Duality for portfolio optimization with short sales

Abstract.

We consider the classical Markowitz portfolio optimization problem with additional constraints representing so-called short sales.  The two objectives of this multiobjective problem are the expected return and the variance of a portfolio combined by a number of risky securities. A multiobjective problem is established which is dual to this classical portfolio problem. Weak and strong duality assertions are verified.  There we consider properly efficient solutions of the portfolio problem and Pareto-efficient solutions of the dual problem, respectively.  The theoretical results are illustrated by means of an example representing the optimization problem for a portfolio containing some German blue chips.