The selection of a high-dimensional optimal portfolio has recently become an important research topic due to the rapid development of computer technology which provides opportunities to invest into a large number of assets simultaneously. In such a situation, the application of the sample estimators for the parameters of the asset return distribution is not recommendable because these estimators work well, only if the number of assets is significantly smaller than the number of observations. If the number of assets in the portfolio is comparable to the sample size, then a modification of the traditional plug-in estimators is needed to ensure reliable results. In order to deal with the curse of dimensionality when a high-dimensional optimal portfolio is constructed, we develop several new results in random matrix theory and apply them to optimal portfolio choice problems. As a result, a distribution-free estimator for the optimal portfolio weights is derived which minimizes the corresponding loss functions. Its finite-sample properties are investigated via an extensive simulation study. Finally, the theoretical findings are applied to high-dimensional data of returns on assets included into the S&P 500 index.