Covariance matrix not positive definite

1
votes

I'm trying to solve a portfolio optimization problem with quadprog library, but solve.QP function returns this:

matrix D in quadratic function is not positive definite!

But, I'm defining Dmat as:

Dmat <- cov(diff(as.matrix(na.locf(prices))))

How can I turn Dmat in a positive definite matrix?

1
rmathematical-optimization
Maybe adding some regularization could help? Something on the lines of Dmat <- Dmat + lambda*diag(ncol(Dmat)) where lambda is a small constant (a hyperparameter you'd want to tune to find a good value). Although it would add some distortion to your results, it might be an acceptable tradeoff to improve the robustness of the method. – Backlin
Try to check the eigenvalues by eigen(Dmat) and if one of them less than zero. There would be some problem with your matrix.Also, sometimes, a high dimension matrix would lead to a huge numerical error if the code is not prepared for such a high dimension problem or just poorly-written. – Stan Yip
Try adding some small values and see what happens to the eigen values, as @galapagos suggested. To properly tune it you'd need some kind of performance metric that you optimize with cross-validation on similar. These type of questions are not really on topic for SO and I think you'd get better help at the cross-validated site though so I'll vote for a migration. – Backlin
The hint by galapagos is along what is actually done. This article could be a good starting point in the literature: repositori.upf.edu/bitstream/handle/10230/560/691.pdf – mts
Portfolio optimization problems use returns, not price differences so, if your prices object contains asset prices, then you'll need to compute returns using something like: temp <- as.matrix(na.locf(prices)); Dmat <- cov( diff(temp)/ head(temp,-1)). Alternatively you could convert prices to log prices and then diff would give you log returns. – WaltS

1 Answers

0
votes

Thanks to the help. I discovered the cov.shrink function from corpcor library, and now I'm defining Dmat as:

cov.shrink(diff(as.matrix(na.locf(precos_mes))))

which works perfectly as a positive definite matrix.