Smoothing Method To Cardinality-constrained Portfolio Selection Problems

In the classical Markowitz mean-variance optimization problem, asset returns and risk are modeled under the assumption of no friction in finance market or dividing stocks freely. So, severeal restrictions, such as transaction cost and round lot trading rule, have added and studied for better application in finance market. In this paper, we consider a cardinality-constrained nonlinear portfolio selection program that minimizes the risk-reward ratio of portfolio subject to a cardinality constraint and linear constraints. It is not necessary for investors to predict the asset returns and risk, the cardinality-constraint can also low down the cost in management and transaction in case of too many securities with small proportion.In this paper, we propose a smoothing method to approximate the cardinality constraint, and use the homotopy method to design the heuristic algorithm. The homotopy algorithm was designed with the smooth parameter, and the convergence proof is given. The computation results show that this method may gain better objective values compared with the1-norm relaxation method. In addition, computation results for Sharpe Ratio cardinality-constrained quadratic program show the effictiveness of this smoothing method.