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In this paper, first we present some drawbacks of the cardinality constrained mean-variance (CCMV) portfolio optimization with short selling and risk-neutral interest rate when the lower and upper bounds of the assets contributions are -1/K and 1/K(K denotes the number of assets in portfolio). Then, we present an improved variant using absolute returns instead of the returns to include short selling in the model. Finally, some numerical results are provided using the data set of the S&P 500 index, Information Technology, and the MIBTEL index in terms of returns and Sharpe ratios to compare the proposed models with those in the literature.

In this paper, first, we study mean-absolute deviation (MAD) portfolio optimization model with cardinality constraints, short selling, and risk-neutral interest rate. Then, in order to insure the investment against unfavorable outcomes, an extension of MAD model that includes options is considered. Moreover, since the data in financial models usually involve uncertainties, we apply robust optimization to the MAD model with options. Finally, a data set of S&P index is used to compare the effectiveness of options in the models in terms of returns and Sharpe ratios.

In this paper, we study mean–variance–Conditional Value-at-Risk (CVaR) portfolio optimization problem with short selling, cardinality constraint and transaction costs. To tackle its mixed-integer quadratic optimization model for large number of scenarios, we take advantage of the penalty decomposition method (PDM). It needs solving a quadratic optimization problem and a mixed-integer linear program at each iteration, where the later one has explicit optimal solution. The convergence of PDM to a partial minimum of original problem is proved. Finally, numerical experiments using the S&P index for 2020 are conducted to evaluate efficiency of the proposed algorithm in terms of return, variance and CVaR gaps and CPU times.