Single period portfolio selection models with transaction costs and initial holdings

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The classical mean-variance model in the portfolio theory does not include transaction costs and initial holdings. In this paper we extend Markowitz’s portfolio selection model to include transaction costs in the presence of initial holdings for the investor. Our approach is new. Our aim is to obtain an optimal portfolio which has a minimum risk or a maximum return. The optimal solution may require both buying and selling a particular asset, which is clearly not an advisable practical strategy. In order to have a good strategy it is necessary to include in the portfolio selection models complementarity constraints. These constraints do not allow that the same asset be bought and sold at the same time. That is why our portfolio selection models include complementarity constraints. This type of constraints increases the difficulty of the problems, which now enter in the category of combinatorial 0ptimization problems. The set of feasible solutions for the problems from the above mentioned class is the union of a set of convex sets but it is no longer convex. We study several approaches for finding solutions of portfolio selection models with complementarity constraints. Several numerical results are discussed.