My question covers a more or less classical portfolio optimization situation with a twist: How to partition assets into minimally correlated portfolios, with and without asset overlap.
I have $N$ assets of which I know their price series and thus also covariance matrix. I would like to create a number of portfolios with them them (each with maximum utility, such as minimum variance), such that the mutual correlation of these portfolios is minimized. (A suitable objective function for the correlation norm could be the sum of the first or third powers of the portfolios' upper triangular covariance matrix.) There's two variants of how to do this: (a) without overlap (i.e., an asset may be appear in at most one portfolio), and (b) with overlap (i.e., an asset may appear in multiple portfolios)
Of course, I could approach this problem by brute force (metaheuristic) optimization, but this will become expensive quickly due to combinatorial complexity as $N$ increases. What I'm hoping is that there's a more statistically guided way, be it analytical or semi-analytical to guide a heuristic optimization approach, to partition the assets into minimally correlated portfolios.