In a quick and easy first step you could add $L_1$-regularization to the Markowitz problem. That is, you add a term $\lambda ||w||$ to the goal function of your optimization problem (where $w$ are the allocation weights to be optimized).

The $L_1$-regularization, which is often termed LASSO in the statistics community, will give you sparse solutions of the weight vectors, i.e. bring several $w_i$ down to zero and leave you with a selected number of remaining asset weights. How many assets exactly will remain depends on the choice of the regularization parameter $\lambda$, which you then should adjust accordingly in order to give 20 assets.

The nice thing about this approach is that you can stay in the same class of optimization algorithms, because the absolute value of the parameters can be incorporated by linear constraints. See for example here.

In a quick and easy first step you could add $L_1$-regularization to the Markowitz problem. That is, you add a term $\lambda ||w||_1$ to the goal function of your optimization problem (where $w$ are the allocation weights to be optimized).

The $L_1$-regularization, which is often termed LASSO in the statistics community, will give you sparse solutions of the weight vectors, i.e. bring several $w_i$ down to zero and leave you with a selected number of remaining asset weights. How many assets exactly will remain depends on the choice of the regularization parameter $\lambda$, which you then should adjust accordingly in order to give 20 assets.

The nice thing about this approach is that you can stay in the same class of optimization algorithms, because the absolute value of the parameters can be incorporated by linear constraints. See for example here.