@YiLiu Thanks for your reply. Haven't tried that. Will investigate more today.

@FedericoPoloni thank you! I will try to do change of basis for the entire problem.

@FedericoPoloni but won't it change the solution of the optimization problem?

Let's say I have a problem $min\, m^T x + 0.5 x^T M x$ - how would you reformulate it fir a deflated matrix M'?

@FedericoPoloni you are right - numerical error is detected as small negative eigenvalues within the optimization package... I will check if there is a way to set a larger tolerance. But if not, how would I deflate the problem? I know how I could deflate the matrix, for example by doing SVD, but then I would need to translate my optimization problem in that space... Hopefully I can just increase the tolerance or error

Thanks @Suvrit a lot for your answer. I will check the details, sorry I'm not too familiar with this subject. All I'm saying is that when I specify [ C C; C C ] matrix in my optimization routine (joptimizer), it complains it's not positive (semi-)definite. That might be a bug in their code too...

please see above comment. also, here it seems like there are additional conditions: math.stackexchange.com/questions/391852/…

Strange, then why is my optimization package complaining it's not either positive definite, nor positive semi-definite? Are you sure? (Sorry for this lame question!)

@Shamisen min (a_T * x + 1/2 * x_T * M * x), where x <= 0, a - real valued vector, M = [ C, -C; -C, C ], and C is some covariance matrix (symmetric positive definite).