Mean - variance optimization. The diversification of the portfolio with the help of $L_2$ regularization.

The mean-variance optimization process often ends with most of the stock weights to be negligibly small. But in order to have a diversified portfolio, it should include a definite number of assets. It can be achieved with the help of the technique known as $L_2$ regularization [1]. To make it happen, we add a small - weight penalty to our objective function - the $\gamma \mathbf{w}^T \mathbf{w}$ (i.e a sum of square of the weights) member. Here $\gamma \gt 0$ is a constant parameter of regularization, the more the value of the $\gamma$, the more regularization (or diversification) effect we will have. This term reduces the number of negligible small weights, because it has a minimum value when all the weights are equally distributed, and the maximum value in the case where the only weight is nonzero and equal to $1$.

For example, to minimize the portfolio volatility using the $L_2$ regularization we should change the objective function as follows: $$ \underset{\mathbf{w}}{\text{minimize}} ~ \left(\mathbf{w}^T \Sigma \mathbf{w} \right) ~~~ \longrightarrow ~~~ \underset{\mathbf{w}}{\text{minimize}} ~ \left(\mathbf{w}^T \Sigma \mathbf{w} + \gamma \mathbf{w}^T \mathbf{w} \right) $$

You can optimize your investment portfolio for a chosen set of stocks on the website for portfolio optimization at https://asset-master.net/. You can minimize portfolio volatility, maximize Sharpe ratio, minimize volatility given some expected return value, etc... It's free for up to 150 stocks in portfolio and uses daily historical prices. Also it shows the historical expected return and volatility for all the stocks in the portfolio.

[1] Boyd, S.; Vandenberghe, L. Convex Optimization (2004).

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