A global optimization problem in portfolio selection

This paper deals with the issue of buy-in thresholds in portfolio optimization using the Markowitz approach. Optimal values of invested fractions calculated using, for instance, the classical minimum-risk problem can be unsatisfactory in practice because they imply that very small amounts of certain assets are purchased. Realistically, we want to impose a discrete restriction on each invested fraction yi such as yi > l or yi = 0. We shall describe an approach which uses a combination of local and global optimization to determine satisfactory solutions. The approach could also be applied to other discrete conditions for instance in dealing with assets that can only be purchased in units of a certain size (roundlots).