Optimization of a functional involving k linear and one quadratic forms with applications to multi-period portfolio selection

We explore the extension of Markowitz's Mean–Variance model into a multi-period model. Such extension allows the assignment of different weights for each time period, allowing the focus on certain periods for obtaining an adequate model of optimal portfolio selection. We show that such a model gives rise to a multivariate constrained optimization problem that involves a function of a system of linear functionals and a quadratic function. We derive the explicit solution for such a model in its most general form, providing us a way to use such a model in practice while avoiding complexities that naturally come from the solution of such an involved multivariate convex problem. We then discuss some of its fundamental features and explore a numerical illustration that shows how one can use the model, in practice, based on a given historical data.