We consider portfolio optimization problems in which the true covariance matrix is misspecified and its value may be obtained by solving a suitably defined learning problem. We consider two types of learning problems to aid in such a resolution: (i) sparse covariance selection; and (ii) sparse precision matrix selection. A tradi- tional sequential approach for addressing such a problem requires first solving the learning problem and then using the solution of this problem in solving the result- ing computational problem. Unfortunately, exact solutions to the learning problem may only be obtained asymptotically; consequently, practical implementations of the sequential approach may provide approximate solutions, at best. Instead, we consider a simultaneous approach that solves both the learning problem and port- folio optimization problems simultaneously. In particular, we use the alternating direction method of multipliers (ADMM) to solve the learning problem while the projected gradient method is used to solve the computational problem. Asymp- totic convergence statements and rate analysis is conducted for the simultaneous scheme. Preliminary numerics on a class of misspecified portfolio optimization problems suggests that the scheme provides accurate solutions with a comparable performance with the sequential approach.
