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Why does the portfolio optimization mean-variance model require the covariance matrix to be positive-definite? Does this requirement have to do with the need to be able to invert the matrix during optimization?

How is positive-definiteness achieved? Does it happen because all matrix elements (variance and covariance) are non-negative? In which cases do asset returns fail to make the covariance matrix positive definite? are there any known work-arounds when this happens

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To supplement the other answer, yes there are optimization reasons for the covariance matrix being symmetric positive definite (SPD). All positive definite matrices are invertible and its inverse is also positive definite.

This guarantees a unique global minimum in a quadratic optimization problem (MVO).

Lots of material available on the topic: https://www.cis.upenn.edu/~cis515/cis515-12-sl14.pdf

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    $\begingroup$ A follow up question for you: when the solution is not unique, it just means there is a set of solutions (portfolios), not just one. How to mathematically characterise this set? I have never seen this clearly explained in finance books. $\endgroup$
    – nbbo2
    Commented Aug 14, 2020 at 14:35
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    $\begingroup$ Here's my guess. If you have a perfectly correlated, identical time series which causes a 0 eigenvalue and thus not a positive definite matrix, the resulting set of portfolio solutions will be cases where the weights for the identical assets can vary as long as they add up to a certain weight since it doesn't matter which you allocate to. $\endgroup$
    – Quantoisseur
    Commented Aug 14, 2020 at 15:51
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    $\begingroup$ Yes, that is correct $\endgroup$
    – Quantoisseur
    Commented Aug 14, 2020 at 16:10
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    $\begingroup$ A zero eigenvalue can happen easily. A negative eigenvalue is theoretically impossible in a true covariance matrix (variance is always positive) but a slight negative could show up because of roundoff error. $\endgroup$
    – nbbo2
    Commented Aug 14, 2020 at 16:12
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    $\begingroup$ @Quantoisseur I have run into zero eigenvalues when one "basket" asset is a linear combination of other assets (e.g., EUR and ECU used to be linear combinations of pre-EUR currencies for a while). The optimal portfolio is not unique because you may be indifferent to holding the basket or its components that replicate it. $\endgroup$
    – Dimitri Vulis
    Commented Aug 14, 2020 at 17:04
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Positive definite matrix $A$ is defined as $x^TAx > 0$ for all vectors $x$.

Since a term $w^T\Sigma w$ in Markowitz (and other models as well) expresses variance in returns, it is a measure of dispersion. Any measure of dispersion has to be positive (or maybe zero but it is a case where there is no uncertainty and hence no risk). Negative dispersion is meaningless.

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