| In this paper we propose to deal with the combinatorial difficulties in mean-variance portfolio selection, caused by various side constraints, via the randomization approach. As examples of such side constraints, we consider in this paper the following two models. In the first model, an investor is interested in holding a `small and compact' portfolio, in the sense that it involves only a small number of securities. The second model explicitly requires that each security involved in the portfolio need to have a substantial presence if it is present at all, thereby avoiding inefficient diversifications. These constraints are motivated by practical considerations, e.g. the management and/or informational costs. By incorporating these side constraints, however, the mean-variance model becomes very hard to solve. We resort to the method of randomization to find good approximation solutions. Extensive numerical experiments show that randomization is indeed a viable alternative for solving such investment models, for which the combinatorial structures in the constraints make it quite hopeless to find an exact optimal solution, whereas a good approximate solution in fact already serves the purpose quite well, given the approximative nature of the models. |
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