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In the portfolio optimization problem at hand, one of the constraints is that the tracking error should not be greater than $\gamma$.

The constraint is therefore:

$(\textbf{x}-\textbf{w})^\mathrm{T}\Sigma(\textbf{x}-\textbf{w})\leq\gamma^2$

where $\Sigma$ is the (sample) covariance matrix, $\textbf{x}=(x_1,\dots, x_n)^\mathrm{T}$ is the vector of decision variables, and $\textbf{w}=(w_1,\dots, w_n)^\mathrm{T}$ are the weights of the benchmark portfolio.

Since in the problem at hand $n=1,000$ and $\Sigma$ was solely calculated on the basis of $T=60$ monthly return observations, the (sample) covariance matrix is unfortunately not positive semi-definite. This is certainly because of $n>T$. During my research I came across this thread. However, finding the nearest positive semi-definite matrix unfortunately did not work in my case. The result is still not positive semi-definite.

Now the question is whether it is advisable and reasonable to consider daily returns instead of monthly returns in order to (possibly) generate positive semi-definiteness as this would result in $T>n$.

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    $\begingroup$ What if you replaced every eigenvalue $<\epsilon=n\%$ of the largest eigenvalue by this $\epsilon$ - surely you'd get a positive definite matrix. $\endgroup$ Commented Jun 15, 2023 at 21:29
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    $\begingroup$ You can use Ledoit-Wolf shrinkage which guarantees positive definiteness or fit the empirical dist of eigenvalues to a Marcenko-Pastur distribution to filter out negative/very small eigenvalues (see NCO). This is similar to what Dimitri suggested. $\endgroup$
    – oronimbus
    Commented Jun 16, 2023 at 12:39

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