3. Robust Markowitz portfolio optimization problem (10 marks) The classical Markowitz portfolio optimization problem can be formulated as (M) maximizer?x subject to xTex 0, where e = (1,...,1)" E R" and 7 > 0. Here, r = (1,...,n)" R" 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 ER":|r-||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 EU} subject to x xy ex=1 x>0. Reformulate the problem (RM) as a conic linear program. Provide explanations for the reformulations. (Hint: You may use the fact that minxern a'x = -||*||2||a||2 for all a R". If you did use this fact, provide justification of it as well.] 3. Robust Markowitz portfolio optimization problem (10 marks) The classical Markowitz portfolio optimization problem can be formulated as (M) maximizer?x subject to xTex 0, where e = (1,...,1)" E R" and 7 > 0. Here, r = (1,...,n)" R" 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 ER":|r-||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 EU} subject to x xy ex=1 x>0. Reformulate the problem (RM) as a conic linear program. Provide explanations for the reformulations. (Hint: You may use the fact that minxern a'x = -||*||2||a||2 for all a R". If you did use this fact, provide justification of it as well.]