A question came up after class on whether GLPK could be used for portfolio optimization.  At first glance the answer is no because the classic Markowitz formulation for portfolio optimization with a  mean-variance criterion leads to a quadratic program.  GLPK is a linear programming solver that doesn't generally handle quadratic programs. There are workarounds of limited utility, which is demonstrated in a post below.  But in general, GLPK is not suited to quadratic programming.

However, changing the optimization criterion leads to some interesting and useful tools for portfolio optimization that can be expressed as a linear programs. An example is a criterion based on the mean of the absolute deviation (MAD) of return as a risk measure, an idea attributed to Konno and Yamazaki (1991).  This approach also has some other interesting features, including the direct use of return data either from simulation or historical records. Solutions for the MAD criterion exhibit second order stochastic dominance - a a theoretically important feature for risk measures that is not shared by the Markowitz formulation.

To demonstrate the concept, I've prepared a sample GMPL file PortfolioMAD.mod that computes an optimal portfolio using the MAD criterion.  The calculation shows how to compute random samples from multivariable Normal distribution in GMPL.  This requires an implementation of a Cholesky factorization in GMPL which is given in the code.