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 disjoint restriction so that each invested fraction is either zero or exceeds a prescribed minimum threshold. 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 disjoint conditions - for instance in dealing with assets that can only be purchased in units of a certain size (roundlots).
TR 335, Numerical Optimization Centre, University of Hertfordshire, Hatfield, UK