3. Robust Markowitz portfolio optimization problem (10 marks) The classical Markowitz portfolio optimization problem can be formulated as (M) maximizer subject to xEx 57 e'x = 1 x 0, where e = (1, ...,1) R" and y > 0. Here, r = ('1,...,P.)" ER" where each r; is the expected return on investment i and is the covariance matrix (and so, is positive semidefinite). In practice, the data r is uncertain due to prediction/estimation error. We assume that r belongs to an uncertainty set U given by U = {r R": ||r-1||2

0 is a given constant scalar. The robust optimization approach then aims at finding the worst case solution of problem (M), that is, solving the following robust optimization problem (RM) maximize min{r"x:r U} subject to ex=1 x > 0 xTex 0. Here, r = ('1,...,P.)" ER" where each r; is the expected return on investment i and is the covariance matrix (and so, is positive semidefinite). In practice, the data r is uncertain due to prediction/estimation error. We assume that r belongs to an uncertainty set U given by U = {r R": ||r-1||2

0 is a given constant scalar. The robust optimization approach then aims at finding the worst case solution of problem (M), that is, solving the following robust optimization problem (RM) maximize min{r"x:r U} subject to ex=1 x > 0 xTex