Explicit methods for portfolio optimization with realistic costs and constraints

We present algorithms for mean–variance optimization with realistic costs, including borrow costs, bid-offer spread cost, and market impact according to the well-known square root law. The algorithm performs coordinate updates, each of which reduces to a univariate subproblem with a closed-form algebraic solution. The technique extends to problems with box constraints and to certain kinds of multi-horizon objectives. We demonstrate an improvement in the computational complexity when the covariance matrix has a factor structure.